Merge pull request #218 from sebcrozet/nexgen

Release v0.11.0

.travis.yml

@@ -1,7 +1,20 @@
+sudo: false
 language: rust
-
+matrix:
+  include:
+    - rust: nightly
+      env: TEST_SUITE=suite_nightly
+    - rust: beta
+      env: TEST_SUITE=suite_beta
+    - rust: stable
+      env: TEST_SUITE=suite_stable
+  allow_failures:
+    - rust: nightly
+    - rust: beta
 script:
     - rustc --version
     - cargo --version
-    - cargo build --verbose --features "arbitrary generic_sizes"
-    - cargo test --verbose --features arbitrary
+    - cargo build --verbose
+    - cargo build --verbose --features arbitrary
+    - cargo build --verbose --features serde-serialize
+    - cargo test --verbose --features "arbitrary serde-serialize"

CHANGELOG.md

@@ -4,6 +4,106 @@ documented here.
 
 This project adheres to [Semantic Versioning](http://semver.org/).
 
+## [0.11.0]
+The [website](http://nalgebra.org) has been fully rewritten and gives a good
+overview of all the added/modified features.
+
+This version is a major rewrite of the library. Major changes are:
+  * Algebraic traits are now defined by the [alga](https://crates.io/crates/alga) crate.
+  All other mathematical traits, except `Axpy` have been removed from
+  **nalgebra**.
+  * Methods are now preferred to free functions because they do not require any
+    trait to be used any more.
+  * Most algebraic entities can be parametrized by type-level integers
+    to specify their dimensions. Using `Dynamic` instead of a type-level
+    integer indicates that the dimension known at run-time only.
+  * Statically-sized **rectangular** matrices.
+  * More transformation types have been added: unit-sized complex numbers (for
+    2D rotations), affine/projective/general transformations with `Affine2/3`,
+    `Projective2/3`, and `Transform2/3`.
+  * Serde serialization is now supported instead of `rustc_serialize`. Enable
+    it with the `serde-serialize` feature.
+  * Matrix **slices** are now implemented.
+
+### Added
+Lots of features including rectangular matrices, slices, and Serde
+serialization. Refer to the brand new [website](http://nalgebra.org) for more
+details. The following free-functions have been added as well:
+  * `::id()` that returns the universal [identity element](http://nalgebra.org/performance_tricks/#the-id-type)
+    of type `Id`.
+  * `::inf_sup()` that returns both the infimum and supremum of a value at the
+    same time.
+  * `::partial_sort2()` that attempts to sort two values in increasing order.
+  * `::wrap()` that moves a value to the given interval by adding or removing
+    the interval width to it.
+
+### Modified
+  * `::cast`            -> `::convert`
+  * `point.as_vector()` -> `point.coords`
+  * `na::origin`        -> `P::origin()`
+  * `na::is_zero`       -> `.is_zero()` (from num::Zero)
+  * `.transform`        -> `.transform_point`/`.transform_vector`
+  * `.translate`        -> `.translate_point`
+  * `::dimension::<P>`  -> `::dimension::<P::Vector>`
+  * `::angle_between`   -> `::angle`
+
+The following free-functions are now replaced by methods (with the same names)
+only:
+`::cross`, `::cholesky`, `::determinant`, `::diagonal`, `::eigen_qr` (becomes
+`.eig`), `::hessenberg`, `::qr`, `::to_homogeneous`, `::to_rotation_matrix`,
+`::transpose`, `::shape`.
+
+
+The following free-functions are now replaced by static methods only:
+  * `::householder_matrix` under the name `::new_householder_generic`
+  * `::identity`
+  * `::new_identity` under the name `::identity`
+  * `::from_homogeneous`
+  * `::repeat` under the name `::from_element`
+
+The following free-function are now replaced methods accessible through traits
+only:
+  * `::transform` -> methods `.transform_point` and `.transform_vector` of the `alga::linear::Transformation` trait.
+  * `::inverse_transform` -> methods `.inverse_transform_point` and
+    `.inverse_transform_vector` of the `alga::linear::ProjectiveTransformation`
+    trait.
+  * `::translate`, `::inverse_translate`, `::rotate`, `::inverse_rotate` ->
+    methods from the `alga::linear::Similarity` trait instead. Those have the
+    same names but end with `_point` or `_vector`, e.g., `.translate_point` and
+    `.translate_vector`.
+  * `::orthonormal_subspace_basis` -> method with the same name from
+    `alga::linear::FiniteDimInnerSpace`.
+  * `::canonical_basis_element` and `::canonical_basis` -> methods with the
+    same names from `alga::linear::FiniteDimVectorSpace`.
+  * `::rotation_between` -> method with the same name from the
+    `alga::linear::Rotation` trait.
+  * `::is_zero` -> method with the same name from `num::Zero`.
+
+
+
+### Removed
+  * The free functions `::prepend_rotation`, `::append_rotation`,
+    `::append_rotation_wrt_center`, `::append_rotation_wrt_point`,
+    `::append_transformation`, and `::append_translation ` have been removed.
+    Instead create the rotation or translation object explicitly and use
+    multiplication to compose it with anything else.
+
+  * The free function `::outer` has been removed. Use column-vector ×
+    row-vector multiplication instead.
+
+  * `::approx_eq`, `::approx_eq_eps` have been removed. Use the `relative_eq!`
+    macro from the [approx](https://crates.io/crates/approx) crate instead.
+
+  * `::covariance` has been removed. There is no replacement for now.
+  * `::mean` has been removed. There is no replacement for now.
+  * `::sample_sphere` has been removed. There is no replacement for now.
+  * `::cross_matrix` has been removed. There is no replacement for now.
+  * `::absolute_rotate` has been removed. There is no replacement for now.
+  * `::rotation`, `::transformation`, `::translation`, `::inverse_rotation`,
+    `::inverse_transformation`, `::inverse_translation` have been removed. Use
+    the appropriate methods/field of each transformation type, e.g.,
+    `rotation.angle()` and `rotation.axis()`.
+
 ## [0.10.0]
 ### Added
 Binary operations are now allowed between references as well. For example

Cargo.toml

@@ -1,7 +1,7 @@
 [package]
 name    = "nalgebra"
-version = "0.10.1"
-authors = [ "Sébastien Crozet <developer@crozet.re>" ] # FIXME: add the contributors.
+version = "0.11.0"
+authors = [ "Sébastien Crozet <developer@crozet.re>" ]
 
 description = "Linear algebra library for computer physics, computer graphics and general low-dimensional linear algebra for Rust."
 documentation = "http://nalgebra.org/doc/nalgebra/index.html"
@@ -16,28 +16,24 @@ name = "nalgebra"
 path = "src/lib.rs"
 
 [features]
-# Generate arbitrary instances of nalgebra types for testing with quickcheck
 arbitrary = [ "quickcheck" ]
-generic_sizes = [ "generic-array", "typenum" ]
-abstract_algebra = [ "algebra" ]
+serde-serialize = [ "serde", "serde_derive" ]
 
 [dependencies]
-rustc-serialize = "0.3.*"
-rand = "0.3.*"
-num  = "0.1.*"
-
-[dependencies.generic-array]
-optional = true
-version  = "0.2.*"
-
-[dependencies.typenum]
-optional = true
-version  = "1.3.*"
+typenum         = "1.4"
+generic-array   = "0.2"
+rand            = "0.3"
+num-traits      = "0.1"
+num-complex     = "0.1"
+approx          = "0.1"
+alga            = "0.5"
+serde           = { version = "0.9", optional = true }
+serde_derive    = { version = "0.9", optional = true }
+# clippy = "*"
 
 [dependencies.quickcheck]
 optional = true
-version  = "0.2.*"
+version  = "0.3"
 
-[dependencies.algebra]
-optional = true
-version = "0.2.*"
+[dev-dependencies]
+serde_json = "0.9"

Makefile

@@ -1,30 +1,11 @@
-tmp=_git_distcheck
-
 all:
-	cargo build --release --features "arbitrary generic_sizes abstract_algebra"
-
-test:
-	cargo test --features "arbitrary generic_sizes abstract_algebra"
-
-
-bench:
-	cargo bench --features "arbitrary generic_sizes abstract_algebra"
-
+	CARGO_INCREMENTAL=1 cargo build --features "arbitrary serde-serialize"
 
 doc:
-	cargo doc --no-deps --features "arbitrary generic_sizes abstract_algebra"
+	CARGO_INCREMENTAL=1 cargo doc --no-deps --features "arbitrary serde-serialize"
 
+bench:
+	cargo bench
 
-clean:
-	cargo clean
-
-distcheck:
-	rm -rf $(tmp)
-	git clone --recursive . $(tmp)
-	make -C $(tmp)
-	make -C $(tmp) test
-	make -C $(tmp) bench
-	rm -rf $(tmp)
-
-.PHONY:doc
-.PHONY:test
+test:
+	CARGO_INCREMENTAL=1 cargo test --features "arbitrary serde-serialize"

README.md

@@ -1,82 +1,26 @@
 <p align="center">
+  <img src="http://nalgebra.org/img/logo_nalgebra.svg" alt="crates.io">
+</p>
+<p align="center">
+    <a href="https://travis-ci.org/sebcrozet/nalgebra">
+        <img src="https://travis-ci.org/sebcrozet/nalgebra.svg?branch=master" alt="Build status">
+    </a>
     <a href="https://crates.io/crates/nalgebra">
          <img src="http://meritbadge.herokuapp.com/nalgebra?style=flat-square" alt="crates.io">
     </a>
-    <a href="https://travis-ci.org/sebcrozet/nalgebra">
-        <img src="https://travis-ci.org/sebcrozet/nalgebra.svg?branch=master" alt="Build status">
+    <a href="https://opensource.org/licenses/BSD-3-Clause">
+        <img src="https://img.shields.io/badge/license-BSD%203--Clause-blue.svg?style=flat">
     </a>
 </p>
 <p align = "center">
     <strong>
-        <a href="http://nalgebra.org/doc/nalgebra">Documentation</a> | <a href="http://users.nphysics.org">Forum</a>
+        <a href="http://nalgebra.org">Users guide</a> | <a href="http://nalgebra.org/rustdoc/nalgebra/index.html">Documentation</a> | <a href="http://users.nphysics.org/c/nalgebra">Forum</a>
     </strong>
 </p>
 
-nalgebra
-========
+-----
 
-**nalgebra** is a low-dimensional linear algebra library written for Rust targeting:
-
-* General-purpose linear algebra (still lacks a lot of features…)
-* Real time computer graphics.
-* Real time computer physics.
-
-## Using **nalgebra**
-You will need the last stable build of the [rust compiler](http://www.rust-lang.org)
-and the official package manager: [cargo](https://github.com/rust-lang/cargo).
-
-Simply add the following to your `Cargo.toml` file:
-
-```.ignore
-[dependencies]
-nalgebra = "0.10.*"
-```
-
-
-All the functionality of **nalgebra** is grouped in one place: the root module `nalgebra::`.  This
-module re-exports everything and includes free functions for all traits methods performing
-out-of-place operations.
-
-Thus, you can import the whole prelude using:
-
-```.ignore
-use nalgebra::*;
-```
-
-However, the recommended way to use **nalgebra** is to import types and traits
-explicitly, and call free-functions using the `na::` prefix:
-
-```.rust
-extern crate nalgebra as na;
-use na::{Vector3, Rotation3, Rotation};
-
-fn main() {
-    let     a = Vector3::new(1.0f64, 1.0, 1.0);
-    let mut b = Rotation3::new(na::zero());
-
-    b.append_rotation_mut(&a);
-
-    assert!(na::approx_eq(&na::rotation(&b), &a));
-}
-```
-
-
-## Features
-**nalgebra** is meant to be a general-purpose, low-dimensional, linear algebra library, with
-an optimized set of tools for computer graphics and physics. Those features include:
-
-* Vectors with predefined static sizes: `Vector1`, `Vector2`, `Vector3`, `Vector4`, `Vector5`, `Vector6`.
-* Vector with a user-defined static size: `VectorN` (available only with the `generic_sizes` feature).
-* Points with static sizes: `Point1`, `Point2`, `Point3`, `Point4`, `Point5`, `Point6`.
-* Square matrices with static sizes: `Matrix1`, `Matrix2`, `Matrix3`, `Matrix4`, `Matrix5`, `Matrix6 `.
-* Rotation matrices: `Rotation2`, `Rotation3`
-* Quaternions: `Quaternion`, `Unit<Quaternion>`.
-* Unit-sized values (unit vectors, unit quaternions, etc.): `Unit<T>`, e.g., `Unit<Vector3<f32>>`.
-* Isometries (translation ⨯ rotation): `Isometry2`, `Isometry3`
-* Similarity transformations (translation ⨯ rotation ⨯ uniform scale): `Similarity2`, `Similarity3`.
-* 3D projections for computer graphics: `Persp3`, `PerspMatrix3`, `Ortho3`, `OrthoMatrix3`.
-* Dynamically sized heap-allocated vector: `DVector`.
-* Dynamically sized stack-allocated vectors with a maximum size: `DVector1` to `DVector6`.
-* Dynamically sized heap-allocated (square or rectangular) matrix: `DMatrix`.
-* Linear algebra and data analysis operators: `Covariance`, `Mean`, `qr`, `cholesky`.
-* Almost one trait per functionality: useful for generic programming.
+<p align = "center">
+<b>Linear algebra library</b>
+<i>for the Rust programming language.</i>
+</p>

benches/common/macros.rs

@@ -16,7 +16,30 @@ macro_rules! bench_binop(
                 i = (i + 1) & (LEN - 1);
 
                 unsafe {
-                    test::black_box((*elems1.get_unchecked(i)).$binop(*elems2.get_unchecked(i)))
+                    test::black_box(elems1.get_unchecked(i).$binop(*elems2.get_unchecked(i)))
+                }
+            })
+        }
+    }
+);
+
+macro_rules! bench_binop_ref(
+    ($name: ident, $t1: ty, $t2: ty, $binop: ident) => {
+        #[bench]
+        fn $name(bh: &mut Bencher) {
+            const LEN: usize = 1 << 13;
+
+            let mut rng = IsaacRng::new_unseeded();
+
+            let elems1: Vec<$t1> = (0usize .. LEN).map(|_| rng.gen::<$t1>()).collect();
+            let elems2: Vec<$t2> = (0usize .. LEN).map(|_| rng.gen::<$t2>()).collect();
+            let mut i = 0;
+
+            bh.iter(|| {
+                i = (i + 1) & (LEN - 1);
+
+                unsafe {
+                    test::black_box(elems1.get_unchecked(i).$binop(elems2.get_unchecked(i)))
                 }
             })
         }
@@ -46,7 +69,7 @@ macro_rules! bench_binop_na(
     }
 );
 
-macro_rules! bench_unop(
+macro_rules! bench_unop_na(
     ($name: ident, $t: ty, $unop: ident) => {
         #[bench]
         fn $name(bh: &mut Bencher) {
@@ -68,7 +91,7 @@ macro_rules! bench_unop(
     }
 );
 
-macro_rules! bench_unop_self(
+macro_rules! bench_unop(
     ($name: ident, $t: ty, $unop: ident) => {
         #[bench]
         fn $name(bh: &mut Bencher) {

benches/construction.rs

@@ -1,19 +0,0 @@
-#![feature(test)]
-
-extern crate test;
-extern crate rand;
-extern crate nalgebra as na;
-
-use rand::{IsaacRng, Rng};
-use test::Bencher;
-use na::{UnitQuaternion, Rotation2, Rotation3, Vector1, Vector3};
-
-#[path="common/macros.rs"]
-mod macros;
-
-bench_construction!(_bench_quaternion_from_axisangle, UnitQuaternion::from_scaled_axis, axisangle: Vector3<f32>);
-bench_construction!(_bench_rot2_from_axisangle, Rotation2::new, axisangle: Vector1<f32>);
-bench_construction!(_bench_rot3_from_axisangle, Rotation3::new, axisangle: Vector3<f32>);
-
-bench_construction!(_bench_quaternion_from_euler_angles, UnitQuaternion::from_euler_angles, roll: f32, pitch: f32, yaw: f32);
-bench_construction!(_bench_rot3_from_euler_angles, Rotation3::from_euler_angles, roll: f32, pitch: f32, yaw: f32);

benches/dmat.rs

@@ -1,90 +0,0 @@
-#![feature(test)]
-
-extern crate test;
-extern crate nalgebra as na;
-
-use test::Bencher;
-use na::{DVector, DMatrix};
-
-macro_rules! bench_mul_dmatrix(
-    ($bh: expr, $nrows: expr, $ncols: expr) => {
-        {
-            $bh.iter(|| {
-                let a:     DMatrix<f64> = DMatrix::new_random($nrows, $ncols);
-                let mut b: DMatrix<f64> = DMatrix::new_random($nrows, $ncols);
-
-                for _ in 0usize .. 1000 {
-                    // XXX: the clone here is highly undesirable!
-                    b = a.clone() * b;
-                }
-            })
-        }
-    }
-);
-
-#[bench]
-fn bench_mul_dmat2(bh: &mut Bencher) {
-    bench_mul_dmatrix!(bh, 2, 2);
-}
-
-#[bench]
-fn bench_mul_dmat3(bh: &mut Bencher) {
-    bench_mul_dmatrix!(bh, 3, 3);
-}
-
-#[bench]
-fn bench_mul_dmat4(bh: &mut Bencher) {
-    bench_mul_dmatrix!(bh, 4, 4);
-}
-
-#[bench]
-fn bench_mul_dmat5(bh: &mut Bencher) {
-    bench_mul_dmatrix!(bh, 5, 5);
-}
-
-#[bench]
-fn bench_mul_dmat6(bh: &mut Bencher) {
-    bench_mul_dmatrix!(bh, 6, 6);
-}
-
-macro_rules! bench_mul_dmat_dvector(
-    ($bh: expr, $nrows: expr, $ncols: expr) => {
-        {
-
-            $bh.iter(|| {
-                let m : DMatrix<f64>     = DMatrix::new_random($nrows, $ncols);
-                let mut v : DVector<f64> = DVector::new_random($ncols);
-
-                for _ in 0usize .. 1000 {
-                    // XXX: the clone here is highly undesirable!
-                    v = m.clone() * v
-                }
-            })
-        }
-    }
-);
-
-#[bench]
-fn bench_mul_dmat_dvec2(bh: &mut Bencher) {
-    bench_mul_dmat_dvector!(bh, 2, 2);
-}
-
-#[bench]
-fn bench_mul_dmat_dvec3(bh: &mut Bencher) {
-    bench_mul_dmat_dvector!(bh, 3, 3);
-}
-
-#[bench]
-fn bench_mul_dmat_dvec4(bh: &mut Bencher) {
-    bench_mul_dmat_dvector!(bh, 4, 4);
-}
-
-#[bench]
-fn bench_mul_dmat_dvec5(bh: &mut Bencher) {
-    bench_mul_dmat_dvector!(bh, 5, 5);
-}
-
-#[bench]
-fn bench_mul_dmat_dvec6(bh: &mut Bencher) {
-    bench_mul_dmat_dvector!(bh, 6, 6);
-}

benches/matrix.rs

@@ -16,6 +16,10 @@ bench_binop!(_bench_mat2_mul_m, Matrix2<f32>, Matrix2<f32>, mul);
 bench_binop!(_bench_mat3_mul_m, Matrix3<f32>, Matrix3<f32>, mul);
 bench_binop!(_bench_mat4_mul_m, Matrix4<f32>, Matrix4<f32>, mul);
 
+bench_binop_ref!(_bench_mat2_tr_mul_m, Matrix2<f32>, Matrix2<f32>, tr_mul);
+bench_binop_ref!(_bench_mat3_tr_mul_m, Matrix3<f32>, Matrix3<f32>, tr_mul);
+bench_binop_ref!(_bench_mat4_tr_mul_m, Matrix4<f32>, Matrix4<f32>, tr_mul);
+
 bench_binop!(_bench_mat2_add_m, Matrix2<f32>, Matrix2<f32>, add);
 bench_binop!(_bench_mat3_add_m, Matrix3<f32>, Matrix3<f32>, add);
 bench_binop!(_bench_mat4_add_m, Matrix4<f32>, Matrix4<f32>, add);
@@ -28,6 +32,10 @@ bench_binop!(_bench_mat2_mul_v, Matrix2<f32>, Vector2<f32>, mul);
 bench_binop!(_bench_mat3_mul_v, Matrix3<f32>, Vector3<f32>, mul);
 bench_binop!(_bench_mat4_mul_v, Matrix4<f32>, Vector4<f32>, mul);
 
+bench_binop_ref!(_bench_mat2_tr_mul_v, Matrix2<f32>, Vector2<f32>, tr_mul);
+bench_binop_ref!(_bench_mat3_tr_mul_v, Matrix3<f32>, Vector3<f32>, tr_mul);
+bench_binop_ref!(_bench_mat4_tr_mul_v, Matrix4<f32>, Vector4<f32>, tr_mul);
+
 bench_binop!(_bench_mat2_mul_s, Matrix2<f32>, f32, mul);
 bench_binop!(_bench_mat3_mul_s, Matrix3<f32>, f32, mul);
 bench_binop!(_bench_mat4_mul_s, Matrix4<f32>, f32, mul);
@@ -36,9 +44,9 @@ bench_binop!(_bench_mat2_div_s, Matrix2<f32>, f32, div);
 bench_binop!(_bench_mat3_div_s, Matrix3<f32>, f32, div);
 bench_binop!(_bench_mat4_div_s, Matrix4<f32>, f32, div);
 
-bench_unop!(_bench_mat2_inv, Matrix2<f32>, inverse);
-bench_unop!(_bench_mat3_inv, Matrix3<f32>, inverse);
-bench_unop!(_bench_mat4_inv, Matrix4<f32>, inverse);
+bench_unop!(_bench_mat2_inv, Matrix2<f32>, try_inverse);
+bench_unop!(_bench_mat3_inv, Matrix3<f32>, try_inverse);
+bench_unop!(_bench_mat4_inv, Matrix4<f32>, try_inverse);
 
 bench_unop!(_bench_mat2_transpose, Matrix2<f32>, transpose);
 bench_unop!(_bench_mat3_transpose, Matrix3<f32>, transpose);

benches/quaternion.rs

@@ -15,10 +15,14 @@ mod macros;
 bench_binop!(_bench_quaternion_add_q, Quaternion<f32>, Quaternion<f32>, add);
 bench_binop!(_bench_quaternion_sub_q, Quaternion<f32>, Quaternion<f32>, sub);
 bench_binop!(_bench_quaternion_mul_q, Quaternion<f32>, Quaternion<f32>, mul);
-// bench_binop!(_bench_quaternion_div_q, Quaternion<f32>, Quaternion<f32>, div)
-bench_binop!(_bench_quaternion_mul_v, UnitQuaternion<f32>, Vector3<f32>, mul);
+
+bench_binop!(_bench_unit_quaternion_mul_v, UnitQuaternion<f32>, Vector3<f32>, mul);
+
 bench_binop!(_bench_quaternion_mul_s, Quaternion<f32>, f32, mul);
 bench_binop!(_bench_quaternion_div_s, Quaternion<f32>, f32, div);
-bench_unop!(_bench_quaternion_inv, Quaternion<f32>, inverse);
-bench_unop_self!(_bench_quaternion_conjugate, Quaternion<f32>, conjugate);
-bench_unop!(_bench_quaternion_normalize, Quaternion<f32>, normalize);
+
+bench_unop!(_bench_quaternion_inv, Quaternion<f32>, try_inverse);
+bench_unop!(_bench_unit_quaternion_inv, UnitQuaternion<f32>, inverse);
+
+// bench_unop_self!(_bench_quaternion_conjugate, Quaternion<f32>, conjugate);
+// bench_unop!(_bench_quaternion_normalize, Quaternion<f32>, normalize);

benches/vec.rs

@@ -1,119 +0,0 @@
-#![feature(test)]
-
-#[cfg(feature = "generic_sizes")]
-extern crate typenum;
-
-extern crate test;
-extern crate rand;
-extern crate nalgebra as na;
-
-use rand::{IsaacRng, Rng};
-use test::Bencher;
-use na::{Vector2, Vector3, Vector4};
-use std::ops::{Add, Sub, Mul, Div};
-
-#[path="common/macros.rs"]
-mod macros;
-
-bench_binop!(_bench_vec2_add_v, Vector2<f32>, Vector2<f32>, add);
-bench_binop!(_bench_vec3_add_v, Vector3<f32>, Vector3<f32>, add);
-bench_binop!(_bench_vec4_add_v, Vector4<f32>, Vector4<f32>, add);
-
-bench_binop!(_bench_vec2_sub_v, Vector2<f32>, Vector2<f32>, sub);
-bench_binop!(_bench_vec3_sub_v, Vector3<f32>, Vector3<f32>, sub);
-bench_binop!(_bench_vec4_sub_v, Vector4<f32>, Vector4<f32>, sub);
-
-bench_binop!(_bench_vec2_mul_v, Vector2<f32>, Vector2<f32>, mul);
-bench_binop!(_bench_vec3_mul_v, Vector3<f32>, Vector3<f32>, mul);
-bench_binop!(_bench_vec4_mul_v, Vector4<f32>, Vector4<f32>, mul);
-
-bench_binop!(_bench_vec2_div_v, Vector2<f32>, Vector2<f32>, div);
-bench_binop!(_bench_vec3_div_v, Vector3<f32>, Vector3<f32>, div);
-bench_binop!(_bench_vec4_div_v, Vector4<f32>, Vector4<f32>, div);
-
-bench_binop!(_bench_vec2_add_s, Vector2<f32>, f32, add);
-bench_binop!(_bench_vec3_add_s, Vector3<f32>, f32, add);
-bench_binop!(_bench_vec4_add_s, Vector4<f32>, f32, add);
-
-bench_binop!(_bench_vec2_sub_s, Vector2<f32>, f32, sub);
-bench_binop!(_bench_vec3_sub_s, Vector3<f32>, f32, sub);
-bench_binop!(_bench_vec4_sub_s, Vector4<f32>, f32, sub);
-
-bench_binop!(_bench_vec2_mul_s, Vector2<f32>, f32, mul);
-bench_binop!(_bench_vec3_mul_s, Vector3<f32>, f32, mul);
-bench_binop!(_bench_vec4_mul_s, Vector4<f32>, f32, mul);
-
-bench_binop!(_bench_vec2_div_s, Vector2<f32>, f32, div);
-bench_binop!(_bench_vec3_div_s, Vector3<f32>, f32, div);
-bench_binop!(_bench_vec4_div_s, Vector4<f32>, f32, div);
-
-bench_binop_na!(_bench_vec2_dot, Vector2<f32>, Vector2<f32>, dot);
-bench_binop_na!(_bench_vec3_dot, Vector3<f32>, Vector3<f32>, dot);
-bench_binop_na!(_bench_vec4_dot, Vector4<f32>, Vector4<f32>, dot);
-
-bench_binop_na!(_bench_vec3_cross, Vector3<f32>, Vector3<f32>, cross);
-
-bench_unop!(_bench_vec2_norm, Vector2<f32>, norm);
-bench_unop!(_bench_vec3_norm, Vector3<f32>, norm);
-bench_unop!(_bench_vec4_norm, Vector4<f32>, norm);
-
-bench_unop!(_bench_vec2_normalize, Vector2<f32>, normalize);
-bench_unop!(_bench_vec3_normalize, Vector3<f32>, normalize);
-bench_unop!(_bench_vec4_normalize, Vector4<f32>, normalize);
-
-#[cfg(feature = "generic_sizes")]
-mod bench_vecn {
-    extern crate test;
-    extern crate rand;
-    extern crate nalgebra as na;
-
-    use rand::{IsaacRng, Rng};
-    use test::Bencher;
-    use std::ops::{Add, Sub, Mul, Div};
-    use typenum::{U2, U3, U4};
-    use na::VectorN;
-
-    bench_binop!(_bench_vecn2_add_v, VectorN<f32, U2>, VectorN<f32, U2>, add);
-    bench_binop!(_bench_vecn3_add_v, VectorN<f32, U3>, VectorN<f32, U3>, add);
-    bench_binop!(_bench_vecn4_add_v, VectorN<f32, U4>, VectorN<f32, U4>, add);
-
-    bench_binop!(_bench_vecn2_sub_v, VectorN<f32, U2>, VectorN<f32, U2>, sub);
-    bench_binop!(_bench_vecn3_sub_v, VectorN<f32, U3>, VectorN<f32, U3>, sub);
-    bench_binop!(_bench_vecn4_sub_v, VectorN<f32, U4>, VectorN<f32, U4>, sub);
-
-    bench_binop!(_bench_vecn2_mul_v, VectorN<f32, U2>, VectorN<f32, U2>, mul);
-    bench_binop!(_bench_vecn3_mul_v, VectorN<f32, U3>, VectorN<f32, U3>, mul);
-    bench_binop!(_bench_vecn4_mul_v, VectorN<f32, U4>, VectorN<f32, U4>, mul);
-
-    bench_binop!(_bench_vecn2_div_v, VectorN<f32, U2>, VectorN<f32, U2>, div);
-    bench_binop!(_bench_vecn3_div_v, VectorN<f32, U3>, VectorN<f32, U3>, div);
-    bench_binop!(_bench_vecn4_div_v, VectorN<f32, U4>, VectorN<f32, U4>, div);
-
-    bench_binop!(_bench_vecn2_add_s, VectorN<f32, U2>, f32, add);
-    bench_binop!(_bench_vecn3_add_s, VectorN<f32, U3>, f32, add);
-    bench_binop!(_bench_vecn4_add_s, VectorN<f32, U4>, f32, add);
-
-    bench_binop!(_bench_vecn2_sub_s, VectorN<f32, U2>, f32, sub);
-    bench_binop!(_bench_vecn3_sub_s, VectorN<f32, U3>, f32, sub);
-    bench_binop!(_bench_vecn4_sub_s, VectorN<f32, U4>, f32, sub);
-
-    bench_binop!(_bench_vecn2_mul_s, VectorN<f32, U2>, f32, mul);
-    bench_binop!(_bench_vecn3_mul_s, VectorN<f32, U3>, f32, mul);
-    bench_binop!(_bench_vecn4_mul_s, VectorN<f32, U4>, f32, mul);
-
-    bench_binop!(_bench_vecn2_div_s, VectorN<f32, U2>, f32, div);
-    bench_binop!(_bench_vecn3_div_s, VectorN<f32, U3>, f32, div);
-    bench_binop!(_bench_vecn4_div_s, VectorN<f32, U4>, f32, div);
-
-    bench_binop_na!(_bench_vecn2_dot, VectorN<f32, U2>, VectorN<f32, U2>, dot);
-    bench_binop_na!(_bench_vecn3_dot, VectorN<f32, U3>, VectorN<f32, U3>, dot);
-    bench_binop_na!(_bench_vecn4_dot, VectorN<f32, U4>, VectorN<f32, U4>, dot);
-
-    bench_unop!(_bench_vecn2_norm, VectorN<f32, U2>, norm);
-    bench_unop!(_bench_vecn3_norm, VectorN<f32, U3>, norm);
-    bench_unop!(_bench_vecn4_norm, VectorN<f32, U4>, norm);
-
-    bench_unop!(_bench_vecn2_normalize, VectorN<f32, U2>, normalize);
-    bench_unop!(_bench_vecn3_normalize, VectorN<f32, U3>, normalize);
-    bench_unop!(_bench_vecn4_normalize, VectorN<f32, U4>, normalize);
-}

benches/vector.rs

@@ -0,0 +1,43 @@
+#![feature(test)]
+
+extern crate test;
+extern crate rand;
+extern crate nalgebra as na;
+
+use rand::{IsaacRng, Rng};
+use test::Bencher;
+use na::{Vector2, Vector3, Vector4};
+use std::ops::{Add, Sub, Mul, Div};
+
+#[path="common/macros.rs"]
+mod macros;
+
+bench_binop!(_bench_vec2_add_v, Vector2<f32>, Vector2<f32>, add);
+bench_binop!(_bench_vec3_add_v, Vector3<f32>, Vector3<f32>, add);
+bench_binop!(_bench_vec4_add_v, Vector4<f32>, Vector4<f32>, add);
+
+bench_binop!(_bench_vec2_sub_v, Vector2<f32>, Vector2<f32>, sub);
+bench_binop!(_bench_vec3_sub_v, Vector3<f32>, Vector3<f32>, sub);
+bench_binop!(_bench_vec4_sub_v, Vector4<f32>, Vector4<f32>, sub);
+
+bench_binop!(_bench_vec2_mul_s, Vector2<f32>, f32, mul);
+bench_binop!(_bench_vec3_mul_s, Vector3<f32>, f32, mul);
+bench_binop!(_bench_vec4_mul_s, Vector4<f32>, f32, mul);
+
+bench_binop!(_bench_vec2_div_s, Vector2<f32>, f32, div);
+bench_binop!(_bench_vec3_div_s, Vector3<f32>, f32, div);
+bench_binop!(_bench_vec4_div_s, Vector4<f32>, f32, div);
+
+bench_binop_ref!(_bench_vec2_dot, Vector2<f32>, Vector2<f32>, dot);
+bench_binop_ref!(_bench_vec3_dot, Vector3<f32>, Vector3<f32>, dot);
+bench_binop_ref!(_bench_vec4_dot, Vector4<f32>, Vector4<f32>, dot);
+
+bench_binop_ref!(_bench_vec3_cross, Vector3<f32>, Vector3<f32>, cross);
+
+bench_unop!(_bench_vec2_norm, Vector2<f32>, norm);
+bench_unop!(_bench_vec3_norm, Vector3<f32>, norm);
+bench_unop!(_bench_vec4_norm, Vector4<f32>, norm);
+
+bench_unop!(_bench_vec2_normalize, Vector2<f32>, normalize);
+bench_unop!(_bench_vec3_normalize, Vector3<f32>, normalize);
+bench_unop!(_bench_vec4_normalize, Vector4<f32>, normalize);

examples/dimensional_genericity.rs

@@ -0,0 +1,66 @@
+extern crate alga;
+extern crate nalgebra as na;
+
+use alga::general::Real;
+use alga::linear::FiniteDimInnerSpace;
+use na::{Unit, ColumnVector, OwnedColumnVector, Vector2, Vector3};
+use na::storage::Storage;
+use na::dimension::{DimName, U1};
+
+/// Reflects a vector wrt. the hyperplane with normal `plane_normal`.
+fn reflect_wrt_hyperplane_with_algebraic_genericity<V>(plane_normal: &Unit<V>, vector: &V) -> V
+    where V: FiniteDimInnerSpace + Copy {
+    let n = plane_normal.as_ref(); // Get the underlying vector of type `V`.
+    *vector - *n * (n.dot(vector) * na::convert(2.0))
+}
+
+
+/// Reflects a vector wrt. the hyperplane with normal `plane_normal`.
+fn reflect_wrt_hyperplane_with_structural_genericity<N, D, S>(plane_normal: &Unit<ColumnVector<N, D, S>>,
+                                                               vector:       &ColumnVector<N, D, S>)
+                                                               -> OwnedColumnVector<N, D, S::Alloc>
+    where N: Real,
+          D: DimName,
+          S: Storage<N, D, U1> {
+    let n = plane_normal.as_ref(); // Get the underlying V.
+    vector - n * (n.dot(vector) * na::convert(2.0))
+}
+
+/// Reflects a 2D vector wrt. the 2D line with normal `plane_normal`.
+fn reflect_wrt_hyperplane2<N>(plane_normal: &Unit<Vector2<N>>,
+                              vector:       &Vector2<N>)
+                              -> Vector2<N>
+    where N: Real {
+    let n = plane_normal.as_ref(); // Get the underlying Vector2
+    vector - n * (n.dot(vector) * na::convert(2.0))
+}
+
+/// Reflects a 3D vector wrt. the 3D plane with normal `plane_normal`.
+/// /!\ This is an exact replicate of `reflect_wrt_hyperplane2, but for 3D.
+fn reflect_wrt_hyperplane3<N>(plane_normal: &Unit<Vector3<N>>,
+                              vector:       &Vector3<N>)
+                              -> Vector3<N>
+    where N: Real {
+    let n = plane_normal.as_ref(); // Get the underlying Vector3
+    vector - n * (n.dot(vector) * na::convert(2.0))
+}
+
+
+fn main() {
+    let plane2 = Vector2::y_axis(); // 2D plane normal.
+    let plane3 = Vector3::y_axis(); // 3D plane normal.
+
+    let v2 = Vector2::new(1.0, 2.0);      // 2D vector to be reflected.
+    let v3 = Vector3::new(1.0, 2.0, 3.0); // 3D vector to be reflected.
+
+    // We can call the same function for 2D and 3D.
+    assert_eq!(reflect_wrt_hyperplane_with_algebraic_genericity(&plane2, &v2).y, -2.0);
+    assert_eq!(reflect_wrt_hyperplane_with_algebraic_genericity(&plane3, &v3).y, -2.0);
+
+    assert_eq!(reflect_wrt_hyperplane_with_structural_genericity(&plane2, &v2).y, -2.0);
+    assert_eq!(reflect_wrt_hyperplane_with_structural_genericity(&plane3, &v3).y, -2.0);
+
+    // Call each specific implementation depending on the dimension.
+    assert_eq!(reflect_wrt_hyperplane2(&plane2, &v2).y, -2.0);
+    assert_eq!(reflect_wrt_hyperplane3(&plane3, &v3).y, -2.0);
+}

examples/homogeneous_coordinates.rs

@@ -0,0 +1,45 @@
+#[macro_use]
+extern crate approx;
+extern crate nalgebra as na;
+
+use std::f32;
+use na::{Vector2, Point2, Isometry2};
+
+
+fn use_dedicated_types() {
+    let iso = Isometry2::new(Vector2::new(1.0, 1.0), f32::consts::PI);
+    let pt  = Point2::new(1.0, 0.0);
+    let vec = Vector2::x();
+
+    let transformed_pt  = iso * pt;
+    let transformed_vec = iso * vec;
+
+    assert_relative_eq!(transformed_pt, Point2::new(0.0, 1.0));
+    assert_relative_eq!(transformed_vec, Vector2::new(-1.0, 0.0));
+}
+
+fn use_homogeneous_coordinates() {
+    let iso = Isometry2::new(Vector2::new(1.0, 1.0), f32::consts::PI);
+    let pt  = Point2::new(1.0, 0.0);
+    let vec = Vector2::x();
+
+    // Compute using homogeneous coordinates.
+    let hom_iso = iso.to_homogeneous();
+    let hom_pt  = pt.to_homogeneous();
+    let hom_vec = vec.to_homogeneous();
+
+    let hom_transformed_pt  = hom_iso * hom_pt;
+    let hom_transformed_vec = hom_iso * hom_vec;
+
+    // Convert back to the cartesian coordinates.
+    let transformed_pt  = Point2::from_homogeneous(hom_transformed_pt).unwrap();
+    let transformed_vec = Vector2::from_homogeneous(hom_transformed_vec).unwrap();
+
+    assert_relative_eq!(transformed_pt, Point2::new(0.0, 1.0));
+    assert_relative_eq!(transformed_vec, Vector2::new(-1.0, 0.0));
+}
+
+fn main() {
+    use_dedicated_types();
+    use_homogeneous_coordinates();
+}

examples/identity.rs

@@ -0,0 +1,42 @@
+extern crate alga;
+extern crate nalgebra as na;
+
+
+use alga::linear::Transformation;
+use na::{Id, Vector3, Point3, Isometry3};
+
+/*
+ * Applies `n` times the transformation `t` to the vector `v` and sum each
+ * intermediate value.
+ */
+fn complicated_algorithm<T>(v: &Vector3<f32>, t: &T, n: usize) -> Vector3<f32>
+  where T: Transformation<Point3<f32>> {
+
+  let mut result = *v;
+
+  // Do lots of operations involving t.
+  for _ in 0 .. n {
+    result = v + t.transform_vector(&result);
+  }
+
+  result
+}
+
+
+/*
+ * The two following calls are equivalent in term of result.
+ */
+fn main() {
+  let v = Vector3::new(1.0, 2.0, 3.0);
+
+  // The specialization generated by the compiler will do vector additions only.
+  let result1 = complicated_algorithm(&v, &Id::new(), 100000);
+
+  // The specialization generated by the compiler will also include matrix multiplications.
+  let iso     = Isometry3::identity();
+  let result2 = complicated_algorithm(&v, &iso, 100000);
+
+  // They both return the same result.
+  assert!(result1 == Vector3::new(100001.0, 200002.0, 300003.0));
+  assert!(result2 == Vector3::new(100001.0, 200002.0, 300003.0));
+}

examples/matrix_construction.rs

@@ -0,0 +1,62 @@
+extern crate nalgebra as na;
+
+use na::{Vector2, RowVector3, Matrix2x3, DMatrix};
+
+
+fn main() {
+    // All the following matrices are equal but constructed in different ways.
+    let m = Matrix2x3::new(1.1, 1.2, 1.3,
+                           2.1, 2.2, 2.3);
+
+    let m1 = Matrix2x3::from_rows(&[
+        RowVector3::new(1.1, 1.2, 1.3),
+        RowVector3::new(2.1, 2.2, 2.3)
+    ]);
+
+    let m2 = Matrix2x3::from_columns(&[
+        Vector2::new(1.1, 2.1),
+        Vector2::new(1.2, 2.2),
+        Vector2::new(1.3, 2.3)
+    ]);
+
+    let m3 = Matrix2x3::from_row_slice(&[
+        1.1, 1.2, 1.3,
+        2.1, 2.2, 2.3
+    ]);
+
+    let m4 = Matrix2x3::from_column_slice(&[
+        1.1, 2.1,
+        1.2, 2.2,
+        1.3, 2.3
+    ]);
+
+    let m5 = Matrix2x3::from_fn(|r, c| (r + 1) as f32 + (c + 1) as f32 / 10.0);
+
+    let m6 = Matrix2x3::from_iterator([ 1.1f32, 2.1, 1.2, 2.2, 1.3, 2.3 ].iter().cloned());
+
+    assert_eq!(m, m1); assert_eq!(m, m2); assert_eq!(m, m3);
+    assert_eq!(m, m4); assert_eq!(m, m5); assert_eq!(m, m6);
+
+    // All the following matrices are equal but constructed in different ways.
+    // This time, we used a dynamically-sized matrix to show the extra arguments
+    // for the matrix shape.
+    let dm = DMatrix::from_row_slice(4, 3, &[
+        1.0, 0.0, 0.0,
+        0.0, 1.0, 0.0,
+        0.0, 0.0, 1.0,
+        0.0, 0.0, 0.0
+    ]);
+
+    let dm1 = DMatrix::from_diagonal_element(4, 3, 1.0);
+    let dm2 = DMatrix::identity(4, 3);
+    let dm3 = DMatrix::from_fn(4, 3, |r, c| if r == c { 1.0 } else { 0.0 });
+    let dm4 = DMatrix::from_iterator(4, 3, [
+        // Components listed column-by-column.
+        1.0, 0.0, 0.0, 0.0,
+        0.0, 1.0, 0.0, 0.0,
+        0.0, 0.0, 1.0, 0.0
+    ].iter().cloned());
+
+    assert_eq!(dm, dm1); assert_eq!(dm, dm2);
+    assert_eq!(dm, dm3); assert_eq!(dm, dm4);
+}

examples/mvp.rs

@@ -0,0 +1,28 @@
+#![allow(unused_variables)]
+
+extern crate nalgebra as na;
+
+use na::{Vector3, Point3, Isometry3, Perspective3};
+
+fn main() {
+    // Our object is translated along the x axis.
+    let model = Isometry3::new(Vector3::x(), na::zero());
+
+    // Our camera looks toward the point (1.0, 0.0, 0.0).
+    // It is located at (0.0, 0.0, 1.0).
+    let eye    = Point3::new(0.0, 0.0, 1.0);
+    let target = Point3::new(1.0, 0.0, 0.0);
+    let view   = Isometry3::look_at_rh(&eye, &target, &Vector3::y());
+
+    // A perspective projection.
+    let projection = Perspective3::new(16.0 / 9.0, 3.14 / 2.0, 1.0, 1000.0);
+
+    // The combination of the model with the view is still an isometry.
+    let model_view = model * view;
+
+    // Convert everything to a `Matrix4` so that they can be combined.
+    let mat_model_view = model_view.to_homogeneous();
+
+    // Combine everything.
+    let model_view_projection = projection.as_matrix() * mat_model_view;
+}

examples/raw_pointer.rs

@@ -0,0 +1,35 @@
+extern crate nalgebra as na;
+
+use na::{Vector3, Point3, Matrix3};
+
+fn main() {
+    let v = Vector3::new(1.0f32, 0.0, 1.0);
+    let p = Point3::new(1.0f32, 0.0, 1.0);
+    let m = na::one::<Matrix3<f32>>();
+
+    // Convert to arrays.
+    let v_array = v.as_slice();
+    let p_array = p.coords.as_slice();
+    let m_array = m.as_slice();
+
+    // Get data pointers.
+    let v_pointer = v_array.as_ptr();
+    let p_pointer = p_array.as_ptr();
+    let m_pointer = m_array.as_ptr();
+
+    /* Then pass the raw pointers to some graphics API. */
+
+    unsafe {
+        assert_eq!(*v_pointer, 1.0);
+        assert_eq!(*v_pointer.offset(1), 0.0);
+        assert_eq!(*v_pointer.offset(2), 1.0);
+
+        assert_eq!(*p_pointer, 1.0);
+        assert_eq!(*p_pointer.offset(1), 0.0);
+        assert_eq!(*p_pointer.offset(2), 1.0);
+
+        assert_eq!(*m_pointer, 1.0);
+        assert_eq!(*m_pointer.offset(4), 1.0);
+        assert_eq!(*m_pointer.offset(8), 1.0);
+    }
+}

examples/scalar_genericity.rs

@@ -0,0 +1,33 @@
+extern crate alga;
+extern crate nalgebra as na;
+
+use alga::general::{RingCommutative, Real};
+use na::{Vector3, Scalar};
+
+fn print_vector<N: Scalar>(m: &Vector3<N>) {
+  println!("{:?}", m)
+}
+
+fn print_squared_norm<N: Scalar + RingCommutative>(v: &Vector3<N>) {
+  // NOTE: alternatively, nalgebra already defines `v.squared_norm()`.
+  let sqnorm = v.dot(v);
+  println!("{:?}", sqnorm);
+}
+
+fn print_norm<N: Real>(v: &Vector3<N>) {
+  // NOTE: alternatively, nalgebra already defines `v.norm()`.
+  let norm = v.dot(v).sqrt();
+
+  // The Real bound implies that N is Display so we can
+  // use "{}" instead of "{:?}" for the format string.
+  println!("{}", norm)
+}
+
+fn main() {
+  let v1 = Vector3::new(1, 2, 3);
+  let v2 = Vector3::new(1.0, 2.0, 3.0);
+
+  print_vector(&v1);
+  print_squared_norm(&v1);
+  print_norm(&v2);
+}

examples/screen_to_view_coords.rs

@@ -0,0 +1,24 @@
+#![allow(unused_variables)]
+
+extern crate nalgebra as na;
+
+use na::{Point2, Point3, Perspective3, Unit};
+
+
+fn main() {
+    let projection   = Perspective3::new(800.0 / 600.0, 3.14 / 2.0, 1.0, 1000.0);
+    let screen_point = Point2::new(10.0f32, 20.0);
+
+    // Compute two points in clip-space.
+    // "ndc" = normalized device coordinates.
+    let near_ndc_point = Point3::new(screen_point.x / 800.0, screen_point.y / 600.0, -1.0);
+    let far_ndc_point  = Point3::new(screen_point.x / 800.0, screen_point.y / 600.0,  1.0);
+
+    // Unproject them to view-space.
+    let near_view_point = projection.unproject_point(&near_ndc_point);
+    let far_view_point  = projection.unproject_point(&far_ndc_point);
+
+    // Compute the view-space line parameters.
+    let line_location  = near_view_point;
+    let line_direction = Unit::new_normalize(far_view_point - near_view_point);
+}

examples/transform_conversion.rs

@@ -0,0 +1,23 @@
+extern crate nalgebra as na;
+
+use na::{Vector2, Isometry2, Similarity2};
+
+fn main() {
+    // Isometry -> Similarity conversion always succeeds.
+    let iso = Isometry2::new(Vector2::new(1.0f32, 2.0), na::zero());
+    let _: Similarity2<f32> = na::convert(iso);
+
+    // Similarity -> Isometry conversion fails if the scaling factor is not 1.0.
+    let sim_without_scaling = Similarity2::new(Vector2::new(1.0f32, 2.0), 3.14, 1.0);
+    let sim_with_scaling    = Similarity2::new(Vector2::new(1.0f32, 2.0), 3.14, 2.0);
+
+    let iso_success: Option<Isometry2<f32>> = na::try_convert(sim_without_scaling);
+    let iso_fail:    Option<Isometry2<f32>> = na::try_convert(sim_with_scaling);
+
+    assert!(iso_success.is_some());
+    assert!(iso_fail.is_none());
+
+    // Similarity -> Isometry conversion can be forced at your own risks.
+    let iso_forced: Isometry2<f32> = unsafe { na::convert_unchecked(sim_with_scaling) };
+    assert_eq!(iso_success.unwrap(), iso_forced);
+}

examples/transform_matrix4.rs

@@ -0,0 +1,39 @@
+#[macro_use]
+extern crate approx;
+extern crate alga;
+extern crate nalgebra as na;
+
+use alga::linear::Transformation;
+use na::{Vector3, Point3, Matrix4};
+
+
+fn main() {
+    // Create a uniform scaling matrix with scaling factor 2.
+    let mut m =  Matrix4::new_scaling(2.0);
+
+    assert_eq!(m.transform_vector(&Vector3::x()), Vector3::x() * 2.0);
+    assert_eq!(m.transform_vector(&Vector3::y()), Vector3::y() * 2.0);
+    assert_eq!(m.transform_vector(&Vector3::z()), Vector3::z() * 2.0);
+
+    // Append a nonuniform scaling in-place.
+    m.append_nonuniform_scaling_mut(&Vector3::new(1.0, 2.0, 3.0));
+
+    assert_eq!(m.transform_vector(&Vector3::x()), Vector3::x() * 2.0);
+    assert_eq!(m.transform_vector(&Vector3::y()), Vector3::y() * 4.0);
+    assert_eq!(m.transform_vector(&Vector3::z()), Vector3::z() * 6.0);
+
+    // Append a translation out-of-place.
+    let m2 = m.append_translation(&Vector3::new(42.0, 0.0, 0.0));
+
+    assert_eq!(m2.transform_point(&Point3::new(1.0, 1.0, 1.0)), Point3::new(42.0 + 2.0, 4.0, 6.0));
+
+    // Create rotation.
+    let rot        = Matrix4::from_scaled_axis(&Vector3::x() * 3.14);
+    let rot_then_m = m * rot; // Right-multiplication is equivalent to prepending `rot` to `m`.
+    let m_then_rot = rot * m; // Left-multiplication is equivalent to appending `rot` to `m`.
+
+    let pt = Point3::new(1.0, 2.0, 3.0);
+
+    assert_relative_eq!(m.transform_point(&rot.transform_point(&pt)), rot_then_m.transform_point(&pt));
+    assert_relative_eq!(rot.transform_point(&m.transform_point(&pt)), m_then_rot.transform_point(&pt));
+}

examples/transform_vector_point.rs

@@ -0,0 +1,20 @@
+#[macro_use]
+extern crate approx;
+extern crate nalgebra as na;
+
+use std::f32;
+use na::{Vector2, Point2, Isometry2};
+
+fn main() {
+    let t = Isometry2::new(Vector2::new(1.0, 1.0), f32::consts::PI);
+    let p = Point2::new(1.0, 0.0); // Will be affected by te rotation and the translation.
+    let v = Vector2::x();          // Will *not* be affected by the translation.
+
+    assert_relative_eq!(t * p, Point2::new(-1.0 + 1.0, 1.0));
+    //                                     ^^^^ │ ^^^^^^^^
+    //                                  rotated │ translated
+
+    assert_relative_eq!(t * v, Vector2::new(-1.0, 0.0));
+    //                                      ^^^^^
+    //                                   rotated only
+}

examples/transform_vector_point3.rs

@@ -0,0 +1,27 @@
+extern crate alga;
+extern crate nalgebra as na;
+
+use alga::linear::Transformation;
+use na::{Vector3, Vector4, Point3, Matrix4};
+
+
+fn main() {
+    let mut m = Matrix4::new_rotation_wrt_point(Vector3::x() * 1.57, Point3::new(1.0, 2.0, 1.0));
+    m.append_scaling_mut(2.0);
+
+    let point1             = Point3::new(2.0, 3.0, 4.0);
+    let homogeneous_point2 = Vector4::new(2.0, 3.0, 4.0, 1.0);
+
+    // First option: use the dedicated `.transform_point(...)` method.
+    let transformed_point1 = m.transform_point(&point1);
+    // Second option: use the homogeneous coordinates of the point.
+    let transformed_homogeneous_point2 = m * homogeneous_point2;
+
+    // Recover the 3D point from its 4D homogeneous coordinates.
+    let transformed_point2 = Point3::from_homogeneous(transformed_homogeneous_point2);
+
+    // Check that transforming the 3D point with the `.transform_point` method is
+    // indeed equivalent to multiplying its 4D homogeneous coordinates by the 4x4
+    // matrix.
+    assert_eq!(transformed_point1, transformed_point2.unwrap());
+}

examples/transformation_pointer.rs

@@ -0,0 +1,25 @@
+extern crate nalgebra as na;
+
+use na::{Vector3, Isometry3};
+
+fn main() {
+    let iso = Isometry3::new(Vector3::new(1.0f32, 0.0, 1.0), na::zero());
+
+    // Compute the homogeneous coordinates first.
+    let iso_matrix  = iso.to_homogeneous();
+    let iso_array   = iso_matrix.as_slice();
+    let iso_pointer = iso_array.as_ptr();
+
+    /* Then pass the raw pointer to some graphics API. */
+
+    unsafe {
+        assert_eq!(*iso_pointer,            1.0);
+        assert_eq!(*iso_pointer.offset(5),  1.0);
+        assert_eq!(*iso_pointer.offset(10), 1.0);
+        assert_eq!(*iso_pointer.offset(15), 1.0);
+
+        assert_eq!(*iso_pointer.offset(12), 1.0);
+        assert_eq!(*iso_pointer.offset(13), 0.0);
+        assert_eq!(*iso_pointer.offset(14), 1.0);
+    }
+}

examples/unit_wrapper.rs

@@ -0,0 +1,30 @@
+extern crate nalgebra as na;
+
+use na::{Unit, Vector3};
+
+fn length_on_direction_with_unit(v: &Vector3<f32>, dir: &Unit<Vector3<f32>>) -> f32 {
+  // No need to normalize `dir`: we know that it is non-zero and normalized.
+  v.dot(dir.as_ref())
+}
+
+
+
+fn length_on_direction_without_unit(v: &Vector3<f32>, dir: &Vector3<f32>) -> f32 {
+  // Obligatory normalization of the direction vector (and test, for robustness).
+  if let Some(unit_dir) = dir.try_normalize(1.0e-6) {
+    v.dot(&unit_dir)
+  }
+  else {
+    // Normalization failed because the norm was too small.
+    panic!("Invalid input direction.")
+  }
+}
+
+fn main() {
+    let v = Vector3::new(1.0, 2.0, 3.0);
+
+    let l1 = length_on_direction_with_unit(&v, &Vector3::y_axis());
+    let l2 = length_on_direction_without_unit(&v, &Vector3::y());
+
+    assert_eq!(l1, l2)
+}

src/core/alias.rs

@@ -0,0 +1,158 @@
+use core::Matrix;
+use core::dimension::{Dynamic, U1, U2, U3, U4, U5, U6};
+use core::matrix_array::MatrixArray;
+use core::matrix_vec::MatrixVec;
+
+/*
+ *
+ *
+ * Column-major matrices.
+ *
+ *
+ */
+/// A dynamically sized column-major matrix.
+pub type DMatrix<N> = Matrix<N, Dynamic, Dynamic, MatrixVec<N, Dynamic, Dynamic>>;
+
+/// A staticaly sized column-major matrix with `R` rows and `C` columns.
+pub type MatrixNM<N, R, C> = Matrix<N, R, C, MatrixArray<N, R, C>>;
+
+/// A staticaly sized column-major square matrix with `D` rows and columns.
+pub type MatrixN<N, D> = MatrixNM<N, D, D>;
+
+/// A stack-allocated, column-major, 1x1 square matrix.
+pub type Matrix1<N> = MatrixN<N, U1>;
+/// A stack-allocated, column-major, 2x2 square matrix.
+pub type Matrix2<N> = MatrixN<N, U2>;
+/// A stack-allocated, column-major, 3x3 square matrix.
+pub type Matrix3<N> = MatrixN<N, U3>;
+/// A stack-allocated, column-major, 4x4 square matrix.
+pub type Matrix4<N> = MatrixN<N, U4>;
+/// A stack-allocated, column-major, 5x5 square matrix.
+pub type Matrix5<N> = MatrixN<N, U5>;
+/// A stack-allocated, column-major, 6x6 square matrix.
+pub type Matrix6<N> = MatrixN<N, U6>;
+
+/// A stack-allocated, column-major, 1x2 square matrix.
+pub type Matrix1x2<N> = MatrixNM<N, U1, U2>;
+/// A stack-allocated, column-major, 1x3 square matrix.
+pub type Matrix1x3<N> = MatrixNM<N, U1, U3>;
+/// A stack-allocated, column-major, 1x4 square matrix.
+pub type Matrix1x4<N> = MatrixNM<N, U1, U4>;
+/// A stack-allocated, column-major, 1x5 square matrix.
+pub type Matrix1x5<N> = MatrixNM<N, U1, U5>;
+/// A stack-allocated, column-major, 1x6 square matrix.
+pub type Matrix1x6<N> = MatrixNM<N, U1, U6>;
+
+/// A stack-allocated, column-major, 2x3 square matrix.
+pub type Matrix2x3<N> = MatrixNM<N, U2, U3>;
+/// A stack-allocated, column-major, 2x4 square matrix.
+pub type Matrix2x4<N> = MatrixNM<N, U2, U4>;
+/// A stack-allocated, column-major, 2x5 square matrix.
+pub type Matrix2x5<N> = MatrixNM<N, U2, U5>;
+/// A stack-allocated, column-major, 2x6 square matrix.
+pub type Matrix2x6<N> = MatrixNM<N, U2, U6>;
+
+/// A stack-allocated, column-major, 3x4 square matrix.
+pub type Matrix3x4<N> = MatrixNM<N, U3, U4>;
+/// A stack-allocated, column-major, 3x5 square matrix.
+pub type Matrix3x5<N> = MatrixNM<N, U3, U5>;
+/// A stack-allocated, column-major, 3x6 square matrix.
+pub type Matrix3x6<N> = MatrixNM<N, U3, U6>;
+
+/// A stack-allocated, column-major, 4x5 square matrix.
+pub type Matrix4x5<N> = MatrixNM<N, U4, U5>;
+/// A stack-allocated, column-major, 4x6 square matrix.
+pub type Matrix4x6<N> = MatrixNM<N, U4, U6>;
+
+/// A stack-allocated, column-major, 5x6 square matrix.
+pub type Matrix5x6<N> = MatrixNM<N, U5, U6>;
+
+
+/// A stack-allocated, column-major, 2x1 square matrix.
+pub type Matrix2x1<N> = MatrixNM<N, U2, U1>;
+/// A stack-allocated, column-major, 3x1 square matrix.
+pub type Matrix3x1<N> = MatrixNM<N, U3, U1>;
+/// A stack-allocated, column-major, 4x1 square matrix.
+pub type Matrix4x1<N> = MatrixNM<N, U4, U1>;
+/// A stack-allocated, column-major, 5x1 square matrix.
+pub type Matrix5x1<N> = MatrixNM<N, U5, U1>;
+/// A stack-allocated, column-major, 6x1 square matrix.
+pub type Matrix6x1<N> = MatrixNM<N, U6, U1>;
+
+/// A stack-allocated, column-major, 3x2 square matrix.
+pub type Matrix3x2<N> = MatrixNM<N, U3, U2>;
+/// A stack-allocated, column-major, 4x2 square matrix.
+pub type Matrix4x2<N> = MatrixNM<N, U4, U2>;
+/// A stack-allocated, column-major, 5x2 square matrix.
+pub type Matrix5x2<N> = MatrixNM<N, U5, U2>;
+/// A stack-allocated, column-major, 6x2 square matrix.
+pub type Matrix6x2<N> = MatrixNM<N, U6, U2>;
+
+/// A stack-allocated, column-major, 4x3 square matrix.
+pub type Matrix4x3<N> = MatrixNM<N, U4, U3>;
+/// A stack-allocated, column-major, 5x3 square matrix.
+pub type Matrix5x3<N> = MatrixNM<N, U5, U3>;
+/// A stack-allocated, column-major, 6x3 square matrix.
+pub type Matrix6x3<N> = MatrixNM<N, U6, U3>;
+
+/// A stack-allocated, column-major, 5x4 square matrix.
+pub type Matrix5x4<N> = MatrixNM<N, U5, U4>;
+/// A stack-allocated, column-major, 6x4 square matrix.
+pub type Matrix6x4<N> = MatrixNM<N, U6, U4>;
+
+/// A stack-allocated, column-major, 6x5 square matrix.
+pub type Matrix6x5<N> = MatrixNM<N, U6, U5>;
+
+
+/*
+ *
+ *
+ * Column vectors.
+ *
+ *
+ */
+/// A dynamically sized column vector.
+pub type DVector<N> = Matrix<N, Dynamic, U1, MatrixVec<N, Dynamic, U1>>;
+
+/// A statically sized D-dimensional column vector.
+pub type VectorN<N, D> = MatrixNM<N, D, U1>;
+
+/// A stack-allocated, 1-dimensional column vector.
+pub type Vector1<N> = VectorN<N, U1>;
+/// A stack-allocated, 2-dimensional column vector.
+pub type Vector2<N> = VectorN<N, U2>;
+/// A stack-allocated, 3-dimensional column vector.
+pub type Vector3<N> = VectorN<N, U3>;
+/// A stack-allocated, 4-dimensional column vector.
+pub type Vector4<N> = VectorN<N, U4>;
+/// A stack-allocated, 5-dimensional column vector.
+pub type Vector5<N> = VectorN<N, U5>;
+/// A stack-allocated, 6-dimensional column vector.
+pub type Vector6<N> = VectorN<N, U6>;
+
+
+/*
+ *
+ *
+ * Row vectors.
+ *
+ *
+ */
+/// A dynamically sized row vector.
+pub type RowDVector<N> = Matrix<N, U1, Dynamic, MatrixVec<N, U1, Dynamic>>;
+
+/// A statically sized D-dimensional row vector.
+pub type RowVectorN<N, D> = MatrixNM<N, U1, D>;
+
+/// A stack-allocated, 1-dimensional row vector.
+pub type RowVector1<N> = RowVectorN<N, U1>;
+/// A stack-allocated, 2-dimensional row vector.
+pub type RowVector2<N> = RowVectorN<N, U2>;
+/// A stack-allocated, 3-dimensional row vector.
+pub type RowVector3<N> = RowVectorN<N, U3>;
+/// A stack-allocated, 4-dimensional row vector.
+pub type RowVector4<N> = RowVectorN<N, U4>;
+/// A stack-allocated, 5-dimensional row vector.
+pub type RowVector5<N> = RowVectorN<N, U5>;
+/// A stack-allocated, 6-dimensional row vector.
+pub type RowVector6<N> = RowVectorN<N, U6>;

src/core/allocator.rs

@@ -0,0 +1,89 @@
+//! Abstract definition of a matrix data storage allocator.
+
+use std::any::Any;
+
+use core::Scalar;
+use core::constraint::{SameNumberOfRows, SameNumberOfColumns, ShapeConstraint};
+use core::dimension::{Dim, U1};
+use core::storage::{Storage, OwnedStorage};
+
+/// A matrix allocator of a memory buffer that may contain `R::to_usize() * C::to_usize()`
+/// elements of type `N`.
+///
+/// An allocator is said to be:
+///   − static:  if `R` and `C` both implement `DimName`.
+///   − dynamic: if either one (or both) of `R` or `C` is equal to `Dynamic`.
+///
+/// Every allocator must be both static and dynamic. Though not all implementations may share the
+/// same `Buffer` type.
+pub trait Allocator<N: Scalar, R: Dim, C: Dim>: Any + Sized {
+    /// The type of buffer this allocator can instanciate.
+    type Buffer: OwnedStorage<N, R, C, Alloc = Self>;
+
+    /// Allocates a buffer with the given number of rows and columns without initializing its content.
+    unsafe fn allocate_uninitialized(nrows: R, ncols: C) -> Self::Buffer;
+
+    /// Allocates a buffer initialized with the content of the given iterator.
+    fn allocate_from_iterator<I: IntoIterator<Item = N>>(nrows: R, ncols: C, iter: I) -> Self::Buffer;
+}
+
+/// A matrix data allocator dedicated to the given owned matrix storage.
+pub trait OwnedAllocator<N: Scalar, R: Dim, C: Dim, S: OwnedStorage<N, R, C, Alloc = Self>>:
+    Allocator<N, R, C, Buffer = S> {
+}
+
+impl<N, R, C, T, S> OwnedAllocator<N, R, C, S> for T
+    where N: Scalar, R: Dim, C: Dim,
+          T: Allocator<N, R, C, Buffer = S>,
+          S: OwnedStorage<N, R, C, Alloc = T> {
+}
+
+/// The number of rows of the result of a componentwise operation on two matrices.
+pub type SameShapeR<R1, R2> = <ShapeConstraint as SameNumberOfRows<R1, R2>>::Representative;
+
+/// The number of columns of the result of a componentwise operation on two matrices.
+pub type SameShapeC<C1, C2> = <ShapeConstraint as SameNumberOfColumns<C1, C2>>::Representative;
+
+// FIXME: Bad name.
+/// Restricts the given number of rows and columns to be respectively the same. Can only be used
+/// when `Self = SA::Alloc`.
+pub trait SameShapeAllocator<N, R1, C1, R2, C2, SA>:
+        Allocator<N, R1, C1> +
+        Allocator<N, SameShapeR<R1, R2>, SameShapeC<C1, C2>>
+    where R1: Dim, R2: Dim, C1: Dim, C2: Dim,
+          N: Scalar,
+          SA: Storage<N, R1, C1, Alloc = Self>,
+          ShapeConstraint: SameNumberOfRows<R1, R2> + SameNumberOfColumns<C1, C2> {
+}
+
+impl<N, R1, R2, C1, C2, SA> SameShapeAllocator<N, R1, C1, R2, C2, SA> for SA::Alloc
+    where R1: Dim, R2: Dim, C1: Dim, C2: Dim,
+          N: Scalar,
+          SA: Storage<N, R1, C1>,
+          SA::Alloc:
+            Allocator<N, R1, C1> +
+            Allocator<N, SameShapeR<R1, R2>, SameShapeC<C1, C2>>,
+          ShapeConstraint: SameNumberOfRows<R1, R2> + SameNumberOfColumns<C1, C2> {
+}
+
+// XXX: Bad name.
+/// Restricts the given number of rows to be equal. Can only be used when `Self = SA::Alloc`.
+pub trait SameShapeColumnVectorAllocator<N, R1, R2, SA>:
+        Allocator<N, R1, U1> +
+        Allocator<N, SameShapeR<R1, R2>, U1> +
+        SameShapeAllocator<N, R1, U1, R2, U1, SA>
+    where R1: Dim, R2: Dim,
+          N: Scalar,
+          SA: Storage<N, R1, U1, Alloc = Self>,
+          ShapeConstraint: SameNumberOfRows<R1, R2> {
+}
+
+impl<N, R1, R2, SA> SameShapeColumnVectorAllocator<N, R1, R2, SA> for SA::Alloc
+    where R1: Dim, R2: Dim,
+          N: Scalar,
+          SA: Storage<N, R1, U1>,
+          SA::Alloc:
+            Allocator<N, R1, U1> +
+            Allocator<N, SameShapeR<R1, R2>, U1>,
+          ShapeConstraint: SameNumberOfRows<R1, R2> {
+}

src/core/cg.rs

@@ -0,0 +1,376 @@
+/*
+ *
+ * Computer-graphics specific implementations.
+ * Currently, it is mostly implemented for homogeneous matrices in 2- and 3-space.
+ *
+ */
+
+use num::One;
+
+use core::{Scalar, SquareMatrix, OwnedSquareMatrix, ColumnVector, Unit};
+use core::dimension::{DimName, DimNameSub, DimNameDiff, U1, U2, U3, U4};
+use core::storage::{Storage, StorageMut, OwnedStorage};
+use core::allocator::{Allocator, OwnedAllocator};
+use geometry::{PointBase, OrthographicBase, PerspectiveBase, IsometryBase, OwnedRotation, OwnedPoint};
+
+use alga::general::{Real, Field};
+use alga::linear::Transformation;
+
+
+impl<N, D: DimName, S> SquareMatrix<N, D, S>
+    where N: Scalar + Field,
+          S: OwnedStorage<N, D, D>,
+          S::Alloc: OwnedAllocator<N, D, D, S> {
+    /// Creates a new homogeneous matrix that applies the same scaling factor on each dimension.
+    #[inline]
+    pub fn new_scaling(scaling: N) -> Self {
+        let mut res = Self::from_diagonal_element(scaling);
+        res[(D::dim() - 1, D::dim() - 1)] = N::one();
+
+        res
+    }
+
+    /// Creates a new homogeneous matrix that applies a distinct scaling factor for each dimension.
+    #[inline]
+    pub fn new_nonuniform_scaling<SB>(scaling: &ColumnVector<N, DimNameDiff<D, U1>, SB>) -> Self
+        where D:  DimNameSub<U1>,
+              SB: Storage<N, DimNameDiff<D, U1>, U1> {
+        let mut res = Self::one();
+        for i in 0 .. scaling.len() {
+            res[(i, i)] = scaling[i];
+        }
+
+        res
+    }
+
+    /// Creates a new homogeneous matrix that applies a pure translation.
+    #[inline]
+    pub fn new_translation<SB>(translation: &ColumnVector<N, DimNameDiff<D, U1>, SB>) -> Self
+        where D:  DimNameSub<U1>,
+              SB: Storage<N, DimNameDiff<D, U1>, U1>,
+              S::Alloc: Allocator<N, DimNameDiff<D, U1>, U1> {
+        let mut res = Self::one();
+        res.fixed_slice_mut::<DimNameDiff<D, U1>, U1>(0, D::dim() - 1).copy_from(translation);
+
+        res
+    }
+}
+
+impl<N, S> SquareMatrix<N, U3, S>
+    where N: Real,
+          S: OwnedStorage<N, U3, U3>,
+          S::Alloc: OwnedAllocator<N, U3, U3, S> {
+    /// Builds a 2 dimensional homogeneous rotation matrix from an angle in radian.
+    #[inline]
+    pub fn new_rotation(angle: N) -> Self
+        where S::Alloc: Allocator<N, U2, U2> {
+        OwnedRotation::<N, U2, S::Alloc>::new(angle).to_homogeneous()
+    }
+}
+
+impl<N, S> SquareMatrix<N, U4, S>
+    where N: Real,
+          S: OwnedStorage<N, U4, U4>,
+          S::Alloc: OwnedAllocator<N, U4, U4, S> {
+
+    /// Builds a 3D homogeneous rotation matrix from an axis and an angle (multiplied together).
+    /// 
+    /// Returns the identity matrix if the given argument is zero.
+    #[inline]
+    pub fn new_rotation<SB>(axisangle: ColumnVector<N, U3, SB>) -> Self
+        where SB: Storage<N, U3, U1>,
+              S::Alloc: Allocator<N, U3, U3> {
+        OwnedRotation::<N, U3, S::Alloc>::new(axisangle).to_homogeneous()
+    }
+
+    /// Builds a 3D homogeneous rotation matrix from an axis and an angle (multiplied together).
+    /// 
+    /// Returns the identity matrix if the given argument is zero.
+    #[inline]
+    pub fn new_rotation_wrt_point<SB>(axisangle: ColumnVector<N, U3, SB>, pt: OwnedPoint<N, U3, S::Alloc>) -> Self
+        where SB: Storage<N, U3, U1>,
+              S::Alloc: Allocator<N, U3, U3> +
+                        Allocator<N, U3, U1> +
+                        Allocator<N, U1, U3> {
+        let rot = OwnedRotation::<N, U3, S::Alloc>::from_scaled_axis(axisangle);
+        IsometryBase::rotation_wrt_point(rot, pt).to_homogeneous()
+    }
+
+    /// Builds a 3D homogeneous rotation matrix from an axis and an angle (multiplied together).
+    /// 
+    /// Returns the identity matrix if the given argument is zero.
+    /// This is identical to `Self::new_rotation`.
+    #[inline]
+    pub fn from_scaled_axis<SB>(axisangle: ColumnVector<N, U3, SB>) -> Self
+        where SB: Storage<N, U3, U1>,
+              S::Alloc: Allocator<N, U3, U3> {
+        OwnedRotation::<N, U3, S::Alloc>::from_scaled_axis(axisangle).to_homogeneous()
+    }
+
+    /// Creates a new rotation from Euler angles.
+    ///
+    /// The primitive rotations are applied in order: 1 roll − 2 pitch − 3 yaw.
+    pub fn from_euler_angles(roll: N, pitch: N, yaw: N) -> Self
+        where S::Alloc: Allocator<N, U3, U3> {
+        OwnedRotation::<N, U3, S::Alloc>::from_euler_angles(roll, pitch, yaw).to_homogeneous()
+    }
+
+    /// Builds a 3D homogeneous rotation matrix from an axis and a rotation angle.
+    pub fn from_axis_angle<SB>(axis: &Unit<ColumnVector<N, U3, SB>>, angle: N) -> Self
+        where SB: Storage<N, U3, U1>,
+              S::Alloc: Allocator<N, U3, U3> {
+        OwnedRotation::<N, U3, S::Alloc>::from_axis_angle(axis, angle).to_homogeneous()
+    }
+
+    /// Creates a new homogeneous matrix for an orthographic projection.
+    #[inline]
+    pub fn new_orthographic(left: N, right: N, bottom: N, top: N, znear: N, zfar: N) -> Self {
+        OrthographicBase::new(left, right, bottom, top, znear, zfar).unwrap()
+    }
+
+    /// Creates a new homogeneous matrix for a perspective projection.
+    #[inline]
+    pub fn new_perspective(aspect: N, fovy: N, znear: N, zfar: N) -> Self {
+        PerspectiveBase::new(aspect, fovy, znear, zfar).unwrap()
+    }
+
+    /// Creates an isometry that corresponds to the local frame of an observer standing at the
+    /// point `eye` and looking toward `target`.
+    ///
+    /// It maps the view direction `target - eye` to the positive `z` axis and the origin to the
+    /// `eye`.
+    #[inline]
+    pub fn new_observer_frame<SB>(eye:    &PointBase<N, U3, SB>,
+                                  target: &PointBase<N, U3, SB>,
+                                  up:     &ColumnVector<N, U3, SB>)
+                                  -> Self
+        where SB: OwnedStorage<N, U3, U1, Alloc = S::Alloc>,
+              SB::Alloc: OwnedAllocator<N, U3, U1, SB> +
+                         Allocator<N, U1, U3> +
+                         Allocator<N, U3, U3> {
+        IsometryBase::<N, U3, SB, OwnedRotation<N, U3, SB::Alloc>>
+                    ::new_observer_frame(eye, target, up).to_homogeneous()
+    }
+
+    /// Builds a right-handed look-at view matrix.
+    #[inline]
+    pub fn look_at_rh<SB>(eye:    &PointBase<N, U3, SB>,
+                          target: &PointBase<N, U3, SB>,
+                          up:     &ColumnVector<N, U3, SB>)
+                          -> Self
+        where SB: OwnedStorage<N, U3, U1, Alloc = S::Alloc>,
+              SB::Alloc: OwnedAllocator<N, U3, U1, SB> +
+                         Allocator<N, U1, U3> +
+                         Allocator<N, U3, U3> {
+        IsometryBase::<N, U3, SB, OwnedRotation<N, U3, SB::Alloc>>
+                    ::look_at_rh(eye, target, up).to_homogeneous()
+    }
+
+    /// Builds a left-handed look-at view matrix.
+    #[inline]
+    pub fn look_at_lh<SB>(eye:    &PointBase<N, U3, SB>,
+                          target: &PointBase<N, U3, SB>,
+                          up:     &ColumnVector<N, U3, SB>)
+                          -> Self
+        where SB: OwnedStorage<N, U3, U1, Alloc = S::Alloc>,
+              SB::Alloc: OwnedAllocator<N, U3, U1, SB> +
+                         Allocator<N, U1, U3> +
+                         Allocator<N, U3, U3> {
+        IsometryBase::<N, U3, SB, OwnedRotation<N, U3, SB::Alloc>>
+                    ::look_at_lh(eye, target, up).to_homogeneous()
+    }
+}
+
+
+impl<N, D: DimName, S> SquareMatrix<N, D, S>
+    where N: Scalar + Field,
+          S: Storage<N, D, D> {
+
+    /// Computes the transformation equal to `self` followed by an uniform scaling factor.
+    #[inline]
+    pub fn append_scaling(&self, scaling: N) -> OwnedSquareMatrix<N, D, S::Alloc>
+        where D: DimNameSub<U1>,
+              S::Alloc: Allocator<N, DimNameDiff<D, U1>, D> {
+        let mut res = self.clone_owned();
+        res.append_scaling_mut(scaling);
+        res
+    }
+
+    /// Computes the transformation equal to an uniform scaling factor followed by `self`.
+    #[inline]
+    pub fn prepend_scaling(&self, scaling: N) -> OwnedSquareMatrix<N, D, S::Alloc>
+        where D: DimNameSub<U1>,
+              S::Alloc: Allocator<N, D, DimNameDiff<D, U1>> {
+        let mut res = self.clone_owned();
+        res.prepend_scaling_mut(scaling);
+        res
+    }
+
+    /// Computes the transformation equal to `self` followed by a non-uniform scaling factor.
+    #[inline]
+    pub fn append_nonuniform_scaling<SB>(&self, scaling: &ColumnVector<N, DimNameDiff<D, U1>, SB>)
+        -> OwnedSquareMatrix<N, D, S::Alloc>
+        where D:  DimNameSub<U1>,
+              SB: Storage<N, DimNameDiff<D, U1>, U1>,
+              S::Alloc: Allocator<N, U1, D> {
+        let mut res = self.clone_owned();
+        res.append_nonuniform_scaling_mut(scaling);
+        res
+    }
+
+    /// Computes the transformation equal to a non-uniform scaling factor followed by `self`.
+    #[inline]
+    pub fn prepend_nonuniform_scaling<SB>(&self, scaling: &ColumnVector<N, DimNameDiff<D, U1>, SB>)
+        -> OwnedSquareMatrix<N, D, S::Alloc>
+        where D:  DimNameSub<U1>,
+              SB: Storage<N, DimNameDiff<D, U1>, U1>,
+              S::Alloc: Allocator<N, D, U1> {
+        let mut res = self.clone_owned();
+        res.prepend_nonuniform_scaling_mut(scaling);
+        res
+    }
+
+    /// Computes the transformation equal to `self` followed by a translation.
+    #[inline]
+    pub fn append_translation<SB>(&self, shift: &ColumnVector<N, DimNameDiff<D, U1>, SB>)
+        -> OwnedSquareMatrix<N, D, S::Alloc>
+        where D: DimNameSub<U1>,
+              SB: Storage<N, DimNameDiff<D, U1>, U1>,
+              S::Alloc: Allocator<N, DimNameDiff<D, U1>, U1> {
+        let mut res = self.clone_owned();
+        res.append_translation_mut(shift);
+        res
+    }
+
+    /// Computes the transformation equal to a translation followed by `self`.
+    #[inline]
+    pub fn prepend_translation<SB>(&self, shift: &ColumnVector<N, DimNameDiff<D, U1>, SB>)
+        -> OwnedSquareMatrix<N, D, S::Alloc>
+        where D: DimNameSub<U1>,
+              SB: Storage<N, DimNameDiff<D, U1>, U1>,
+              S::Alloc: Allocator<N, DimNameDiff<D, U1>, U1> +
+                        Allocator<N, DimNameDiff<D, U1>, DimNameDiff<D, U1>> +
+                        Allocator<N, U1, DimNameDiff<D, U1>> {
+        let mut res = self.clone_owned();
+        res.prepend_translation_mut(shift);
+        res
+    }
+}
+
+impl<N, D: DimName, S> SquareMatrix<N, D, S>
+    where N: Scalar + Field,
+          S: StorageMut<N, D, D> {
+
+    /// Computes in-place the transformation equal to `self` followed by an uniform scaling factor.
+    #[inline]
+    pub fn append_scaling_mut(&mut self, scaling: N)
+        where D: DimNameSub<U1>,
+              S::Alloc: Allocator<N, DimNameDiff<D, U1>, D> {
+        let mut to_scale = self.fixed_rows_mut::<DimNameDiff<D, U1>>(0);
+        to_scale *= scaling;
+    }
+
+    /// Computes in-place the transformation equal to an uniform scaling factor followed by `self`.
+    #[inline]
+    pub fn prepend_scaling_mut(&mut self, scaling: N)
+        where D: DimNameSub<U1>,
+              S::Alloc: Allocator<N, D, DimNameDiff<D, U1>> {
+        let mut to_scale = self.fixed_columns_mut::<DimNameDiff<D, U1>>(0);
+        to_scale *= scaling;
+    }
+
+    /// Computes in-place the transformation equal to `self` followed by a non-uniform scaling factor.
+    #[inline]
+    pub fn append_nonuniform_scaling_mut<SB>(&mut self, scaling: &ColumnVector<N, DimNameDiff<D, U1>, SB>)
+        where D:  DimNameSub<U1>,
+              SB: Storage<N, DimNameDiff<D, U1>, U1>,
+              S::Alloc: Allocator<N, U1, D> {
+        for i in 0 .. scaling.len() {
+            let mut to_scale = self.fixed_rows_mut::<U1>(i);
+            to_scale *= scaling[i];
+        }
+    }
+
+    /// Computes in-place the transformation equal to a non-uniform scaling factor followed by `self`.
+    #[inline]
+    pub fn prepend_nonuniform_scaling_mut<SB>(&mut self, scaling: &ColumnVector<N, DimNameDiff<D, U1>, SB>)
+        where D:  DimNameSub<U1>,
+              SB: Storage<N, DimNameDiff<D, U1>, U1>,
+              S::Alloc: Allocator<N, D, U1> {
+        for i in 0 .. scaling.len() {
+            let mut to_scale = self.fixed_columns_mut::<U1>(i);
+            to_scale *= scaling[i];
+        }
+    }
+
+    /// Computes the transformation equal to `self` followed by a translation.
+    #[inline]
+    pub fn append_translation_mut<SB>(&mut self, shift: &ColumnVector<N, DimNameDiff<D, U1>, SB>)
+        where D: DimNameSub<U1>,
+              SB: Storage<N, DimNameDiff<D, U1>, U1>,
+              S::Alloc: Allocator<N, DimNameDiff<D, U1>, U1> {
+        for i in 0 .. D::dim() {
+            for j in 0 .. D::dim() - 1 {
+                self[(j, i)] += shift[j] * self[(D::dim() - 1, i)];
+            }
+        }
+    }
+
+    /// Computes the transformation equal to a translation followed by `self`.
+    #[inline]
+    pub fn prepend_translation_mut<SB>(&mut self, shift: &ColumnVector<N, DimNameDiff<D, U1>, SB>)
+        where D: DimNameSub<U1>,
+              SB: Storage<N, DimNameDiff<D, U1>, U1>,
+              S::Alloc: Allocator<N, DimNameDiff<D, U1>, U1> +
+                        Allocator<N, DimNameDiff<D, U1>, DimNameDiff<D, U1>> +
+                        Allocator<N, U1, DimNameDiff<D, U1>> {
+        let scale = self.fixed_slice::<U1, DimNameDiff<D, U1>>(D::dim() - 1, 0).tr_dot(&shift);
+        let post_translation = self.fixed_slice::<DimNameDiff<D, U1>, DimNameDiff<D, U1>>(0, 0) * shift;
+
+        self[(D::dim() - 1, D::dim() - 1)] += scale;
+
+        let mut translation = self.fixed_slice_mut::<DimNameDiff<D, U1>, U1>(0, D::dim() - 1);
+        translation += post_translation;
+    }
+}
+
+
+impl<N, D, SA, SB> Transformation<PointBase<N, DimNameDiff<D, U1>, SB>> for SquareMatrix<N, D, SA>
+    where N:  Real,
+          D:  DimNameSub<U1>,
+          SA: OwnedStorage<N, D, D>,
+          SB: OwnedStorage<N, DimNameDiff<D, U1>, U1, Alloc = SA::Alloc>,
+          SA::Alloc: OwnedAllocator<N, D, D, SA> +
+                     Allocator<N, DimNameDiff<D, U1>, DimNameDiff<D, U1>> +
+                     Allocator<N, DimNameDiff<D, U1>, U1> +
+                     Allocator<N, U1, DimNameDiff<D, U1>>,
+          SB::Alloc: OwnedAllocator<N, DimNameDiff<D, U1>, U1, SB> {
+    #[inline]
+    fn transform_vector(&self, v: &ColumnVector<N, DimNameDiff<D, U1>, SB>)
+        -> ColumnVector<N, DimNameDiff<D, U1>, SB> {
+        let transform  = self.fixed_slice::<DimNameDiff<D, U1>, DimNameDiff<D, U1>>(0, 0);
+        let normalizer = self.fixed_slice::<U1, DimNameDiff<D, U1>>(D::dim() - 1, 0);
+        let n = normalizer.tr_dot(&v);
+
+        if !n.is_zero() {
+            return transform * (v / n);
+        }
+
+        transform * v
+    }
+
+    #[inline]
+    fn transform_point(&self, pt: &PointBase<N, DimNameDiff<D, U1>, SB>)
+        -> PointBase<N, DimNameDiff<D, U1>, SB> {
+        let transform   = self.fixed_slice::<DimNameDiff<D, U1>, DimNameDiff<D, U1>>(0, 0);
+        let translation = self.fixed_slice::<DimNameDiff<D, U1>, U1>(0, D::dim() - 1);
+        let normalizer  = self.fixed_slice::<U1, DimNameDiff<D, U1>>(D::dim() - 1, 0);
+        let n = normalizer.tr_dot(&pt.coords) + unsafe { *self.get_unchecked(D::dim() - 1, D::dim() - 1) };
+
+        if !n.is_zero() {
+            return transform * (pt / n) + translation;
+        }
+
+        transform * pt + translation
+    }
+}

src/core/componentwise.rs

@@ -0,0 +1,77 @@
+// Non-convensional componentwise operators.
+
+use num::Signed;
+
+use alga::general::{ClosedMul, ClosedDiv};
+
+use core::{Scalar, Matrix, OwnedMatrix, MatrixSum};
+use core::dimension::Dim;
+use core::storage::{Storage, StorageMut};
+use core::allocator::SameShapeAllocator;
+use core::constraint::{ShapeConstraint, SameNumberOfRows, SameNumberOfColumns};
+
+
+/// The type of the result of a matrix componentwise operation.
+pub type MatrixComponentOp<N, R1, C1, R2, C2, SA> = MatrixSum<N, R1, C1, R2, C2, SA>;
+
+impl<N: Scalar, R: Dim, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S> {
+    /// Computes the componentwise absolute value.
+    #[inline]
+    pub fn abs(&self) -> OwnedMatrix<N, R, C, S::Alloc>
+        where N: Signed {
+        let mut res = self.clone_owned();
+
+        for e in res.iter_mut() {
+            *e = e.abs();
+        }
+
+        res
+    }
+
+    // FIXME: add other operators like component_ln, component_pow, etc. ?
+}
+
+macro_rules! component_binop_impl(
+    ($($binop: ident, $binop_mut: ident, $Trait: ident . $binop_assign: ident);* $(;)*) => {$(
+        impl<N: Scalar, R: Dim, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S> {
+            /// Componentwise matrix multiplication.
+            #[inline]
+            pub fn $binop<R2, C2, SB>(&self, rhs: &Matrix<N, R2, C2, SB>) -> MatrixComponentOp<N, R, C, R2, C2, S>
+                where N: $Trait,
+                      R2: Dim, C2: Dim,
+                      SB: Storage<N, R2, C2>,
+                      S::Alloc: SameShapeAllocator<N, R, C, R2, C2, S>,
+                      ShapeConstraint: SameNumberOfRows<R, R2> + SameNumberOfColumns<C, C2> {
+        
+                let mut res = self.clone_owned_sum();
+        
+                for (res, rhs) in res.iter_mut().zip(rhs.iter()) {
+                    res.$binop_assign(*rhs);
+                }
+        
+                res
+            }
+        }
+        
+        impl<N: Scalar, R: Dim, C: Dim, S: StorageMut<N, R, C>> Matrix<N, R, C, S> {
+            /// Componentwise matrix multiplication.
+            #[inline]
+            pub fn $binop_mut<R2, C2, SB>(&mut self, rhs: &Matrix<N, R2, C2, SB>)
+                where N: $Trait,
+                      R2: Dim,
+                      C2: Dim,
+                      SB: Storage<N, R2, C2>,
+                      ShapeConstraint: SameNumberOfRows<R, R2> + SameNumberOfColumns<C, C2> {
+                for (me, rhs) in self.iter_mut().zip(rhs.iter()) {
+                    me.$binop_assign(*rhs);
+                }
+            }
+        }
+    )*}
+);
+
+component_binop_impl!(
+    component_mul, component_mul_mut, ClosedMul.mul_assign;
+    component_div, component_div_mut, ClosedDiv.div_assign;
+    // FIXME: add other operators like bitshift, etc. ?
+);

src/core/constraint.rs

@@ -0,0 +1,75 @@
+//! Compatibility constraints between matrix shapes, e.g., for addition or multiplication.
+
+use core::dimension::{Dim, DimName, Dynamic};
+
+/// A type used in `where` clauses for enforcing constraints.
+pub struct ShapeConstraint;
+
+/// Constraints `C1` and `R2` to be equivalent.
+pub trait AreMultipliable<R1: Dim, C1: Dim,
+                          R2: Dim, C2: Dim> {
+}
+
+
+impl<R1: Dim, C1: Dim, R2: Dim, C2: Dim> AreMultipliable<R1, C1, R2, C2> for ShapeConstraint
+where ShapeConstraint: DimEq<C1, R2> {
+}
+
+macro_rules! equality_trait_decl(
+    ($($doc: expr, $Trait: ident),* $(,)*) => {$(
+        // XXX: we can't do something like `DimEq<D1> for D2` because we would require a blancket impl…
+        #[doc = $doc]
+        pub trait $Trait<D1: Dim, D2: Dim> {
+            /// This is either equal to `D1` or `D2`, always choosing the one (if any) which is a type-level
+            /// constant.
+            type Representative: Dim;
+        }
+
+        impl<D: Dim> $Trait<D, D> for ShapeConstraint {
+            type Representative = D;
+        }
+
+        impl<D: DimName> $Trait<D, Dynamic> for ShapeConstraint {
+            type Representative = D;
+        }
+
+        impl<D: DimName> $Trait<Dynamic, D> for ShapeConstraint {
+            type Representative = D;
+        }
+    )*}
+);
+
+equality_trait_decl!(
+    "Constraints `D1` and `D2` to be equivalent.",
+    DimEq,
+
+    "Constraints `D1` and `D2` to be equivalent. \
+     They are both assumed to be the number of \
+     rows of a matrix.",
+    SameNumberOfRows,
+
+    "Constraints `D1` and `D2` to be equivalent. \
+     They are both assumed to be the number of \
+     columns of a matrix.",
+    SameNumberOfColumns
+);
+
+/// Constraints D1 and D2 to be equivalent, where they both designate dimensions of algebraic
+/// entities (e.g. square matrices).
+pub trait SameDimension<D1: Dim, D2: Dim>: SameNumberOfRows<D1, D2> + SameNumberOfColumns<D1, D2> {
+    /// This is either equal to `D1` or `D2`, always choosing the one (if any) which is a type-level
+    /// constant.
+    type Representative: Dim;
+}
+
+impl<D: Dim> SameDimension<D, D> for ShapeConstraint {
+    type Representative = D;
+}
+
+impl<D: DimName> SameDimension<D, Dynamic> for ShapeConstraint {
+    type Representative = D;
+}
+
+impl<D: DimName> SameDimension<Dynamic, D> for ShapeConstraint {
+    type Representative = D;
+}

src/core/construction.rs

@@ -0,0 +1,657 @@
+#[cfg(feature = "arbitrary")]
+use quickcheck::{Arbitrary, Gen};
+
+use std::iter;
+use num::{Zero, One, Bounded};
+use rand::{self, Rand, Rng};
+use typenum::{self, Cmp, Greater};
+
+use alga::general::{ClosedAdd, ClosedMul};
+
+use core::{Scalar, Matrix, SquareMatrix, ColumnVector, Unit};
+use core::dimension::{Dim, DimName, Dynamic, U1, U2, U3, U4, U5, U6};
+use core::allocator::{Allocator, OwnedAllocator};
+use core::storage::{Storage, OwnedStorage};
+
+/*
+ *
+ * Generic constructors.
+ *
+ */
+impl<N: Scalar, R: Dim, C: Dim, S: OwnedStorage<N, R, C>> Matrix<N, R, C, S>
+    // XXX: needed because of a compiler bug. See the rust compiler issue #26026.
+    where S::Alloc: OwnedAllocator<N, R, C, S> {
+    /// Creates a new uninitialized matrix. If the matrix has a compile-time dimension, this panics
+    /// if `nrows != R::to_usize()` or `ncols != C::to_usize()`.
+    #[inline]
+    pub unsafe fn new_uninitialized_generic(nrows: R, ncols: C) -> Matrix<N, R, C, S> {
+        Matrix::from_data(S::Alloc::allocate_uninitialized(nrows, ncols))
+    }
+
+    /// Creates a matrix with all its elements set to `elem`.
+    #[inline]
+    pub fn from_element_generic(nrows: R, ncols: C, elem: N) -> Matrix<N, R, C, S> {
+        let len = nrows.value() * ncols.value();
+        Matrix::from_iterator_generic(nrows, ncols, iter::repeat(elem).take(len))
+    }
+
+    /// Creates a matrix with all its elements filled by an iterator.
+    #[inline]
+    pub fn from_iterator_generic<I>(nrows: R, ncols: C, iter: I) -> Matrix<N, R, C, S>
+        where I: IntoIterator<Item = N> {
+        Matrix::from_data(S::Alloc::allocate_from_iterator(nrows, ncols, iter))
+    }
+
+    /// Creates a matrix with its elements filled with the components provided by a slice in
+    /// row-major order.
+    ///
+    /// The order of elements in the slice must follow the usual mathematic writing, i.e.,
+    /// row-by-row.
+    #[inline]
+    pub fn from_row_slice_generic(nrows: R, ncols: C, slice: &[N]) -> Matrix<N, R, C, S> {
+        assert!(slice.len() == nrows.value() * ncols.value(),
+                "Matrix init. error: the slice did not contain the right number of elements.");
+
+        let mut res  = unsafe { Self::new_uninitialized_generic(nrows, ncols) };
+        let mut iter = slice.iter();
+
+        for i in 0 .. nrows.value() {
+            for j in 0 .. ncols.value() {
+                unsafe {
+                    *res.get_unchecked_mut(i, j) = *iter.next().unwrap()
+                }
+            }
+        }
+
+        res
+    }
+
+    /// Creates a matrix with its elements filled with the components provided by a slice. The
+    /// components must have the same layout as the matrix data storage (i.e. row-major or column-major).
+    #[inline]
+    pub fn from_column_slice_generic(nrows: R, ncols: C, slice: &[N]) -> Matrix<N, R, C, S> {
+        Matrix::from_iterator_generic(nrows, ncols, slice.iter().cloned())
+    }
+
+    /// Creates a matrix filled with the results of a function applied to each of its component
+    /// coordinates.
+    #[inline]
+    pub fn from_fn_generic<F>(nrows: R, ncols: C, mut f: F) -> Matrix<N, R, C, S>
+        where F: FnMut(usize, usize) -> N {
+        let mut res = unsafe { Self::new_uninitialized_generic(nrows, ncols) };
+
+        for i in 0 .. nrows.value() {
+            for j in 0 .. ncols.value() {
+                unsafe { *res.get_unchecked_mut(i, j) = f(i, j) }
+            }
+        }
+
+        res
+    }
+
+    /// Creates a new indentity matrix.
+    ///
+    /// If the matrix is not square, the largest square submatrix starting at index `(0, 0)` is set
+    /// to the identity matrix. All other entries are set to zero.
+    #[inline]
+    pub fn identity_generic(nrows: R, ncols: C) -> Matrix<N, R, C, S>
+        where N: Zero + One {
+        Self::from_diagonal_element_generic(nrows, ncols, N::one())
+    }
+
+    /// Creates a new matrix with its diagonal filled with copies of `elt`.
+    ///
+    /// If the matrix is not square, the largest square submatrix starting at index `(0, 0)` is set
+    /// to the identity matrix. All other entries are set to zero.
+    #[inline]
+    pub fn from_diagonal_element_generic(nrows: R, ncols: C, elt: N) -> Matrix<N, R, C, S>
+        where N: Zero + One {
+        let mut res = unsafe { Self::new_uninitialized_generic(nrows, ncols) };
+        res.fill(N::zero());
+
+        for i in 0 .. ::min(nrows.value(), ncols.value()) {
+            unsafe { *res.get_unchecked_mut(i, i) = elt }
+        }
+
+        res
+    }
+
+    /// Builds a new matrix from its rows.
+    ///
+    /// Panics if not enough rows are provided (for statically-sized matrices), or if all rows do
+    /// not have the same dimensions.
+    #[inline]
+    pub fn from_rows<SB>(rows: &[Matrix<N, U1, C, SB>]) -> Matrix<N, R, C, S>
+        where SB: Storage<N, U1, C> {
+
+        assert!(rows.len() > 0, "At least one row must be given.");
+        let nrows = R::try_to_usize().unwrap_or(rows.len());
+        let ncols = rows[0].len();
+        assert!(rows.len() == nrows, "Invalid number of rows provided to build this matrix.");
+
+        if C::try_to_usize().is_none() {
+            assert!(rows.iter().all(|r| r.len() == ncols),
+            "The provided rows must all have the same dimension.");
+        }
+
+        // FIXME: optimize that.
+        Self::from_fn_generic(R::from_usize(nrows), C::from_usize(ncols), |i, j| rows[i][(0, j)])
+    }
+
+
+    /// Builds a new matrix from its columns.
+    ///
+    /// Panics if not enough columns are provided (for statically-sized matrices), or if all
+    /// columns do not have the same dimensions.
+    #[inline]
+    pub fn from_columns<SB>(columns: &[ColumnVector<N, R, SB>]) -> Matrix<N, R, C, S>
+        where SB: Storage<N, R, U1> {
+
+        assert!(columns.len() > 0, "At least one column must be given.");
+        let ncols = C::try_to_usize().unwrap_or(columns.len());
+        let nrows = columns[0].len();
+        assert!(columns.len() == ncols, "Invalid number of columns provided to build this matrix.");
+
+        if R::try_to_usize().is_none() {
+            assert!(columns.iter().all(|r| r.len() == nrows),
+            "The columns provided must all have the same dimension.");
+        }
+
+        // FIXME: optimize that.
+        Self::from_fn_generic(R::from_usize(nrows), C::from_usize(ncols), |i, j| columns[j][i])
+    }
+}
+
+impl<N, R: Dim, C: Dim, S> Matrix<N, R, C, S>
+    where N: Scalar + Rand,
+          S: OwnedStorage<N, R, C>,
+          S::Alloc: OwnedAllocator<N, R, C, S> {
+    /// Creates a matrix filled with random values.
+    #[inline]
+    pub fn new_random_generic(nrows: R, ncols: C) -> Matrix<N, R, C, S> {
+        Matrix::from_fn_generic(nrows, ncols, |_, _| rand::random())
+    }
+}
+
+impl<N, D: Dim, S> SquareMatrix<N, D, S>
+    where N: Scalar + Zero,
+          S: OwnedStorage<N, D, D>,
+          S::Alloc: OwnedAllocator<N, D, D, S> {
+    /// Creates a square matrix with its diagonal set to `diag` and all other entries set to 0.
+    #[inline]
+    pub fn from_diagonal<SB: Storage<N, D, U1>>(diag: &ColumnVector<N, D, SB>) -> Self {
+        let (dim, _) = diag.data.shape();
+        let mut res = Self::from_element_generic(dim, dim, N::zero());
+
+        for i in 0 .. diag.len() {
+            unsafe { *res.get_unchecked_mut(i, i) = *diag.get_unchecked(i, 0); }
+        }
+
+        res
+    }
+}
+
+
+/*
+ *
+ * Generate constructors with varying number of arguments, depending on the object type.
+ *
+ */
+macro_rules! impl_constructors(
+    ($($Dims: ty),*; $(=> $DimIdent: ident: $DimBound: ident),*; $($gargs: expr),*; $($args: ident),*) => {
+        impl<N: Scalar, $($DimIdent: $DimBound, )* S> Matrix<N $(, $Dims)*, S>
+            where S: OwnedStorage<N $(, $Dims)*>,
+                  S::Alloc: OwnedAllocator<N $(, $Dims)*, S> {
+
+            /// Creates a new uninitialized matrix.
+            #[inline]
+            pub unsafe fn new_uninitialized($($args: usize),*) -> Matrix<N $(, $Dims)*, S> {
+                Self::new_uninitialized_generic($($gargs),*)
+            }
+
+            /// Creates a matrix with all its elements set to `elem`.
+            #[inline]
+            pub fn from_element($($args: usize,)* elem: N) -> Matrix<N $(, $Dims)*, S> {
+                Self::from_element_generic($($gargs, )* elem)
+            }
+
+            /// Creates a matrix with all its elements filled by an iterator.
+            #[inline]
+            pub fn from_iterator<I>($($args: usize,)* iter: I) -> Matrix<N $(, $Dims)*, S>
+                where I: IntoIterator<Item = N> {
+                Self::from_iterator_generic($($gargs, )* iter)
+            }
+
+            /// Creates a matrix with its elements filled with the components provided by a slice
+            /// in row-major order.
+            ///
+            /// The order of elements in the slice must follow the usual mathematic writing, i.e.,
+            /// row-by-row.
+            #[inline]
+            pub fn from_row_slice($($args: usize,)* slice: &[N]) -> Matrix<N $(, $Dims)*, S> {
+                Self::from_row_slice_generic($($gargs, )* slice)
+            }
+
+            /// Creates a matrix with its elements filled with the components provided by a slice
+            /// in column-major order.
+            #[inline]
+            pub fn from_column_slice($($args: usize,)* slice: &[N]) -> Matrix<N $(, $Dims)*, S> {
+                Self::from_column_slice_generic($($gargs, )* slice)
+            }
+
+            /// Creates a matrix filled with the results of a function applied to each of its
+            /// component coordinates.
+            // FIXME: don't take a dimension of the matrix is statically sized.
+            #[inline]
+            pub fn from_fn<F>($($args: usize,)* f: F) -> Matrix<N $(, $Dims)*, S>
+                where F: FnMut(usize, usize) -> N {
+                Self::from_fn_generic($($gargs, )* f)
+            }
+
+            /// Creates an identity matrix. If the matrix is not square, the largest square
+            /// submatrix (starting at the first row and column) is set to the identity while all
+            /// other entries are set to zero.
+            #[inline]
+            pub fn identity($($args: usize,)*) -> Matrix<N $(, $Dims)*, S>
+                where N: Zero + One {
+                Self::identity_generic($($gargs),* )
+            }
+
+            /// Creates a matrix filled with its diagonal filled with `elt` and all other
+            /// components set to zero.
+            #[inline]
+            pub fn from_diagonal_element($($args: usize,)* elt: N) -> Matrix<N $(, $Dims)*, S>
+                where N: Zero + One {
+                Self::from_diagonal_element_generic($($gargs, )* elt)
+            }
+        }
+
+        impl<N: Scalar + Rand, $($DimIdent: $DimBound, )* S> Matrix<N $(, $Dims)*, S>
+            where S: OwnedStorage<N $(, $Dims)*>,
+                  S::Alloc: OwnedAllocator<N $(, $Dims)*, S> {
+
+            /// Creates a matrix filled with random values.
+            #[inline]
+            pub fn new_random($($args: usize),*) -> Matrix<N $(, $Dims)*, S> {
+                Self::new_random_generic($($gargs),*)
+            }
+        }
+    }
+);
+
+// FIXME: this is not very pretty. We could find a better call syntax.
+impl_constructors!(R, C;                         // Arguments for Matrix<N, ..., S>
+                   => R: DimName, => C: DimName; // Type parameters for impl<N, ..., S>
+                   R::name(), C::name();         // Arguments for `_generic` constructors.
+                   );                            // Arguments for non-generic constructors.
+
+impl_constructors!(R, Dynamic;
+                   => R: DimName;
+                   R::name(), Dynamic::new(ncols);
+                   ncols);
+
+impl_constructors!(Dynamic, C;
+                   => C: DimName;
+                   Dynamic::new(nrows), C::name();
+                   nrows);
+
+impl_constructors!(Dynamic, Dynamic;
+                   ;
+                   Dynamic::new(nrows), Dynamic::new(ncols);
+                   nrows, ncols);
+
+/*
+ *
+ * Zero, One, Rand traits.
+ *
+ */
+impl<N, R: DimName, C: DimName, S> Zero for Matrix<N, R, C, S>
+    where N: Scalar + Zero + ClosedAdd,
+          S: OwnedStorage<N, R, C>,
+          S::Alloc: OwnedAllocator<N, R, C, S> {
+    #[inline]
+    fn zero() -> Self {
+        Self::from_element(N::zero())
+    }
+
+    #[inline]
+    fn is_zero(&self) -> bool {
+        self.iter().all(|e| e.is_zero())
+    }
+}
+
+impl<N, D: DimName, S> One for Matrix<N, D, D, S>
+    where N: Scalar + Zero + One + ClosedMul + ClosedAdd,
+          S: OwnedStorage<N, D, D>,
+          S::Alloc: OwnedAllocator<N, D, D, S> {
+    #[inline]
+    fn one() -> Self {
+        Self::identity()
+    }
+}
+
+impl<N, R: DimName, C: DimName, S> Bounded for Matrix<N, R, C, S>
+    where N: Scalar + Bounded,
+          S: OwnedStorage<N, R, C>,
+          S::Alloc: OwnedAllocator<N, R, C, S> {
+    #[inline]
+    fn max_value() -> Self {
+        Self::from_element(N::max_value())
+    }
+
+    #[inline]
+    fn min_value() -> Self {
+        Self::from_element(N::min_value())
+    }
+}
+
+impl<N: Scalar + Rand, R: Dim, C: Dim, S> Rand for Matrix<N, R, C, S>
+    where S: OwnedStorage<N, R, C>,
+          S::Alloc: OwnedAllocator<N, R, C, S> {
+    #[inline]
+    fn rand<G: Rng>(rng: &mut G) -> Self {
+        let nrows = R::try_to_usize().unwrap_or(rng.gen_range(0, 10));
+        let ncols = C::try_to_usize().unwrap_or(rng.gen_range(0, 10));
+
+        Self::from_fn_generic(R::from_usize(nrows), C::from_usize(ncols), |_, _| rng.gen())
+    }
+}
+
+
+#[cfg(feature = "arbitrary")]
+impl<N, R, C, S> Arbitrary for Matrix<N, R, C, S>
+    where R: Dim, C: Dim,
+          N: Scalar + Arbitrary + Send,
+          S: OwnedStorage<N, R, C> + Send,
+          S::Alloc: OwnedAllocator<N, R, C, S> {
+    #[inline]
+    fn arbitrary<G: Gen>(g: &mut G) -> Self {
+        let nrows = R::try_to_usize().unwrap_or(g.gen_range(0, 10));
+        let ncols = C::try_to_usize().unwrap_or(g.gen_range(0, 10));
+
+        Self::from_fn_generic(R::from_usize(nrows), C::from_usize(ncols), |_, _| N::arbitrary(g))
+    }
+}
+
+
+/*
+ *
+ * Constructors for small matrices and vectors.
+ *
+ */
+macro_rules! componentwise_constructors_impl(
+    ($($R: ty, $C: ty, $($args: ident:($irow: expr,$icol: expr)),*);* $(;)*) => {$(
+        impl<N, S> Matrix<N, $R, $C, S>
+            where N: Scalar,
+                  S: OwnedStorage<N, $R, $C>,
+                  S::Alloc: OwnedAllocator<N, $R, $C, S> {
+            /// Initializes this matrix from its components.
+            #[inline]
+            pub fn new($($args: N),*) -> Matrix<N, $R, $C, S> {
+                unsafe {
+                    let mut res = Self::new_uninitialized();
+                    $( *res.get_unchecked_mut($irow, $icol) = $args; )*
+
+                    res
+                }
+            }
+        }
+    )*}
+);
+
+componentwise_constructors_impl!(
+    /*
+     * Square matrices 1 .. 6.
+     */
+    U2, U2, m11:(0,0), m12:(0,1),
+            m21:(1,0), m22:(1,1);
+    U3, U3, m11:(0,0), m12:(0,1), m13:(0,2),
+            m21:(1,0), m22:(1,1), m23:(1,2),
+            m31:(2,0), m32:(2,1), m33:(2,2);
+    U4, U4, m11:(0,0), m12:(0,1), m13:(0,2), m14:(0,3),
+            m21:(1,0), m22:(1,1), m23:(1,2), m24:(1,3),
+            m31:(2,0), m32:(2,1), m33:(2,2), m34:(2,3),
+            m41:(3,0), m42:(3,1), m43:(3,2), m44:(3,3);
+    U5, U5, m11:(0,0), m12:(0,1), m13:(0,2), m14:(0,3), m15:(0,4),
+            m21:(1,0), m22:(1,1), m23:(1,2), m24:(1,3), m25:(1,4),
+            m31:(2,0), m32:(2,1), m33:(2,2), m34:(2,3), m35:(2,4),
+            m41:(3,0), m42:(3,1), m43:(3,2), m44:(3,3), m45:(3,4),
+            m51:(4,0), m52:(4,1), m53:(4,2), m54:(4,3), m55:(4,4);
+    U6, U6, m11:(0,0), m12:(0,1), m13:(0,2), m14:(0,3), m15:(0,4), m16:(0,5),
+            m21:(1,0), m22:(1,1), m23:(1,2), m24:(1,3), m25:(1,4), m26:(1,5),
+            m31:(2,0), m32:(2,1), m33:(2,2), m34:(2,3), m35:(2,4), m36:(2,5),
+            m41:(3,0), m42:(3,1), m43:(3,2), m44:(3,3), m45:(3,4), m46:(3,5),
+            m51:(4,0), m52:(4,1), m53:(4,2), m54:(4,3), m55:(4,4), m56:(4,5),
+            m61:(5,0), m62:(5,1), m63:(5,2), m64:(5,3), m65:(5,4), m66:(5,5);
+
+    /*
+     * Rectangular matrices with 2 rows.
+     */
+    U2, U3, m11:(0,0), m12:(0,1), m13:(0,2),
+            m21:(1,0), m22:(1,1), m23:(1,2);
+    U2, U4, m11:(0,0), m12:(0,1), m13:(0,2), m14:(0,3),
+            m21:(1,0), m22:(1,1), m23:(1,2), m24:(1,3);
+    U2, U5, m11:(0,0), m12:(0,1), m13:(0,2), m14:(0,3), m15:(0,4),
+            m21:(1,0), m22:(1,1), m23:(1,2), m24:(1,3), m25:(1,4);
+    U2, U6, m11:(0,0), m12:(0,1), m13:(0,2), m14:(0,3), m15:(0,4), m16:(0,5),
+            m21:(1,0), m22:(1,1), m23:(1,2), m24:(1,3), m25:(1,4), m26:(1,5);
+
+    /*
+     * Rectangular matrices with 3 rows.
+     */
+    U3, U2, m11:(0,0), m12:(0,1),
+            m21:(1,0), m22:(1,1),
+            m31:(2,0), m32:(2,1);
+    U3, U4, m11:(0,0), m12:(0,1), m13:(0,2), m14:(0,3),
+            m21:(1,0), m22:(1,1), m23:(1,2), m24:(1,3),
+            m31:(2,0), m32:(2,1), m33:(2,2), m34:(2,3);
+    U3, U5, m11:(0,0), m12:(0,1), m13:(0,2), m14:(0,3), m15:(0,4),
+            m21:(1,0), m22:(1,1), m23:(1,2), m24:(1,3), m25:(1,4),
+            m31:(2,0), m32:(2,1), m33:(2,2), m34:(2,3), m35:(2,4);
+    U3, U6, m11:(0,0), m12:(0,1), m13:(0,2), m14:(0,3), m15:(0,4), m16:(0,5),
+            m21:(1,0), m22:(1,1), m23:(1,2), m24:(1,3), m25:(1,4), m26:(1,5),
+            m31:(2,0), m32:(2,1), m33:(2,2), m34:(2,3), m35:(2,4), m36:(2,5);
+
+    /*
+     * Rectangular matrices with 4 rows.
+     */
+    U4, U2, m11:(0,0), m12:(0,1),
+            m21:(1,0), m22:(1,1),
+            m31:(2,0), m32:(2,1),
+            m41:(3,0), m42:(3,1);
+    U4, U3, m11:(0,0), m12:(0,1), m13:(0,2),
+            m21:(1,0), m22:(1,1), m23:(1,2),
+            m31:(2,0), m32:(2,1), m33:(2,2),
+            m41:(3,0), m42:(3,1), m43:(3,2);
+    U4, U5, m11:(0,0), m12:(0,1), m13:(0,2), m14:(0,3), m15:(0,4),
+            m21:(1,0), m22:(1,1), m23:(1,2), m24:(1,3), m25:(1,4),
+            m31:(2,0), m32:(2,1), m33:(2,2), m34:(2,3), m35:(2,4),
+            m41:(3,0), m42:(3,1), m43:(3,2), m44:(3,3), m45:(3,4);
+    U4, U6, m11:(0,0), m12:(0,1), m13:(0,2), m14:(0,3), m15:(0,4), m16:(0,5),
+            m21:(1,0), m22:(1,1), m23:(1,2), m24:(1,3), m25:(1,4), m26:(1,5),
+            m31:(2,0), m32:(2,1), m33:(2,2), m34:(2,3), m35:(2,4), m36:(2,5),
+            m41:(3,0), m42:(3,1), m43:(3,2), m44:(3,3), m45:(3,4), m46:(3,5);
+
+    /*
+     * Rectangular matrices with 5 rows.
+     */
+    U5, U2, m11:(0,0), m12:(0,1),
+            m21:(1,0), m22:(1,1),
+            m31:(2,0), m32:(2,1),
+            m41:(3,0), m42:(3,1),
+            m51:(4,0), m52:(4,1);
+    U5, U3, m11:(0,0), m12:(0,1), m13:(0,2),
+            m21:(1,0), m22:(1,1), m23:(1,2),
+            m31:(2,0), m32:(2,1), m33:(2,2),
+            m41:(3,0), m42:(3,1), m43:(3,2),
+            m51:(4,0), m52:(4,1), m53:(4,2);
+    U5, U4, m11:(0,0), m12:(0,1), m13:(0,2), m14:(0,3),
+            m21:(1,0), m22:(1,1), m23:(1,2), m24:(1,3),
+            m31:(2,0), m32:(2,1), m33:(2,2), m34:(2,3),
+            m41:(3,0), m42:(3,1), m43:(3,2), m44:(3,3),
+            m51:(4,0), m52:(4,1), m53:(4,2), m54:(4,3);
+    U5, U6, m11:(0,0), m12:(0,1), m13:(0,2), m14:(0,3), m15:(0,4), m16:(0,5),
+            m21:(1,0), m22:(1,1), m23:(1,2), m24:(1,3), m25:(1,4), m26:(1,5),
+            m31:(2,0), m32:(2,1), m33:(2,2), m34:(2,3), m35:(2,4), m36:(2,5),
+            m41:(3,0), m42:(3,1), m43:(3,2), m44:(3,3), m45:(3,4), m46:(3,5),
+            m51:(4,0), m52:(4,1), m53:(4,2), m54:(4,3), m55:(4,4), m56:(4,5);
+
+    /*
+     * Rectangular matrices with 6 rows.
+     */
+    U6, U2, m11:(0,0), m12:(0,1),
+            m21:(1,0), m22:(1,1),
+            m31:(2,0), m32:(2,1),
+            m41:(3,0), m42:(3,1),
+            m51:(4,0), m52:(4,1),
+            m61:(5,0), m62:(5,1);
+    U6, U3, m11:(0,0), m12:(0,1), m13:(0,2),
+            m21:(1,0), m22:(1,1), m23:(1,2),
+            m31:(2,0), m32:(2,1), m33:(2,2),
+            m41:(3,0), m42:(3,1), m43:(3,2),
+            m51:(4,0), m52:(4,1), m53:(4,2),
+            m61:(5,0), m62:(5,1), m63:(5,2);
+    U6, U4, m11:(0,0), m12:(0,1), m13:(0,2), m14:(0,3),
+            m21:(1,0), m22:(1,1), m23:(1,2), m24:(1,3),
+            m31:(2,0), m32:(2,1), m33:(2,2), m34:(2,3),
+            m41:(3,0), m42:(3,1), m43:(3,2), m44:(3,3),
+            m51:(4,0), m52:(4,1), m53:(4,2), m54:(4,3),
+            m61:(5,0), m62:(5,1), m63:(5,2), m64:(5,3);
+    U6, U5, m11:(0,0), m12:(0,1), m13:(0,2), m14:(0,3), m15:(0,4),
+            m21:(1,0), m22:(1,1), m23:(1,2), m24:(1,3), m25:(1,4),
+            m31:(2,0), m32:(2,1), m33:(2,2), m34:(2,3), m35:(2,4),
+            m41:(3,0), m42:(3,1), m43:(3,2), m44:(3,3), m45:(3,4),
+            m51:(4,0), m52:(4,1), m53:(4,2), m54:(4,3), m55:(4,4),
+            m61:(5,0), m62:(5,1), m63:(5,2), m64:(5,3), m65:(5,4);
+
+    /*
+     * Row vectors 1 .. 6.
+     */
+    U1, U1, x:(0,0);
+    U1, U2, x:(0,0), y:(0,1);
+    U1, U3, x:(0,0), y:(0,1), z:(0,2);
+    U1, U4, x:(0,0), y:(0,1), z:(0,2), w:(0,3);
+    U1, U5, x:(0,0), y:(0,1), z:(0,2), w:(0,3), a:(0,4);
+    U1, U6, x:(0,0), y:(0,1), z:(0,2), w:(0,3), a:(0,4), b:(0,5);
+
+    /*
+     * Column vectors 1 .. 6.
+     */
+    U2, U1, x:(0,0), y:(1,0);
+    U3, U1, x:(0,0), y:(1,0), z:(2,0);
+    U4, U1, x:(0,0), y:(1,0), z:(2,0), w:(3,0);
+    U5, U1, x:(0,0), y:(1,0), z:(2,0), w:(3,0), a:(4,0);
+    U6, U1, x:(0,0), y:(1,0), z:(2,0), w:(3,0), a:(4,0), b:(5,0);
+);
+
+/*
+ *
+ * Axis constructors.
+ *
+ */
+impl<N, R: DimName, S> ColumnVector<N, R, S>
+where N: Scalar + Zero + One,
+      S: OwnedStorage<N, R, U1>,
+      S::Alloc: OwnedAllocator<N, R, U1, S> {
+    /// The column vector with a 1 as its first component, and zero elsewhere.
+    #[inline]
+    pub fn x() -> Self
+      where R::Value: Cmp<typenum::U0, Output = Greater> {
+        let mut res = Self::from_element(N::zero());
+        unsafe { *res.get_unchecked_mut(0, 0) = N::one(); }
+
+        res
+    }
+
+    /// The column vector with a 1 as its second component, and zero elsewhere.
+    #[inline]
+    pub fn y() -> Self
+      where R::Value: Cmp<typenum::U1, Output = Greater> {
+        let mut res = Self::from_element(N::zero());
+        unsafe { *res.get_unchecked_mut(1, 0) = N::one(); }
+
+        res
+    }
+
+    /// The column vector with a 1 as its third component, and zero elsewhere.
+    #[inline]
+    pub fn z() -> Self
+      where R::Value: Cmp<typenum::U2, Output = Greater> {
+        let mut res = Self::from_element(N::zero());
+        unsafe { *res.get_unchecked_mut(2, 0) = N::one(); }
+
+        res
+    }
+
+    /// The column vector with a 1 as its fourth component, and zero elsewhere.
+    #[inline]
+    pub fn w() -> Self
+      where R::Value: Cmp<typenum::U3, Output = Greater> {
+        let mut res = Self::from_element(N::zero());
+        unsafe { *res.get_unchecked_mut(3, 0) = N::one(); }
+
+        res
+    }
+
+    /// The column vector with a 1 as its fifth component, and zero elsewhere.
+    #[inline]
+    pub fn a() -> Self
+      where R::Value: Cmp<typenum::U4, Output = Greater> {
+        let mut res = Self::from_element(N::zero());
+        unsafe { *res.get_unchecked_mut(4, 0) = N::one(); }
+
+        res
+    }
+
+    /// The column vector with a 1 as its sixth component, and zero elsewhere.
+    #[inline]
+    pub fn b() -> Self
+      where R::Value: Cmp<typenum::U5, Output = Greater> {
+        let mut res = Self::from_element(N::zero());
+        unsafe { *res.get_unchecked_mut(5, 0) = N::one(); }
+
+        res
+    }
+
+    /// The unit column vector with a 1 as its first component, and zero elsewhere.
+    #[inline]
+    pub fn x_axis() -> Unit<Self>
+      where R::Value: Cmp<typenum::U0, Output = Greater> {
+          Unit::new_unchecked(Self::x())
+    }
+
+    /// The unit column vector with a 1 as its second component, and zero elsewhere.
+    #[inline]
+    pub fn y_axis() -> Unit<Self>
+      where R::Value: Cmp<typenum::U1, Output = Greater> {
+          Unit::new_unchecked(Self::y())
+    }
+
+    /// The unit column vector with a 1 as its third component, and zero elsewhere.
+    #[inline]
+    pub fn z_axis() -> Unit<Self>
+      where R::Value: Cmp<typenum::U2, Output = Greater> {
+          Unit::new_unchecked(Self::z())
+    }
+
+    /// The unit column vector with a 1 as its fourth component, and zero elsewhere.
+    #[inline]
+    pub fn w_axis() -> Unit<Self>
+      where R::Value: Cmp<typenum::U3, Output = Greater> {
+          Unit::new_unchecked(Self::w())
+    }
+
+    /// The unit column vector with a 1 as its fifth component, and zero elsewhere.
+    #[inline]
+    pub fn a_axis() -> Unit<Self>
+      where R::Value: Cmp<typenum::U4, Output = Greater> {
+          Unit::new_unchecked(Self::a())
+    }
+
+    /// The unit column vector with a 1 as its sixth component, and zero elsewhere.
+    #[inline]
+    pub fn b_axis() -> Unit<Self>
+      where R::Value: Cmp<typenum::U5, Output = Greater> {
+          Unit::new_unchecked(Self::b())
+    }
+}

src/core/conversion.rs

@@ -0,0 +1,59 @@
+use alga::general::{SubsetOf, SupersetOf};
+
+use core::{Scalar, Matrix};
+use core::dimension::Dim;
+use core::constraint::{ShapeConstraint, SameNumberOfRows, SameNumberOfColumns};
+use core::storage::OwnedStorage;
+use core::allocator::{OwnedAllocator, SameShapeAllocator};
+
+
+// FIXME: too bad this won't work allo slice conversions.
+impl<N1, N2, R1, C1, R2, C2, SA, SB> SubsetOf<Matrix<N2, R2, C2, SB>> for Matrix<N1, R1, C1, SA>
+    where R1: Dim, C1: Dim, R2: Dim, C2: Dim,
+          N1: Scalar,
+          N2: Scalar + SupersetOf<N1>,
+          SA: OwnedStorage<N1, R1, C1>,
+          SB: OwnedStorage<N2, R2, C2>,
+          SB::Alloc: OwnedAllocator<N2, R2, C2, SB>,
+          SA::Alloc: OwnedAllocator<N1, R1, C1, SA> +
+                     SameShapeAllocator<N1, R1, C1, R2, C2, SA>,
+          ShapeConstraint: SameNumberOfRows<R1, R2> + SameNumberOfColumns<C1, C2> {
+    #[inline]
+    fn to_superset(&self) -> Matrix<N2, R2, C2, SB> {
+        let (nrows, ncols) = self.shape();
+        let nrows2 = R2::from_usize(nrows);
+        let ncols2 = C2::from_usize(ncols);
+
+        let mut res = unsafe { Matrix::<N2, R2, C2, SB>::new_uninitialized_generic(nrows2, ncols2) };
+        for i in 0 .. nrows {
+            for j in 0 .. ncols {
+                unsafe {
+                    *res.get_unchecked_mut(i, j) = N2::from_subset(self.get_unchecked(i, j))
+                }
+            }
+        }
+
+        res
+    }
+
+    #[inline]
+    fn is_in_subset(m: &Matrix<N2, R2, C2, SB>) -> bool {
+        m.iter().all(|e| e.is_in_subset())
+    }
+
+    #[inline]
+    unsafe fn from_superset_unchecked(m: &Matrix<N2, R2, C2, SB>) -> Self {
+        let (nrows2, ncols2) = m.shape();
+        let nrows = R1::from_usize(nrows2);
+        let ncols = C1::from_usize(ncols2);
+
+        let mut res = Self::new_uninitialized_generic(nrows, ncols);
+        for i in 0 .. nrows2 {
+            for j in 0 .. ncols2 {
+                *res.get_unchecked_mut(i, j) = m.get_unchecked(i, j).to_subset_unchecked()
+            }
+        }
+
+        res
+    }
+}

src/core/coordinates.rs

@@ -0,0 +1,239 @@
+#![allow(missing_docs)]
+
+//! Structures to which matrices and vector can be auto-dereferenced (through `Deref`) to access
+//! components using their names. For example, if `v` is a 3D vector, one can write `v.z` instead
+//! of `v[2]`.
+
+use std::mem;
+use std::ops::{Deref, DerefMut};
+
+use core::{Scalar, Matrix};
+use core::dimension::{U1, U2, U3, U4, U5, U6};
+use core::storage::OwnedStorage;
+use core::allocator::OwnedAllocator;
+
+/*
+ *
+ * Give coordinates to owned Vector{1 .. 6} and Matrix{1 .. 6}
+ *
+ */
+
+macro_rules! coords_impl(
+    ($T: ident; $($comps: ident),*) => {
+        /// Data structure used to provide access to matrix and vector coordinates with the dot
+        /// notation, e.g., `v.x` is the same as `v[0]` for a vector.
+        #[repr(C)]
+        #[derive(Eq, PartialEq, Clone, Hash, Debug, Copy)]
+        #[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
+        pub struct $T<N: Scalar> {
+            $(pub $comps: N),*
+        }
+    }
+);
+
+
+macro_rules! deref_impl(
+    ($R: ty, $C: ty; $Target: ident) => {
+        impl<N: Scalar, S> Deref for Matrix<N, $R, $C, S>
+            where S: OwnedStorage<N, $R, $C>,
+                  S::Alloc: OwnedAllocator<N, $R, $C, S> {
+            type Target = $Target<N>;
+
+            #[inline]
+            fn deref(&self) -> &Self::Target {
+                unsafe { mem::transmute(self) }
+            }
+        }
+
+        impl<N: Scalar, S> DerefMut for Matrix<N, $R, $C, S>
+            where S: OwnedStorage<N, $R, $C>,
+                  S::Alloc: OwnedAllocator<N, $R, $C, S> {
+            #[inline]
+            fn deref_mut(&mut self) -> &mut Self::Target {
+                unsafe { mem::transmute(self) }
+            }
+        }
+    }
+);
+
+/*
+ *
+ * Vector coordinates.
+ *
+ */
+coords_impl!(X; x);
+coords_impl!(XY; x, y);
+coords_impl!(XYZ; x, y, z);
+coords_impl!(XYZW; x, y, z, w);
+coords_impl!(XYZWA; x, y, z, w, a);
+coords_impl!(XYZWAB; x, y, z, w, a, b);
+coords_impl!(IJKW; i, j, k, w);
+
+/*
+ * Rectangular matrices with 2 rows.
+ */
+coords_impl!(M2x2; m11, m21,
+                   m12, m22);
+coords_impl!(M2x3; m11, m21,
+                   m12, m22,
+                   m13, m23);
+coords_impl!(M2x4; m11, m21,
+                   m12, m22,
+                   m13, m23,
+                   m14, m24);
+coords_impl!(M2x5; m11, m21,
+                   m12, m22,
+                   m13, m23,
+                   m14, m24,
+                   m15, m25);
+coords_impl!(M2x6; m11, m21,
+                   m12, m22,
+                   m13, m23,
+                   m14, m24,
+                   m15, m25,
+                   m16, m26);
+
+/*
+ * Rectangular matrices with 3 rows.
+ */
+coords_impl!(M3x2; m11, m21, m31,
+                   m12, m22, m32);
+coords_impl!(M3x3; m11, m21, m31,
+                   m12, m22, m32,
+                   m13, m23, m33);
+coords_impl!(M3x4; m11, m21, m31,
+                   m12, m22, m32,
+                   m13, m23, m33,
+                   m14, m24, m34);
+coords_impl!(M3x5; m11, m21, m31,
+                   m12, m22, m32,
+                   m13, m23, m33,
+                   m14, m24, m34,
+                   m15, m25, m35);
+coords_impl!(M3x6; m11, m21, m31,
+                   m12, m22, m32,
+                   m13, m23, m33,
+                   m14, m24, m34,
+                   m15, m25, m35,
+                   m16, m26, m36);
+
+/*
+ * Rectangular matrices with 4 rows.
+ */
+coords_impl!(M4x2; m11, m21, m31, m41,
+                   m12, m22, m32, m42);
+coords_impl!(M4x3; m11, m21, m31, m41,
+                   m12, m22, m32, m42,
+                   m13, m23, m33, m43);
+coords_impl!(M4x4; m11, m21, m31, m41,
+                   m12, m22, m32, m42,
+                   m13, m23, m33, m43,
+                   m14, m24, m34, m44);
+coords_impl!(M4x5; m11, m21, m31, m41,
+                   m12, m22, m32, m42,
+                   m13, m23, m33, m43,
+                   m14, m24, m34, m44,
+                   m15, m25, m35, m45);
+coords_impl!(M4x6; m11, m21, m31, m41,
+                   m12, m22, m32, m42,
+                   m13, m23, m33, m43,
+                   m14, m24, m34, m44,
+                   m15, m25, m35, m45,
+                   m16, m26, m36, m46);
+
+/*
+ * Rectangular matrices with 5 rows.
+ */
+coords_impl!(M5x2; m11, m21, m31, m41, m51,
+                   m12, m22, m32, m42, m52);
+coords_impl!(M5x3; m11, m21, m31, m41, m51,
+                   m12, m22, m32, m42, m52,
+                   m13, m23, m33, m43, m53);
+coords_impl!(M5x4; m11, m21, m31, m41, m51,
+                   m12, m22, m32, m42, m52,
+                   m13, m23, m33, m43, m53,
+                   m14, m24, m34, m44, m54);
+coords_impl!(M5x5; m11, m21, m31, m41, m51,
+                   m12, m22, m32, m42, m52,
+                   m13, m23, m33, m43, m53,
+                   m14, m24, m34, m44, m54,
+                   m15, m25, m35, m45, m55);
+coords_impl!(M5x6; m11, m21, m31, m41, m51,
+                   m12, m22, m32, m42, m52,
+                   m13, m23, m33, m43, m53,
+                   m14, m24, m34, m44, m54,
+                   m15, m25, m35, m45, m55,
+                   m16, m26, m36, m46, m56);
+
+/*
+ * Rectangular matrices with 6 rows.
+ */
+
+coords_impl!(M6x2; m11, m21, m31, m41, m51, m61,
+                   m12, m22, m32, m42, m52, m62);
+coords_impl!(M6x3; m11, m21, m31, m41, m51, m61,
+                   m12, m22, m32, m42, m52, m62,
+                   m13, m23, m33, m43, m53, m63);
+coords_impl!(M6x4; m11, m21, m31, m41, m51, m61,
+                   m12, m22, m32, m42, m52, m62,
+                   m13, m23, m33, m43, m53, m63,
+                   m14, m24, m34, m44, m54, m64);
+coords_impl!(M6x5; m11, m21, m31, m41, m51, m61,
+                   m12, m22, m32, m42, m52, m62,
+                   m13, m23, m33, m43, m53, m63,
+                   m14, m24, m34, m44, m54, m64,
+                   m15, m25, m35, m45, m55, m65);
+coords_impl!(M6x6; m11, m21, m31, m41, m51, m61,
+                   m12, m22, m32, m42, m52, m62,
+                   m13, m23, m33, m43, m53, m63,
+                   m14, m24, m34, m44, m54, m64,
+                   m15, m25, m35, m45, m55, m65,
+                   m16, m26, m36, m46, m56, m66);
+
+/*
+ *
+ * Attach coordinates to matrices.
+ *
+ */
+deref_impl!(U1, U1; X);
+deref_impl!(U2, U1; XY);
+deref_impl!(U3, U1; XYZ);
+deref_impl!(U4, U1; XYZW);
+deref_impl!(U5, U1; XYZWA);
+deref_impl!(U6, U1; XYZWAB);
+
+deref_impl!(U1, U2; XY);
+deref_impl!(U1, U3; XYZ);
+deref_impl!(U1, U4; XYZW);
+deref_impl!(U1, U5; XYZWA);
+deref_impl!(U1, U6; XYZWAB);
+
+deref_impl!(U2, U2; M2x2);
+deref_impl!(U2, U3; M2x3);
+deref_impl!(U2, U4; M2x4);
+deref_impl!(U2, U5; M2x5);
+deref_impl!(U2, U6; M2x6);
+
+deref_impl!(U3, U2; M3x2);
+deref_impl!(U3, U3; M3x3);
+deref_impl!(U3, U4; M3x4);
+deref_impl!(U3, U5; M3x5);
+deref_impl!(U3, U6; M3x6);
+
+deref_impl!(U4, U2; M4x2);
+deref_impl!(U4, U3; M4x3);
+deref_impl!(U4, U4; M4x4);
+deref_impl!(U4, U5; M4x5);
+deref_impl!(U4, U6; M4x6);
+
+deref_impl!(U5, U2; M5x2);
+deref_impl!(U5, U3; M5x3);
+deref_impl!(U5, U4; M5x4);
+deref_impl!(U5, U5; M5x5);
+deref_impl!(U5, U6; M5x6);
+
+deref_impl!(U6, U2; M6x2);
+deref_impl!(U6, U3; M6x3);
+deref_impl!(U6, U4; M6x4);
+deref_impl!(U6, U5; M6x5);
+deref_impl!(U6, U6; M6x6);

src/core/decompositions.rs

@@ -0,0 +1,373 @@
+use std::cmp;
+
+use alga::general::Real;
+use core::{SquareMatrix, OwnedSquareMatrix, ColumnVector, OwnedColumnVector};
+use dimension::{Dim, Dynamic, U1};
+use storage::{Storage, OwnedStorage};
+use allocator::{Allocator, OwnedAllocator};
+
+
+
+impl<N, D: Dim, S> SquareMatrix<N, D, S>
+    where N: Real,
+          S: OwnedStorage<N, D, D>,
+          S::Alloc: OwnedAllocator<N, D, D, S> {
+    /// Get the householder matrix corresponding to a reflexion to the hyperplane
+    /// defined by `vector`. It can be a reflexion contained in a subspace.
+    ///
+    /// # Arguments
+    /// * `dimension` - the dimension of the space the resulting matrix operates in
+    /// * `start` - the starting dimension of the subspace of the reflexion
+    /// * `vector` - the vector defining the reflection.
+    pub fn new_householder_generic<SB, D2>(dimension: D, start: usize, vector: &ColumnVector<N, D2, SB>)
+        -> OwnedSquareMatrix<N, D, S::Alloc>
+        where D2: Dim,
+              SB: Storage<N, D2, U1> {
+        let mut qk = Self::identity_generic(dimension, dimension);
+        let subdim = vector.shape().0;
+
+        let stop = subdim + start;
+
+        assert!(dimension.value() >= stop, "Householder matrix creation: subspace dimension index out of bounds.");
+
+        for j in start .. stop {
+            for i in start .. stop {
+                unsafe {
+                    let vv   = *vector.get_unchecked(i - start, 0) * *vector.get_unchecked(j - start, 0);
+                    let qkij = *qk.get_unchecked(i, j);
+                    *qk.get_unchecked_mut(i, j) = qkij - vv - vv;
+                }
+            }
+        }
+
+        qk
+    }
+}
+
+
+impl<N: Real, D: Dim, S: Storage<N, D, D>> SquareMatrix<N, D, S> {
+    /// QR decomposition using Householder reflections.
+    pub fn qr(self) -> (OwnedSquareMatrix<N, D, S::Alloc>, OwnedSquareMatrix<N, D, S::Alloc>)
+        where S::Alloc: Allocator<N, Dynamic, U1> +
+                        Allocator<N, D, U1> {
+
+        let (nrows, ncols) = self.data.shape();
+
+        // XXX: too restrictive.
+        assert!(nrows.value() >= ncols.value(), "");
+
+        let mut q = OwnedSquareMatrix::<N, D, S::Alloc>::identity_generic(nrows, ncols);
+        let mut r = self.into_owned();
+
+        // Temporary buffer that contains a column.
+        let mut col = unsafe {
+            OwnedColumnVector::<N, D, S::Alloc>::new_uninitialized_generic(nrows, U1)
+        };
+
+        for ite in 0 .. cmp::min(nrows.value() - 1, ncols.value()) {
+            let subdim = Dynamic::new(nrows.value() - ite);
+            let mut v  = col.rows_mut(0, subdim.value());
+            v.copy_from(&r.generic_slice((ite, ite), (subdim, U1)));
+
+            let alpha =
+                if unsafe { *v.get_unchecked(ite, 0) } >= ::zero() {
+                    -v.norm()
+                }
+                else {
+                    v.norm()
+                };
+
+            unsafe {
+                let x = *v.get_unchecked(0, 0);
+                *v.get_unchecked_mut(0, 0) = x - alpha;
+            }
+
+            if !v.normalize_mut().is_zero() {
+                let mut qk = OwnedSquareMatrix::<N, D, S::Alloc>::new_householder_generic(nrows, ite, &v);
+                r = &qk * r;
+
+                // FIXME: add a method `q.mul_tr(qk) := q * qk.transpose` ?
+                qk.transpose_mut();
+                q = q * qk;
+            }
+        }
+
+        (q, r)
+    }
+
+    /// Eigendecomposition of a square symmetric matrix.
+    pub fn eig(&self, eps: N, niter: usize)
+               -> (OwnedSquareMatrix<N, D, S::Alloc>, OwnedColumnVector<N, D, S::Alloc>)
+        where S::Alloc: Allocator<N, D, U1> +
+                        Allocator<N, Dynamic, U1> {
+
+        assert!(self.is_square(),
+                "Unable to compute the eigenvectors and eigenvalues of a non-square matrix.");
+
+        let dim = self.data.shape().0;
+
+        let (mut eigenvectors, mut eigenvalues) = self.hessenberg();
+
+        if dim.value() == 1 {
+            return (eigenvectors, eigenvalues.diagonal());
+        }
+
+        // Allocate arrays for Givens rotation components
+        let mut c = unsafe { OwnedColumnVector::<N, D, S::Alloc>::new_uninitialized_generic(dim, U1) };
+        let mut s = unsafe { OwnedColumnVector::<N, D, S::Alloc>::new_uninitialized_generic(dim, U1) };
+
+        let mut iter = 0;
+        let mut curdim = dim.value() - 1;
+
+        for _ in 0 .. dim.value() {
+
+            let mut stop = false;
+
+            while !stop && iter < niter {
+
+                let lambda;
+
+                unsafe {
+                    let a = *eigenvalues.get_unchecked(curdim - 1, curdim - 1);
+                    let b = *eigenvalues.get_unchecked(curdim - 1, curdim);
+                    let c = *eigenvalues.get_unchecked(curdim, curdim - 1);
+                    let d = *eigenvalues.get_unchecked(curdim, curdim);
+
+                    let trace = a + d;
+                    let determinant = a * d - b * c;
+
+                    let constquarter: N = ::convert(0.25f64);
+                    let consthalf:    N = ::convert(0.5f64);
+
+                    let e = (constquarter * trace * trace - determinant).sqrt();
+
+                    let lambda1 = consthalf * trace + e;
+                    let lambda2 = consthalf * trace - e;
+
+                    if (lambda1 - d).abs() < (lambda2 - d).abs() {
+                        lambda = lambda1;
+                    }
+                    else {
+                        lambda = lambda2;
+                    }
+
+                }
+
+                // Shift matrix
+                for k in 0 .. curdim + 1 {
+                    unsafe {
+                        let a = *eigenvalues.get_unchecked(k, k);
+                        *eigenvalues.get_unchecked_mut(k, k) = a - lambda;
+                    }
+                }
+
+
+                // Givens rotation from left
+                for k in 0 .. curdim {
+                    let x_i = unsafe { *eigenvalues.get_unchecked(k, k) };
+                    let x_j = unsafe { *eigenvalues.get_unchecked(k + 1, k) };
+
+                    let ctmp;
+                    let stmp;
+
+                    if x_j.abs() < eps {
+                        ctmp = N::one();
+                        stmp = N::zero();
+                    }
+                    else if x_i.abs() < eps {
+                        ctmp = N::zero();
+                        stmp = -N::one();
+                    }
+                    else {
+                        let r = x_i.hypot(x_j);
+                        ctmp = x_i / r;
+                        stmp = -x_j / r;
+                    }
+
+                    c[k] = ctmp;
+                    s[k] = stmp;
+
+                    for j in k .. (curdim + 1) {
+                        unsafe {
+                            let a = *eigenvalues.get_unchecked(k, j);
+                            let b = *eigenvalues.get_unchecked(k + 1, j);
+
+                            *eigenvalues.get_unchecked_mut(k, j)     = ctmp * a - stmp * b;
+                            *eigenvalues.get_unchecked_mut(k + 1, j) = stmp * a + ctmp * b;
+                        }
+
+                    }
+                }
+
+                // Givens rotation from right applied to eigenvalues
+                for k in 0 .. curdim {
+                    for i in 0 .. (k + 2) {
+                        unsafe {
+                            let a = *eigenvalues.get_unchecked(i, k);
+                            let b = *eigenvalues.get_unchecked(i, k + 1);
+
+                            *eigenvalues.get_unchecked_mut(i, k)     = c[k] * a - s[k] * b;
+                            *eigenvalues.get_unchecked_mut(i, k + 1) = s[k] * a + c[k] * b;
+                        }
+                    }
+                }
+
+
+                // Shift back
+                for k in 0 .. curdim + 1 {
+                    unsafe {
+                        let a = *eigenvalues.get_unchecked(k, k);
+                        *eigenvalues.get_unchecked_mut(k, k) = a + lambda;
+                    }
+                }
+
+
+                // Givens rotation from right applied to eigenvectors
+                for k in 0 .. curdim {
+                    for i in 0 .. dim.value() {
+
+                        unsafe {
+                            let a = *eigenvectors.get_unchecked(i, k);
+                            let b = *eigenvectors.get_unchecked(i, k + 1);
+
+                            *eigenvectors.get_unchecked_mut(i, k)     = c[k] * a - s[k] * b;
+                            *eigenvectors.get_unchecked_mut(i, k + 1) = s[k] * a + c[k] * b;
+                        }
+                    }
+                }
+
+                iter = iter + 1;
+                stop = true;
+
+                for j in 0 .. curdim {
+                    // Check last row.
+                    if unsafe { *eigenvalues.get_unchecked(curdim, j) }.abs() >= eps {
+                        stop = false;
+                        break;
+                    }
+
+                    // Check last column.
+                    if unsafe { *eigenvalues.get_unchecked(j, curdim) }.abs() >= eps {
+                        stop = false;
+                        break;
+                    }
+                }
+            }
+
+
+            if stop {
+                if curdim > 1 {
+                    curdim = curdim - 1;
+                }
+                else {
+                    break;
+                }
+            }
+        }
+
+        (eigenvectors, eigenvalues.diagonal())
+    }
+
+    /// Cholesky decomposition G of a square symmetric positive definite matrix A, such that A = G * G^T
+    ///
+    /// Matrix symmetricness is not checked. Returns `None` if `self` is not definite positive.
+    #[inline]
+    pub fn cholesky(&self) -> Result<OwnedSquareMatrix<N, D, S::Alloc>, &'static str> {
+        let out = self.transpose();
+
+        if !out.relative_eq(self, N::default_epsilon(), N::default_max_relative()) {
+            return Err("Cholesky: Input matrix is not symmetric");
+        }
+
+        self.do_cholesky(out)
+    }
+
+    /// Cholesky decomposition G of a square symmetric positive definite matrix A, such that A = G * G^T
+    #[inline]
+    pub fn cholesky_unchecked(&self) -> Result<OwnedSquareMatrix<N, D, S::Alloc>, &'static str> {
+        let out = self.transpose();
+        self.do_cholesky(out)
+    }
+
+    #[inline(always)]
+    fn do_cholesky(&self, mut out: OwnedSquareMatrix<N, D, S::Alloc>)
+                   -> Result<OwnedSquareMatrix<N, D, S::Alloc>, &'static str> {
+        assert!(self.is_square(), "The input matrix must be square.");
+
+        for i in 0 .. out.nrows() {
+            for j in 0 .. (i + 1) {
+
+                let mut sum = out[(i, j)];
+
+                for k in 0 .. j {
+                    sum = sum - out[(i, k)] * out[(j, k)];
+                }
+
+                if i > j {
+                    out[(i, j)] = sum / out[(j, j)];
+                }
+                else if sum > N::zero() {
+                    out[(i, i)] = sum.sqrt();
+                }
+                else {
+                    return Err("Cholesky: Input matrix is not positive definite to machine precision.");
+                }
+            }
+        }
+
+        for i in 0 .. out.nrows() {
+            for j in i + 1 .. out.ncols() {
+                out[(i, j)] = N::zero();
+            }
+        }
+
+        Ok(out)
+    }
+
+    /// Hessenberg
+    /// Returns the matrix `self` in Hessenberg form and the corresponding similarity transformation
+    ///
+    /// # Returns
+    /// The tuple (`q`, `h`) that `q * h * q^T = self`
+    pub fn hessenberg(&self) -> (OwnedSquareMatrix<N, D, S::Alloc>, OwnedSquareMatrix<N, D, S::Alloc>)
+        where S::Alloc: Allocator<N, D, U1> + Allocator<N, Dynamic, U1> {
+
+        let (nrows, ncols) = self.data.shape();
+        let mut h = self.clone_owned();
+
+        let mut q = OwnedSquareMatrix::<N, D, S::Alloc>::identity_generic(nrows, ncols);
+
+        if ncols.value() <= 2 {
+            return (q, h);
+        }
+
+        // Temporary buffer that contains a column.
+        let mut col = unsafe {
+            OwnedColumnVector::<N, D, S::Alloc>::new_uninitialized_generic(nrows, U1)
+        };
+
+        for ite in 0 .. (ncols.value() - 2) {
+            let subdim = Dynamic::new(nrows.value() - (ite + 1));
+            let mut v = col.rows_mut(0, subdim.value());
+            v.copy_from(&h.generic_slice((ite + 1, ite), (subdim, U1)));
+
+            let alpha = v.norm();
+
+            unsafe {
+                let x = *v.get_unchecked(0, 0);
+                *v.get_unchecked_mut(0, 0) = x - alpha;
+            }
+
+            if !v.normalize_mut().is_zero() {
+                // XXX: we output the householder matrix to a pre-allocated matrix instead of
+                // return a value to `p`. This would avoid allocation at each iteration.
+                let p = OwnedSquareMatrix::<N, D, S::Alloc>::new_householder_generic(nrows, ite + 1, &v);
+
+                q = q * &p;
+                h = &p * h * p;
+            }
+        }
+
+        (q, h)
+    }
+}

src/core/default_allocator.rs

@@ -0,0 +1,109 @@
+//! The default matrix data storage allocator.
+//!
+//! This will use stack-allocated buffers for matrices with dimensions known at compile-time, and
+//! heap-allocated buffers for matrices with at least one dimension unknown at compile-time.
+
+use std::mem;
+use std::ops::Mul;
+
+use typenum::Prod;
+use generic_array::ArrayLength;
+
+use core::Scalar;
+use core::dimension::{Dim, DimName, Dynamic};
+use core::allocator::Allocator;
+use core::matrix_array::MatrixArray;
+use core::matrix_vec::MatrixVec;
+
+/*
+ *
+ * Allocator.
+ *
+ */
+/// An allocator based on `GenericArray` and `MatrixVec` for statically-sized and dynamically-sized
+/// matrices respectively.
+pub struct DefaultAllocator;
+
+// Static - Static
+impl<N, R, C> Allocator<N, R, C> for DefaultAllocator
+    where N: Scalar,
+          R: DimName,
+          C: DimName,
+          R::Value: Mul<C::Value>,
+          Prod<R::Value, C::Value>: ArrayLength<N> {
+    type Buffer = MatrixArray<N, R, C>;
+
+    #[inline]
+    unsafe fn allocate_uninitialized(_: R, _: C) -> Self::Buffer {
+        mem::uninitialized()
+    }
+
+    #[inline]
+    fn allocate_from_iterator<I: IntoIterator<Item = N>>(nrows: R, ncols: C, iter: I) -> Self::Buffer {
+        let mut res = unsafe { Self::allocate_uninitialized(nrows, ncols) };
+        let mut count = 0;
+
+        for (res, e) in res.iter_mut().zip(iter.into_iter()) {
+            *res = e;
+            count += 1;
+        }
+
+        assert!(count == nrows.value() * ncols.value(),
+                "Matrix init. from iterator: iterator not long enough.");
+
+        res
+    }
+}
+
+
+// Dynamic - Static
+// Dynamic - Dynamic
+impl<N: Scalar, C: Dim> Allocator<N, Dynamic, C> for DefaultAllocator {
+    type Buffer = MatrixVec<N, Dynamic, C>;
+
+    #[inline]
+    unsafe fn allocate_uninitialized(nrows: Dynamic, ncols: C) -> Self::Buffer {
+        let mut res = Vec::new();
+        let length = nrows.value() * ncols.value();
+        res.reserve_exact(length);
+        res.set_len(length);
+
+        MatrixVec::new(nrows, ncols, res)
+    }
+
+    #[inline]
+    fn allocate_from_iterator<I: IntoIterator<Item = N>>(nrows: Dynamic, ncols: C, iter: I) -> Self::Buffer {
+        let it = iter.into_iter();
+        let res: Vec<N> = it.collect();
+        assert!(res.len() == nrows.value() * ncols.value(),
+                "Allocation from iterator error: the iterator did not yield the correct number of elements.");
+
+        MatrixVec::new(nrows, ncols, res)
+    }
+}
+
+
+// Static - Dynamic
+impl<N: Scalar, R: DimName> Allocator<N, R, Dynamic> for DefaultAllocator {
+    type Buffer = MatrixVec<N, R, Dynamic>;
+
+    #[inline]
+    unsafe fn allocate_uninitialized(nrows: R, ncols: Dynamic) -> Self::Buffer {
+        let mut res = Vec::new();
+        let length = nrows.value() * ncols.value();
+        res.reserve_exact(length);
+        res.set_len(length);
+
+        MatrixVec::new(nrows, ncols, res)
+    }
+
+    #[inline]
+    fn allocate_from_iterator<I: IntoIterator<Item = N>>(nrows: R, ncols: Dynamic, iter: I) -> Self::Buffer {
+        let it = iter.into_iter();
+        let res: Vec<N> = it.collect();
+        assert!(res.len() == nrows.value() * ncols.value(),
+                "Allocation from iterator error: the iterator did not yield the correct number of elements.");
+
+        MatrixVec::new(nrows, ncols, res)
+    }
+}

src/core/determinant.rs

@@ -0,0 +1,56 @@
+use num::One;
+
+use alga::general::{ClosedMul, ClosedSub, ClosedAdd};
+
+use core::{Scalar, SquareMatrix};
+use core::dimension::Dim;
+use core::storage::Storage;
+
+
+impl<N, D: Dim, S> SquareMatrix<N, D, S>
+    where N: Scalar + One + ClosedMul + ClosedAdd + ClosedSub,
+          S: Storage<N, D, D> {
+    /// This matrix determinant.
+    #[inline]
+    pub fn determinant(&self) -> N {
+        assert!(self.is_square(), "Unable to invert a non-square matrix.");
+        let dim = self.shape().0;
+
+        unsafe {
+            match dim {
+                0 => N::one(),
+                1 => {
+                    *self.get_unchecked(0, 0)
+                },
+                2 => {
+                    let m11 = *self.get_unchecked(0, 0); let m12 = *self.get_unchecked(0, 1);
+                    let m21 = *self.get_unchecked(1, 0); let m22 = *self.get_unchecked(1, 1);
+
+                    m11 * m22 - m21 * m12
+                },
+                3 => {
+                    let m11 = *self.get_unchecked(0, 0);
+                    let m12 = *self.get_unchecked(0, 1);
+                    let m13 = *self.get_unchecked(0, 2);
+
+                    let m21 = *self.get_unchecked(1, 0);
+                    let m22 = *self.get_unchecked(1, 1);
+                    let m23 = *self.get_unchecked(1, 2);
+
+                    let m31 = *self.get_unchecked(2, 0);
+                    let m32 = *self.get_unchecked(2, 1);
+                    let m33 = *self.get_unchecked(2, 2);
+
+                    let minor_m12_m23 = m22 * m33 - m32 * m23;
+                    let minor_m11_m23 = m21 * m33 - m31 * m23;
+                    let minor_m11_m22 = m21 * m32 - m31 * m22;
+
+                    m11 * minor_m12_m23 - m12 * minor_m11_m23 + m13 * minor_m11_m22
+                },
+                _ => {
+                    unimplemented!()
+                }
+            }
+        }
+    }
+}

src/core/dimension.rs

@@ -0,0 +1,360 @@
+#![allow(missing_docs)]
+
+//! Traits and tags for identifying the dimension of all algebraic entities.
+
+use std::fmt::Debug;
+use std::any::Any;
+use std::ops::{Add, Sub, Mul, Div};
+use typenum::{self, Unsigned, UInt, B1, Bit, UTerm, Sum, Prod, Diff, Quot};
+
+/// Dim of dynamically-sized algebraic entities.
+#[derive(Clone, Copy, Eq, PartialEq, Debug)]
+#[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
+pub struct Dynamic {
+    value: usize
+}
+
+impl Dynamic {
+    /// A dynamic size equal to `value`.
+    #[inline]
+    pub fn new(value: usize) -> Dynamic {
+        Dynamic {
+            value: value
+        }
+    }
+}
+
+/// Trait implemented by `Dynamic`.
+pub trait IsDynamic { }
+/// Trait implemented by `Dynamic` and type-level integers different from `U1`.
+pub trait IsNotStaticOne { }
+
+impl IsDynamic for Dynamic { }
+impl IsNotStaticOne for Dynamic { }
+
+/// Trait implemented by any type that can be used as a dimension. This includes type-level
+/// integers and `Dynamic` (for dimensions not known at compile-time).
+pub trait Dim: Any + Debug + Copy + PartialEq + Send {
+    /// Gets the compile-time value of `Self`. Returns `None` if it is not known, i.e., if `Self =
+    /// Dynamic`.
+    fn try_to_usize() -> Option<usize>;
+
+    /// Gets the run-time value of `self`. For type-level integers, this is the same as
+    /// `Self::try_to_usize().unwrap()`.
+    fn value(&self) -> usize;
+
+    /// Builds an instance of `Self` from a run-time value. Panics if `Self` is a type-level
+    /// integer and `dim != Self::try_to_usize().unwrap()`.
+    fn from_usize(dim: usize) -> Self;
+}
+
+impl Dim for Dynamic {
+    #[inline]
+    fn try_to_usize() -> Option<usize> {
+        None
+    }
+
+    #[inline]
+    fn from_usize(dim: usize) -> Self {
+        Dynamic::new(dim)
+    }
+
+    #[inline]
+    fn value(&self) -> usize {
+        self.value
+    }
+}
+
+/*
+ *
+ * Operations.
+ *
+ */
+
+macro_rules! dim_ops(
+    ($($DimOp:    ident, $DimNameOp: ident,
+       $Op:       ident, $op: ident,
+       $DimResOp: ident, $DimNameResOp: ident,
+       $ResOp: ident);* $(;)*) => {$(
+        pub type $DimResOp<D1, D2> = <D1 as $DimOp<D2>>::Output;
+
+        pub trait $DimOp<D: Dim>: Dim {
+            type Output: Dim;
+
+            fn $op(self, other: D) -> Self::Output;
+        }
+
+        impl<D1: DimName, D2: DimName> $DimOp<D2> for D1
+            where D1::Value: $Op<D2::Value>,
+                  $ResOp<D1::Value, D2::Value>: NamedDim {
+            type Output = <$ResOp<D1::Value, D2::Value> as NamedDim>::Name;
+
+            #[inline]
+            fn $op(self, _: D2) -> Self::Output {
+                Self::Output::name()
+            }
+        }
+
+        impl<D: Dim> $DimOp<D> for Dynamic {
+            type Output = Dynamic;
+
+            #[inline]
+            fn $op(self, other: D) -> Dynamic {
+                Dynamic::new(self.value.$op(other.value()))
+            }
+        }
+
+        impl<D: DimName> $DimOp<Dynamic> for D {
+            type Output = Dynamic;
+
+            #[inline]
+            fn $op(self, other: Dynamic) -> Dynamic {
+                Dynamic::new(self.value().$op(other.value))
+            }
+        }
+
+        pub type $DimNameResOp<D1, D2> = <D1 as $DimNameOp<D2>>::Output;
+
+        pub trait $DimNameOp<D: DimName>: DimName {
+            type Output: DimName;
+
+            fn $op(self, other: D) -> Self::Output;
+        }
+
+        impl<D1: DimName, D2: DimName> $DimNameOp<D2> for D1
+            where D1::Value: $Op<D2::Value>,
+                  $ResOp<D1::Value, D2::Value>: NamedDim {
+            type Output = <$ResOp<D1::Value, D2::Value> as NamedDim>::Name;
+
+            #[inline]
+            fn $op(self, _: D2) -> Self::Output {
+                Self::Output::name()
+            }
+        }
+   )*}
+);
+
+dim_ops!(
+    DimAdd, DimNameAdd, Add, add, DimSum,  DimNameSum,  Sum;
+    DimMul, DimNameMul, Mul, mul, DimProd, DimNameProd, Prod;
+    DimSub, DimNameSub, Sub, sub, DimDiff, DimNameDiff, Diff;
+    DimDiv, DimNameDiv, Div, div, DimQuot, DimNameQuot, Quot;
+);
+
+
+/// Trait implemented exclusively by type-level integers.
+pub trait DimName: Dim {
+    type Value: NamedDim<Name = Self>;
+
+    /// The name of this dimension, i.e., the singleton `Self`.
+    #[inline]
+    fn name() -> Self;
+
+    // FIXME: this is not a very idiomatic name.
+    /// The value of this dimension.
+    #[inline]
+    fn dim() -> usize {
+        Self::Value::to_usize()
+    }
+}
+
+pub trait NamedDim: Sized + Any + Unsigned {
+    type Name: DimName<Value = Self>;
+}
+
+#[derive(Debug, Copy, Clone, Hash, PartialEq, Eq)]
+#[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
+pub struct U1;
+
+impl Dim for U1 {
+    #[inline]
+    fn try_to_usize() -> Option<usize> {
+        Some(1)
+    }
+
+    #[inline]
+    fn from_usize(dim: usize) -> Self {
+        assert!(dim == 1, "Mismatched dimension.");
+        U1
+    }
+
+    #[inline]
+    fn value(&self) -> usize {
+        1
+    }
+}
+
+impl DimName for U1 {
+    type Value = typenum::U1;
+
+    #[inline]
+    fn name() -> Self {
+        U1
+    }
+}
+
+impl NamedDim for typenum::U1{
+    type Name = U1;
+}
+
+macro_rules! named_dimension(
+    ($($D: ident),* $(,)*) => {$(
+        #[derive(Debug, Copy, Clone, Hash, PartialEq, Eq)]
+        #[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
+        pub struct $D;
+
+        impl Dim for $D {
+            #[inline]
+            fn try_to_usize() -> Option<usize> {
+                Some(typenum::$D::to_usize())
+            }
+
+            #[inline]
+            fn from_usize(dim: usize) -> Self {
+                assert!(dim == typenum::$D::to_usize(), "Mismatched dimension.");
+                $D
+            }
+
+            #[inline]
+            fn value(&self) -> usize {
+                typenum::$D::to_usize()
+            }
+        }
+
+        impl DimName for $D {
+            type Value = typenum::$D;
+
+            #[inline]
+            fn name() -> Self {
+                $D
+            }
+        }
+
+        impl NamedDim for typenum::$D {
+            type Name = $D;
+        }
+
+        impl IsNotStaticOne for $D { }
+    )*}
+);
+
+
+// We give explicit names to all Unsigned in [0, 128[
+named_dimension!(
+    U0, /*U1,*/ U2, U3, U4, U5, U6, U7, U8, U9,
+    U10, U11, U12, U13, U14, U15, U16, U17, U18, U19,
+    U20, U21, U22, U23, U24, U25, U26, U27, U28, U29,
+    U30, U31, U32, U33, U34, U35, U36, U37, U38, U39,
+    U40, U41, U42, U43, U44, U45, U46, U47, U48, U49,
+    U50, U51, U52, U53, U54, U55, U56, U57, U58, U59,
+    U60, U61, U62, U63, U64, U65, U66, U67, U68, U69,
+    U70, U71, U72, U73, U74, U75, U76, U77, U78, U79,
+    U80, U81, U82, U83, U84, U85, U86, U87, U88, U89,
+    U90, U91, U92, U93, U94, U95, U96, U97, U98, U99,
+    U100, U101, U102, U103, U104, U105, U106, U107, U108, U109,
+    U110, U111, U112, U113, U114, U115, U116, U117, U118, U119,
+    U120, U121, U122, U123, U124, U125, U126, U127
+);
+
+// For values greater than U1023, just use the typenum binary representation directly.
+impl<A: Bit + Any + Debug + Copy + PartialEq + Send,
+     B: Bit + Any + Debug + Copy + PartialEq + Send,
+     C: Bit + Any + Debug + Copy + PartialEq + Send,
+     D: Bit + Any + Debug + Copy + PartialEq + Send,
+     E: Bit + Any + Debug + Copy + PartialEq + Send,
+     F: Bit + Any + Debug + Copy + PartialEq + Send,
+     G: Bit + Any + Debug + Copy + PartialEq + Send>
+NamedDim
+for UInt<UInt<UInt<UInt<UInt<UInt<UInt<UInt<UTerm, B1>, A>, B>, C>, D>, E>, F>, G> {
+    type Name = Self;
+}
+
+impl<A: Bit + Any + Debug + Copy + PartialEq + Send,
+     B: Bit + Any + Debug + Copy + PartialEq + Send,
+     C: Bit + Any + Debug + Copy + PartialEq + Send,
+     D: Bit + Any + Debug + Copy + PartialEq + Send,
+     E: Bit + Any + Debug + Copy + PartialEq + Send,
+     F: Bit + Any + Debug + Copy + PartialEq + Send,
+     G: Bit + Any + Debug + Copy + PartialEq + Send>
+Dim
+for UInt<UInt<UInt<UInt<UInt<UInt<UInt<UInt<UTerm, B1>, A>, B>, C>, D>, E>, F>, G> {
+    #[inline]
+    fn try_to_usize() -> Option<usize> {
+        Some(Self::to_usize())
+    }
+
+    #[inline]
+    fn from_usize(dim: usize) -> Self {
+        assert!(dim == Self::to_usize(), "Mismatched dimension.");
+        Self::new()
+    }
+
+    #[inline]
+    fn value(&self) -> usize {
+        Self::to_usize()
+    }
+}
+
+impl<A: Bit + Any + Debug + Copy + PartialEq + Send,
+     B: Bit + Any + Debug + Copy + PartialEq + Send,
+     C: Bit + Any + Debug + Copy + PartialEq + Send,
+     D: Bit + Any + Debug + Copy + PartialEq + Send,
+     E: Bit + Any + Debug + Copy + PartialEq + Send,
+     F: Bit + Any + Debug + Copy + PartialEq + Send,
+     G: Bit + Any + Debug + Copy + PartialEq + Send>
+DimName
+for UInt<UInt<UInt<UInt<UInt<UInt<UInt<UInt<UTerm, B1>, A>, B>, C>, D>, E>, F>, G> {
+    type Value = Self;
+
+    #[inline]
+    fn name() -> Self {
+        Self::new()
+    }
+}
+
+impl<A: Bit + Any + Debug + Copy + PartialEq + Send,
+     B: Bit + Any + Debug + Copy + PartialEq + Send,
+     C: Bit + Any + Debug + Copy + PartialEq + Send,
+     D: Bit + Any + Debug + Copy + PartialEq + Send,
+     E: Bit + Any + Debug + Copy + PartialEq + Send,
+     F: Bit + Any + Debug + Copy + PartialEq + Send,
+     G: Bit + Any + Debug + Copy + PartialEq + Send>
+IsNotStaticOne
+for UInt<UInt<UInt<UInt<UInt<UInt<UInt<UInt<UTerm, B1>, A>, B>, C>, D>, E>, F>, G> {
+}
+
+
+
+impl<U: Unsigned + DimName, B: Bit + Any + Debug + Copy + PartialEq + Send> NamedDim for UInt<U, B> {
+    type Name = UInt<U, B>;
+}
+
+impl<U: Unsigned + DimName, B: Bit + Any + Debug + Copy + PartialEq + Send> Dim for UInt<U, B> {
+    #[inline]
+    fn try_to_usize() -> Option<usize> {
+        Some(Self::to_usize())
+    }
+
+    #[inline]
+    fn from_usize(dim: usize) -> Self {
+        assert!(dim == Self::to_usize(), "Mismatched dimension.");
+        Self::new()
+    }
+
+    #[inline]
+    fn value(&self) -> usize {
+        Self::to_usize()
+    }
+}
+
+impl<U: Unsigned + DimName, B: Bit + Any + Debug + Copy + PartialEq + Send> DimName for UInt<U, B> {
+    type Value = UInt<U, B>;
+
+    #[inline]
+    fn name() -> Self {
+        Self::new()
+    }
+}
+
+impl<U: Unsigned + DimName, B: Bit + Any + Debug + Copy + PartialEq + Send> IsNotStaticOne for UInt<U, B> {
+}

src/core/helper.rs

@@ -0,0 +1,19 @@
+#[cfg(feature="arbitrary")]
+use quickcheck::{Arbitrary, Gen};
+use rand::{Rand, Rng};
+
+/// Simple helper function for rejection sampling
+#[cfg(feature="arbitrary")]
+#[doc(hidden)]
+#[inline]
+pub fn reject<G: Gen, F: FnMut(&T) -> bool, T: Arbitrary>(g: &mut G, f: F) -> T {
+    use std::iter;
+    iter::repeat(()).map(|_| Arbitrary::arbitrary(g)).find(f).unwrap()
+}
+
+#[doc(hidden)]
+#[inline]
+pub fn reject_rand<G: Rng, F: FnMut(&T) -> bool, T: Rand>(g: &mut G, f: F) -> T {
+    use std::iter;
+    iter::repeat(()).map(|_| Rand::rand(g)).find(f).unwrap()
+}

src/core/inverse.rs

@@ -0,0 +1,202 @@
+use approx::ApproxEq;
+
+use alga::general::Field;
+
+use core::{Scalar, Matrix, SquareMatrix, OwnedSquareMatrix};
+use core::dimension::Dim;
+use core::storage::{Storage, StorageMut};
+
+
+impl<N, D: Dim, S> SquareMatrix<N, D, S>
+    where N: Scalar + Field + ApproxEq,
+          S: Storage<N, D, D> {
+    /// Attempts to invert this matrix.
+    #[inline]
+    pub fn try_inverse(self) -> Option<OwnedSquareMatrix<N, D, S::Alloc>> {
+        let mut res = self.into_owned();
+
+        if res.shape().0 <= 3 {
+            if res.try_inverse_mut() {
+                Some(res)
+            }
+            else {
+                None
+            }
+        }
+        else {
+            gauss_jordan_inverse(res)
+        }
+    }
+}
+
+
+impl<N, D: Dim, S> SquareMatrix<N, D, S>
+    where N: Scalar + Field + ApproxEq,
+          S: StorageMut<N, D, D> {
+    /// Attempts to invert this matrix in-place. Returns `false` and leaves `self` untouched if
+    /// inversion fails.
+    #[inline]
+    pub fn try_inverse_mut(&mut self) -> bool {
+        assert!(self.is_square(), "Unable to invert a non-square matrix.");
+
+        let dim = self.shape().0;
+
+        unsafe {
+            match dim {
+                0 => true,
+                1 => {
+                    if relative_eq!(self.get_unchecked(0, 0), &N::zero()) {
+                        false
+                    }
+                    else {
+                        *self.get_unchecked_mut(0, 0) = N::one() / self.determinant();
+                        true
+                    }
+                },
+                2 => {
+                    let determinant = self.determinant();
+
+                    if relative_eq!(&determinant, &N::zero()) {
+                        false
+                    }
+                    else {
+                        let m11 = *self.get_unchecked(0, 0); let m12 = *self.get_unchecked(0, 1);
+                        let m21 = *self.get_unchecked(1, 0); let m22 = *self.get_unchecked(1, 1);
+
+                        *self.get_unchecked_mut(0, 0) =  m22 / determinant;
+                        *self.get_unchecked_mut(0, 1) = -m12 / determinant;
+
+                        *self.get_unchecked_mut(1, 0) = -m21 / determinant;
+                        *self.get_unchecked_mut(1, 1) =  m11 / determinant;
+
+                        true
+                    }
+                },
+                3 => {
+                    let m11 = *self.get_unchecked(0, 0);
+                    let m12 = *self.get_unchecked(0, 1);
+                    let m13 = *self.get_unchecked(0, 2);
+
+                    let m21 = *self.get_unchecked(1, 0);
+                    let m22 = *self.get_unchecked(1, 1);
+                    let m23 = *self.get_unchecked(1, 2);
+
+                    let m31 = *self.get_unchecked(2, 0);
+                    let m32 = *self.get_unchecked(2, 1);
+                    let m33 = *self.get_unchecked(2, 2);
+
+
+                    let minor_m12_m23 = m22 * m33 - m32 * m23;
+                    let minor_m11_m23 = m21 * m33 - m31 * m23;
+                    let minor_m11_m22 = m21 * m32 - m31 * m22;
+
+                    let determinant = m11 * minor_m12_m23 -
+                                      m12 * minor_m11_m23 +
+                                      m13 * minor_m11_m22;
+
+                    if relative_eq!(&determinant, &N::zero()) {
+                        false
+                    }
+                    else {
+                        *self.get_unchecked_mut(0, 0) = minor_m12_m23 / determinant;
+                        *self.get_unchecked_mut(0, 1) = (m13 * m32 - m33 * m12) / determinant;
+                        *self.get_unchecked_mut(0, 2) = (m12 * m23 - m22 * m13) / determinant;
+
+                        *self.get_unchecked_mut(1, 0) = -minor_m11_m23 / determinant;
+                        *self.get_unchecked_mut(1, 1) = (m11 * m33 - m31 * m13) / determinant;
+                        *self.get_unchecked_mut(1, 2) = (m13 * m21 - m23 * m11) / determinant;
+
+                        *self.get_unchecked_mut(2, 0) = minor_m11_m22  / determinant;
+                        *self.get_unchecked_mut(2, 1) = (m12 * m31 - m32 * m11) / determinant;
+                        *self.get_unchecked_mut(2, 2) = (m11 * m22 - m21 * m12) / determinant;
+
+                        true
+                    }
+                },
+                _ => {
+                    let oself = self.clone_owned();
+                    if let Some(res) = gauss_jordan_inverse(oself) {
+                        self.copy_from(&res);
+                        true
+                    }
+                    else {
+                        false
+                    }
+                }
+            }
+        }
+    }
+}
+
+
+/// Inverts the given matrix using Gauss-Jordan Ellimitation.
+fn gauss_jordan_inverse<N, D, S>(mut matrix: SquareMatrix<N, D, S>) -> Option<OwnedSquareMatrix<N, D, S::Alloc>>
+    where D: Dim,
+          N: Scalar + Field + ApproxEq,
+          S: StorageMut<N, D, D> {
+
+    assert!(matrix.is_square(), "Unable to invert a non-square matrix.");
+    let dim = matrix.data.shape().0;
+    let mut res: OwnedSquareMatrix<N, D, S::Alloc> = Matrix::identity_generic(dim, dim);
+    let dim = dim.value();
+
+    unsafe {
+        for k in 0 .. dim {
+            // Search a non-zero value on the k-th column.
+            // FIXME: would it be worth it to spend some more time searching for the
+            // max instead?
+
+            let mut n0 = k; // index of a non-zero entry.
+
+            while n0 != dim {
+                if !matrix.get_unchecked(n0, k).is_zero() {
+                    break;
+                }
+
+                n0 += 1;
+            }
+
+            if n0 == dim {
+                return None
+            }
+
+            // Swap pivot line.
+            if n0 != k {
+                for j in 0 .. dim {
+                    matrix.swap_unchecked((n0, j), (k, j));
+                    res.swap_unchecked((n0, j), (k, j));
+                }
+            }
+
+            let pivot = *matrix.get_unchecked(k, k);
+
+            for j in k .. dim {
+                let selfval = *matrix.get_unchecked(k, j) / pivot;
+                *matrix.get_unchecked_mut(k, j) = selfval;
+            }
+
+            for j in 0 .. dim {
+                let resval = *res.get_unchecked(k, j) / pivot;
+                *res.get_unchecked_mut(k, j) = resval;
+            }
+
+            for l in 0 .. dim {
+                if l != k {
+                    let normalizer = *matrix.get_unchecked(l, k);
+
+                    for j in k .. dim {
+                        let selfval = *matrix.get_unchecked(l, j) - *matrix.get_unchecked(k, j) * normalizer;
+                        *matrix.get_unchecked_mut(l, j) = selfval;
+                    }
+
+                    for j in 0 .. dim {
+                        let resval  = *res.get_unchecked(l, j) - *res.get_unchecked(k, j) * normalizer;
+                        *res.get_unchecked_mut(l, j) = resval;
+                    }
+                }
+            }
+        }
+
+        Some(res)
+    }
+}

src/core/iter.rs

@@ -0,0 +1,88 @@
+//! Matrix iterators.
+
+use std::marker::PhantomData;
+use std::mem;
+
+use core::Scalar;
+use core::dimension::Dim;
+use core::storage::{Storage, StorageMut};
+
+macro_rules! iterator {
+    (struct $Name:ident for $Storage:ident.$ptr: ident -> $Ptr:ty, $Ref:ty, $SRef: ty) => {
+
+        /// An iterator through a dense matrix with arbitrary strides matrix.
+        pub struct $Name<'a, N: Scalar, R: Dim, C: Dim, S: 'a + $Storage<N, R, C>> {
+            ptr:       $Ptr,
+            inner_ptr: $Ptr,
+            inner_end: $Ptr,
+            size:      usize, // We can't use an end pointer here because a stride might be zero.
+            strides:   (S::RStride, S::CStride),
+            _phantoms: PhantomData<($Ref, R, C, S)>
+        }
+
+        // FIXME: we need to specialize for the case where the matrix storage is owned (in which
+        // case the iterator is trivial because it does not have any stride).
+        impl<'a, N: Scalar, R: Dim, C: Dim, S: 'a + $Storage<N, R, C>> $Name<'a, N, R, C, S> {
+            /// Creates a new iterator for the given matrix storage.
+            pub fn new(storage: $SRef) -> $Name<'a, N, R, C, S> {
+                let shape   = storage.shape();
+                let strides = storage.strides();
+                let inner_offset = shape.0.value() * strides.0.value();
+                let ptr = storage.$ptr();
+
+                $Name {
+                    ptr:       ptr,
+                    inner_ptr: ptr,
+                    inner_end: unsafe { ptr.offset(inner_offset as isize) },
+                    size:      shape.0.value() * shape.1.value(),
+                    strides:   strides,
+                    _phantoms: PhantomData
+                }
+            }
+        }
+
+        impl<'a, N: Scalar, R: Dim, C: Dim, S: 'a + $Storage<N, R, C>> Iterator for $Name<'a, N, R, C, S> {
+            type Item = $Ref;
+
+            #[inline]
+            fn next(&mut self) -> Option<$Ref> {
+                unsafe {
+                    if self.size == 0 {
+                        None
+                    } else {
+                        self.size -= 1;
+
+                        // Jump to the next outer dimension if needed.
+                        if self.ptr == self.inner_end {
+                            let stride = self.strides.1.value() as isize;
+                            self.inner_end = self.ptr.offset(stride);
+                            self.ptr = self.inner_ptr.offset(stride);
+                            self.inner_ptr = self.ptr;
+                        }
+
+                        // Go to the next element.
+                        let old = self.ptr;
+
+                        let stride = self.strides.0.value() as isize;
+                        self.ptr = self.ptr.offset(stride);
+
+                        Some(mem::transmute(old))
+                    }
+                }
+            }
+
+            #[inline]
+            fn size_hint(&self) -> (usize, Option<usize>) {
+                (self.size, Some(self.size))
+            }
+
+            #[inline]
+            fn count(self) -> usize {
+                self.size_hint().0
+            }
+        }
+    }
+}
+
+iterator!(struct MatrixIter for Storage.ptr -> *const N, &'a N, &'a S);
+iterator!(struct MatrixIterMut for StorageMut.ptr_mut -> *mut N, &'a mut N, &'a mut S);

src/core/matrix_alga.rs

@@ -0,0 +1,788 @@
+use num::{Zero, One};
+
+use alga::general::{AbstractMagma, AbstractGroupAbelian, AbstractGroup, AbstractLoop,
+                    AbstractMonoid, AbstractQuasigroup, AbstractSemigroup, AbstractModule,
+                    Module, Field, RingCommutative, Real, Inverse, Additive, Multiplicative,
+                    MeetSemilattice, JoinSemilattice, Lattice, Identity,
+                    ClosedAdd, ClosedNeg, ClosedMul};
+use alga::linear::{VectorSpace, NormedSpace, InnerSpace, FiniteDimVectorSpace, FiniteDimInnerSpace};
+
+use core::{Scalar, Matrix, SquareMatrix};
+use core::dimension::{Dim, DimName};
+use core::storage::OwnedStorage;
+use core::allocator::OwnedAllocator;
+
+/*
+ *
+ * Additive structures.
+ *
+ */
+impl<N, R: DimName, C: DimName, S> Identity<Additive> for Matrix<N, R, C, S>
+    where N: Scalar + Zero,
+          S: OwnedStorage<N, R, C>,
+          S::Alloc: OwnedAllocator<N, R, C, S> {
+    #[inline]
+    fn identity() -> Self {
+        Self::from_element(N::zero())
+    }
+}
+
+impl<N, R: DimName, C: DimName, S> AbstractMagma<Additive> for Matrix<N, R, C, S>
+    where N: Scalar + ClosedAdd,
+          S: OwnedStorage<N, R, C>,
+          S::Alloc: OwnedAllocator<N, R, C, S> {
+    #[inline]
+    fn operate(&self, other: &Self) -> Self {
+        self + other
+    }
+}
+
+impl<N, R: DimName, C: DimName, S> Inverse<Additive> for Matrix<N, R, C, S>
+    where N: Scalar + ClosedNeg,
+          S: OwnedStorage<N, R, C>,
+          S::Alloc: OwnedAllocator<N, R, C, S> {
+    #[inline]
+    fn inverse(&self) -> Matrix<N, R, C, S> {
+        -self
+    }
+
+    #[inline]
+    fn inverse_mut(&mut self) {
+        *self = -self.clone()
+    }
+}
+
+macro_rules! inherit_additive_structure(
+    ($($marker: ident<$operator: ident> $(+ $bounds: ident)*),* $(,)*) => {$(
+        impl<N, R: DimName, C: DimName, S> $marker<$operator> for Matrix<N, R, C, S>
+            where N: Scalar + $marker<$operator> $(+ $bounds)*,
+                  S: OwnedStorage<N, R, C>,
+                  S::Alloc: OwnedAllocator<N, R, C, S> { }
+    )*}
+);
+
+inherit_additive_structure!(
+    AbstractSemigroup<Additive>    + ClosedAdd,
+    AbstractMonoid<Additive>       + Zero + ClosedAdd,
+    AbstractQuasigroup<Additive>   + ClosedAdd + ClosedNeg,
+    AbstractLoop<Additive>         + Zero + ClosedAdd + ClosedNeg,
+    AbstractGroup<Additive>        + Zero + ClosedAdd + ClosedNeg,
+    AbstractGroupAbelian<Additive> + Zero + ClosedAdd + ClosedNeg
+);
+
+impl<N, R: DimName, C: DimName, S> AbstractModule for Matrix<N, R, C, S>
+    where N: Scalar + RingCommutative,
+          S: OwnedStorage<N, R, C>,
+          S::Alloc: OwnedAllocator<N, R, C, S> {
+    type AbstractRing = N;
+
+    #[inline]
+    fn multiply_by(&self, n: N) -> Self {
+        self * n
+    }
+}
+
+impl<N, R: DimName, C: DimName, S> Module for Matrix<N, R, C, S>
+    where N: Scalar + RingCommutative,
+          S: OwnedStorage<N, R, C>,
+          S::Alloc: OwnedAllocator<N, R, C, S> {
+    type Ring = N;
+}
+
+impl<N, R: DimName, C: DimName, S> VectorSpace for Matrix<N, R, C, S>
+    where N: Scalar + Field,
+          S: OwnedStorage<N, R, C>,
+          S::Alloc: OwnedAllocator<N, R, C, S> {
+    type Field = N;
+}
+
+impl<N, R: DimName, C: DimName, S> FiniteDimVectorSpace for Matrix<N, R, C, S>
+    where N: Scalar + Field,
+          S: OwnedStorage<N, R, C>,
+          S::Alloc: OwnedAllocator<N, R, C, S> {
+    #[inline]
+    fn dimension() -> usize {
+        R::dim() * C::dim()
+    }
+
+    #[inline]
+    fn canonical_basis_element(i: usize) -> Self {
+        assert!(i < Self::dimension(), "Index out of bound.");
+
+        let mut res = Self::zero();
+        unsafe { *res.data.get_unchecked_linear_mut(i) = N::one(); }
+
+        res
+    }
+
+    #[inline]
+    fn dot(&self, other: &Self) -> N {
+        self.dot(other)
+    }
+
+    #[inline]
+    unsafe fn component_unchecked(&self, i: usize) -> &N {
+        self.data.get_unchecked_linear(i)
+    }
+
+    #[inline]
+    unsafe fn component_unchecked_mut(&mut self, i: usize) -> &mut N {
+        self.data.get_unchecked_linear_mut(i)
+    }
+}
+
+impl<N, R: DimName, C: DimName, S> NormedSpace for Matrix<N, R, C, S>
+    where N: Real,
+          S: OwnedStorage<N, R, C>,
+          S::Alloc: OwnedAllocator<N, R, C, S> {
+    #[inline]
+    fn norm_squared(&self) -> N {
+        self.norm_squared()
+    }
+
+    #[inline]
+    fn norm(&self) -> N {
+        self.norm()
+    }
+
+    #[inline]
+    fn normalize(&self) -> Self {
+        self.normalize()
+    }
+
+    #[inline]
+    fn normalize_mut(&mut self) -> N {
+        self.normalize_mut()
+    }
+
+    #[inline]
+    fn try_normalize(&self, min_norm: N) -> Option<Self> {
+        self.try_normalize(min_norm)
+    }
+
+    #[inline]
+    fn try_normalize_mut(&mut self, min_norm: N) -> Option<N> {
+        self.try_normalize_mut(min_norm)
+    }
+}
+
+impl<N, R: DimName, C: DimName, S> InnerSpace for Matrix<N, R, C, S>
+    where N: Real,
+          S: OwnedStorage<N, R, C>,
+          S::Alloc: OwnedAllocator<N, R, C, S> {
+    type Real = N;
+
+    #[inline]
+    fn angle(&self, other: &Self) -> N {
+        self.angle(other)
+    }
+
+    #[inline]
+    fn inner_product(&self, other: &Self) -> N {
+        self.dot(other)
+    }
+}
+
+// FIXME: specialization will greatly simplify this implementation in the future.
+// In particular:
+//   − use `x()` instead of `::canonical_basis_element`
+//   − use `::new(x, y, z)` instead of `::from_slice`
+impl<N, R: DimName, C: DimName, S> FiniteDimInnerSpace for Matrix<N, R, C, S>
+    where N: Real,
+          S: OwnedStorage<N, R, C>,
+          S::Alloc: OwnedAllocator<N, R, C, S> {
+    #[inline]
+    fn orthonormalize(vs: &mut [Matrix<N, R, C, S>]) -> usize {
+        let mut nbasis_elements = 0;
+
+        for i in 0 .. vs.len() {
+            {
+                let (elt, basis) = vs[.. i + 1].split_last_mut().unwrap();
+
+                for basis_element in &basis[.. nbasis_elements] {
+                    *elt -= &*basis_element * elt.dot(basis_element)
+                }
+            }
+
+            if vs[i].try_normalize_mut(N::zero()).is_some() {
+                // FIXME: this will be efficient on dynamically-allocated vectors but for
+                // statically-allocated ones, `.clone_from` would be better.
+                vs.swap(nbasis_elements, i);
+                nbasis_elements += 1;
+
+                // All the other vectors will be dependent.
+                if nbasis_elements == Self::dimension() {
+                    break;
+                }
+            }
+        }
+
+        nbasis_elements
+    }
+
+    #[inline]
+    fn orthonormal_subspace_basis<F>(vs: &[Self], mut f: F)
+        where F: FnMut(&Self) -> bool {
+        // FIXME: is this necessary?
+        assert!(vs.len() <= Self::dimension(), "The given set of vectors has no chance of being a free family.");
+
+        match Self::dimension() {
+            1 => {
+                if vs.len() == 0 {
+                    f(&Self::canonical_basis_element(0));
+                }
+            },
+            2 => {
+                if vs.len() == 0 {
+                    let _ = f(&Self::canonical_basis_element(0)) &&
+                            f(&Self::canonical_basis_element(1));
+                }
+                else if vs.len() == 1 {
+                    let v = &vs[0];
+                    let res = Self::from_column_slice(&[-v[1], v[0]]);
+
+                    f(&res.normalize());
+                }
+
+                // Otherwise, nothing.
+            },
+            3 => {
+                if vs.len() == 0 {
+                    let _ = f(&Self::canonical_basis_element(0)) &&
+                            f(&Self::canonical_basis_element(1)) &&
+                            f(&Self::canonical_basis_element(2));
+                }
+                else if vs.len() == 1 {
+                    let v = &vs[0];
+                    let mut a;
+
+                    if v[0].abs() > v[1].abs() {
+                        a = Self::from_column_slice(&[v[2], N::zero(), -v[0]]);
+                    }
+                    else {
+                        a = Self::from_column_slice(&[N::zero(), -v[2], v[1]]);
+                    };
+
+                    let _ = a.normalize_mut();
+
+                    if f(&a.cross(v)) {
+                        f(&a);
+                    }
+                }
+                else if vs.len() == 2 {
+                    f(&vs[0].cross(&vs[1]).normalize());
+                }
+            },
+            _ => {
+                // XXX: use a GenericArray instead.
+                let mut known_basis = Vec::new();
+
+                for v in vs.iter() {
+                    known_basis.push(v.normalize())
+                }
+
+                for i in 0 .. Self::dimension() - vs.len() {
+                    let mut elt = Self::canonical_basis_element(i);
+
+                    for v in &known_basis {
+                        elt -= v * elt.dot(v)
+                    };
+
+                    if let Some(subsp_elt) = elt.try_normalize(N::zero()) {
+                        if !f(&subsp_elt) { return };
+
+                        known_basis.push(subsp_elt);
+                    }
+                }
+            }
+        }
+    }
+}
+
+
+/*
+ *
+ *
+ * Multiplicative structures.
+ *
+ *
+ */
+impl<N, D: DimName, S> Identity<Multiplicative> for SquareMatrix<N, D, S>
+    where N: Scalar + Zero + One,
+          S: OwnedStorage<N, D, D>,
+          S::Alloc: OwnedAllocator<N, D, D, S> {
+    #[inline]
+    fn identity() -> Self {
+        Self::identity()
+    }
+}
+
+impl<N, D: DimName, S> AbstractMagma<Multiplicative> for SquareMatrix<N, D, S>
+    where N: Scalar + Zero + ClosedAdd + ClosedMul,
+          S: OwnedStorage<N, D, D>,
+          S::Alloc: OwnedAllocator<N, D, D, S> {
+    #[inline]
+    fn operate(&self, other: &Self) -> Self {
+        self * other
+    }
+}
+
+macro_rules! impl_multiplicative_structure(
+    ($($marker: ident<$operator: ident> $(+ $bounds: ident)*),* $(,)*) => {$(
+        impl<N, D: DimName, S> $marker<$operator> for SquareMatrix<N, D, S>
+            where N: Scalar + Zero + ClosedAdd + ClosedMul + $marker<$operator> $(+ $bounds)*,
+                  S: OwnedStorage<N, D, D>,
+                  S::Alloc: OwnedAllocator<N, D, D, S> { }
+    )*}
+);
+
+impl_multiplicative_structure!(
+    AbstractSemigroup<Multiplicative>,
+    AbstractMonoid<Multiplicative> + One
+);
+
+// // FIXME: Field too strong?
+// impl<N, S> Matrix for Matrix<N, S>
+//     where N: Scalar + Field,
+//           S: Storage<N> {
+//     type Field     = N;
+//     type Row       = OwnedMatrix<N, Static<U1>, S::C, S::Alloc>;
+//     type Column    = OwnedMatrix<N, S::R, Static<U1>, S::Alloc>;
+//     type Transpose = OwnedMatrix<N, S::C, S::R, S::Alloc>;
+
+//     #[inline]
+//     fn nrows(&self) -> usize {
+//         self.shape().0
+//     }
+
+//     #[inline]
+//     fn ncolumns(&self) -> usize {
+//         self.shape().1
+//     }
+
+//     #[inline]
+//     fn row(&self, row: usize) -> Self::Row {
+//         let mut res: Self::Row = ::zero();
+
+//         for (column, e) in res.iter_mut().enumerate() {
+//             *e = self[(row, column)];
+//         }
+
+//         res
+//     }
+
+//     #[inline]
+//     fn column(&self, column: usize) -> Self::Column {
+//         let mut res: Self::Column = ::zero();
+
+//         for (row, e) in res.iter_mut().enumerate() {
+//             *e = self[(row, column)];
+//         }
+
+//         res
+//     }
+
+//     #[inline]
+//     unsafe fn get_unchecked(&self, i: usize, j: usize) -> Self::Field {
+//         self.get_unchecked(i, j)
+//     }
+
+//     #[inline]
+//     fn transpose(&self) -> Self::Transpose {
+//         self.transpose()
+//     }
+// }
+
+// impl<N, S> MatrixMut for Matrix<N, S>
+//     where N: Scalar + Field,
+//           S: StorageMut<N> {
+//     #[inline]
+//     fn set_row_mut(&mut self, irow: usize, row: &Self::Row) {
+//         assert!(irow < self.shape().0, "Row index out of bounds.");
+
+//         for (icol, e) in row.iter().enumerate() {
+//             unsafe { self.set_unchecked(irow, icol, *e) }
+//         }
+//     }
+
+//     #[inline]
+//     fn set_column_mut(&mut self, icol: usize, col: &Self::Column) {
+//         assert!(icol < self.shape().1, "Column index out of bounds.");
+//         for (irow, e) in col.iter().enumerate() {
+//             unsafe { self.set_unchecked(irow, icol, *e) }
+//         }
+//     }
+
+//     #[inline]
+//     unsafe fn set_unchecked(&mut self, i: usize, j: usize, val: Self::Field) {
+//         *self.get_unchecked_mut(i, j) = val
+//     }
+// }
+
+// // FIXME: Real is needed here only for invertibility...
+// impl<N: Real> SquareMatrixMut for $t<N> {
+//     #[inline]
+//     fn from_diagonal(diag: &Self::Coordinates) -> Self {
+//         let mut res: $t<N> = ::zero();
+//         res.set_diagonal_mut(diag);
+//         res
+//     }
+
+//     #[inline]
+//     fn set_diagonal_mut(&mut self, diag: &Self::Coordinates) {
+//         for (i, e) in diag.iter().enumerate() {
+//             unsafe { self.set_unchecked(i, i, *e) }
+//         }
+//     }
+// }
+
+
+
+// Specializations depending on the dimension.
+// matrix_group_approx_impl!(common: $t, 1, $vector, $($compN),+);
+
+// // FIXME: Real is needed here only for invertibility...
+// impl<N: Real> SquareMatrix for $t<N> {
+//     type Vector = $vector<N>;
+
+//     #[inline]
+//     fn diagonal(&self) -> Self::Coordinates {
+//         $vector::new(self.m11)
+//     }
+
+//     #[inline]
+//     fn determinant(&self) -> Self::Field {
+//         self.m11
+//     }
+
+//     #[inline]
+//     fn try_inverse(&self) -> Option<Self> {
+//         let mut res = *self;
+//         if res.try_inverse_mut() {
+//             Some(res)
+//         }
+//         else {
+//             None
+//         }
+//     }
+
+//     #[inline]
+//     fn try_inverse_mut(&mut self) -> bool {
+//         if relative_eq!(&self.m11, &::zero()) {
+//             false
+//         }
+//         else {
+//             self.m11 = ::one::<N>() / ::determinant(self);
+
+//             true
+//         }
+//     }
+
+//     #[inline]
+//     fn transpose_mut(&mut self) {
+//         // no-op
+//     }
+// }
+
+//  ident, 2, $vector: ident, $($compN: ident),+) => {
+// matrix_group_approx_impl!(common: $t, 2, $vector, $($compN),+);
+
+// // FIXME: Real is needed only for inversion here.
+// impl<N: Real> SquareMatrix for $t<N> {
+//     type Vector = $vector<N>;
+
+//     #[inline]
+//     fn diagonal(&self) -> Self::Coordinates {
+//         $vector::new(self.m11, self.m22)
+//     }
+
+//     #[inline]
+//     fn determinant(&self) -> Self::Field {
+//         self.m11 * self.m22 - self.m21 * self.m12
+//     }
+
+//     #[inline]
+//     fn try_inverse(&self) -> Option<Self> {
+//         let mut res = *self;
+//         if res.try_inverse_mut() {
+//             Some(res)
+//         }
+//         else {
+//             None
+//         }
+//     }
+
+//     #[inline]
+//     fn try_inverse_mut(&mut self) -> bool {
+//         let determinant = ::determinant(self);
+
+//         if relative_eq!(&determinant, &::zero()) {
+//             false
+//         }
+//         else {
+//             *self = Matrix2::new(
+//                 self.m22 / determinant , -self.m12 / determinant,
+//                 -self.m21 / determinant, self.m11 / determinant);
+
+//             true
+//         }
+//     }
+
+//     #[inline]
+//     fn transpose_mut(&mut self) {
+//         mem::swap(&mut self.m12, &mut self.m21)
+//     }
+// }
+
+//  ident, 3, $vector: ident, $($compN: ident),+) => {
+// matrix_group_approx_impl!(common: $t, 3, $vector, $($compN),+);
+
+// // FIXME: Real is needed only for inversion here.
+// impl<N: Real> SquareMatrix for $t<N> {
+//     type Vector = $vector<N>;
+
+//     #[inline]
+//     fn diagonal(&self) -> Self::Coordinates {
+//         $vector::new(self.m11, self.m22, self.m33)
+//     }
+
+//     #[inline]
+//     fn determinant(&self) -> Self::Field {
+//         let minor_m12_m23 = self.m22 * self.m33 - self.m32 * self.m23;
+//         let minor_m11_m23 = self.m21 * self.m33 - self.m31 * self.m23;
+//         let minor_m11_m22 = self.m21 * self.m32 - self.m31 * self.m22;
+
+//         self.m11 * minor_m12_m23 - self.m12 * minor_m11_m23 + self.m13 * minor_m11_m22
+//     }
+
+//     #[inline]
+//     fn try_inverse(&self) -> Option<Self> {
+//         let mut res = *self;
+//         if res.try_inverse_mut() {
+//             Some(res)
+//         }
+//         else {
+//             None
+//         }
+//     }
+
+//     #[inline]
+//     fn try_inverse_mut(&mut self) -> bool {
+//         let minor_m12_m23 = self.m22 * self.m33 - self.m32 * self.m23;
+//         let minor_m11_m23 = self.m21 * self.m33 - self.m31 * self.m23;
+//         let minor_m11_m22 = self.m21 * self.m32 - self.m31 * self.m22;
+
+//         let determinant = self.m11 * minor_m12_m23 -
+//                           self.m12 * minor_m11_m23 +
+//                           self.m13 * minor_m11_m22;
+
+//         if relative_eq!(&determinant, &::zero()) {
+//             false
+//         }
+//         else {
+//             *self = Matrix3::new(
+//                 (minor_m12_m23 / determinant),
+//                 ((self.m13 * self.m32 - self.m33 * self.m12) / determinant),
+//                 ((self.m12 * self.m23 - self.m22 * self.m13) / determinant),
+
+//                 (-minor_m11_m23 / determinant),
+//                 ((self.m11 * self.m33 - self.m31 * self.m13) / determinant),
+//                 ((self.m13 * self.m21 - self.m23 * self.m11) / determinant),
+
+//                 (minor_m11_m22  / determinant),
+//                 ((self.m12 * self.m31 - self.m32 * self.m11) / determinant),
+//                 ((self.m11 * self.m22 - self.m21 * self.m12) / determinant)
+//                 );
+
+//             true
+//         }
+//     }
+
+//     #[inline]
+//     fn transpose_mut(&mut self) {
+//         mem::swap(&mut self.m12, &mut self.m21);
+//         mem::swap(&mut self.m13, &mut self.m31);
+//         mem::swap(&mut self.m23, &mut self.m32);
+//     }
+// }
+
+//  ident, $dimension: expr, $vector: ident, $($compN: ident),+) => {
+// matrix_group_approx_impl!(common: $t, $dimension, $vector, $($compN),+);
+
+// // FIXME: Real is needed only for inversion here.
+// impl<N: Real> SquareMatrix for $t<N> {
+//     type Vector = $vector<N>;
+
+//     #[inline]
+//     fn diagonal(&self) -> Self::Coordinates {
+//         let mut diagonal: $vector<N> = ::zero();
+
+//         for i in 0 .. $dimension {
+//             unsafe { diagonal.unsafe_set(i, self.get_unchecked(i, i)) }
+//         }
+
+//         diagonal
+//     }
+
+//     #[inline]
+//     fn determinant(&self) -> Self::Field {
+//         // FIXME: extremely naive implementation.
+//         let mut det = ::zero();
+
+//         for icol in 0 .. $dimension {
+//             let e = unsafe { self.unsafe_at((0, icol)) };
+
+//             if e != ::zero() {
+//                 let minor_mat = self.delete_row_column(0, icol);
+//                 let minor     = minor_mat.determinant();
+
+//                 if icol % 2 == 0 {
+//                     det += minor;
+//                 }
+//                 else {
+//                     det -= minor;
+//                 }
+//             }
+//         }
+
+//         det
+//     }
+
+//     #[inline]
+//     fn try_inverse(&self) -> Option<Self> {
+//         let mut res = *self;
+//         if res.try_inverse_mut() {
+//             Some(res)
+//         }
+//         else {
+//             None
+//         }
+//     }
+
+//     #[inline]
+//     fn try_inverse_mut(&mut self) -> bool {
+//         let mut res: $t<N> = ::one();
+
+//         // Inversion using Gauss-Jordan elimination
+//         for k in 0 .. $dimension {
+//             // search a non-zero value on the k-th column
+//             // FIXME: would it be worth it to spend some more time searching for the
+//             // max instead?
+
+//             let mut n0 = k; // index of a non-zero entry
+
+//             while n0 != $dimension {
+//                 if self[(n0, k)] != ::zero() {
+//                     break;
+//                 }
+
+//                 n0 = n0 + 1;
+//             }
+
+//             if n0 == $dimension {
+//                 return false
+//             }
+
+//             // swap pivot line
+//             if n0 != k {
+//                 for j in 0 .. $dimension {
+//                     self.swap((n0, j), (k, j));
+//                     res.swap((n0, j), (k, j));
+//                 }
+//             }
+
+//             let pivot = self[(k, k)];
+
+//             for j in k .. $dimension {
+//                 let selfval = self[(k, j)] / pivot;
+//                 self[(k, j)] = selfval;
+//             }
+
+//             for j in 0 .. $dimension {
+//                 let resval = res[(k, j)] / pivot;
+//                 res[(k, j)] = resval;
+//             }
+
+//             for l in 0 .. $dimension {
+//                 if l != k {
+//                     let normalizer = self[(l, k)];
+
+//                     for j in k .. $dimension {
+//                         let selfval = self[(l, j)] - self[(k, j)] * normalizer;
+//                         self[(l, j)] = selfval;
+//                     }
+
+//                     for j in 0 .. $dimension {
+//                         let resval  = res[(l, j)] - res[(k, j)] * normalizer;
+//                         res[(l, j)] = resval;
+//                     }
+//                 }
+//             }
+//         }
+
+//         *self = res;
+
+//         true
+//     }
+
+//     #[inline]
+//     fn transpose_mut(&mut self) {
+//         for i in 1 .. $dimension {
+//             for j in 0 .. i {
+//                 self.swap((i, j), (j, i))
+//             }
+//         }
+//     }
+
+
+
+
+/*
+ *
+ * Ordering
+ *
+ */
+impl<N, R: Dim, C: Dim, S> MeetSemilattice for Matrix<N, R, C, S>
+    where N: Scalar + MeetSemilattice,
+          S: OwnedStorage<N, R, C>,
+          S::Alloc: OwnedAllocator<N, R, C, S> {
+    #[inline]
+    fn meet(&self, other: &Self) -> Self {
+        self.zip_map(other, |a, b| a.meet(&b))
+    }
+}
+
+impl<N, R: Dim, C: Dim, S> JoinSemilattice for Matrix<N, R, C, S>
+    where N: Scalar + JoinSemilattice,
+          S: OwnedStorage<N, R, C>,
+          S::Alloc: OwnedAllocator<N, R, C, S> {
+    #[inline]
+    fn join(&self, other: &Self) -> Self {
+        self.zip_map(other, |a, b| a.join(&b))
+    }
+}
+
+
+impl<N, R: Dim, C: Dim, S> Lattice for Matrix<N, R, C, S>
+    where N: Scalar + Lattice,
+          S: OwnedStorage<N, R, C>,
+          S::Alloc: OwnedAllocator<N, R, C, S> {
+    #[inline]
+    fn meet_join(&self, other: &Self) -> (Self, Self) {
+        let shape = self.data.shape();
+        assert!(shape == other.data.shape(), "Matrix meet/join error: mismatched dimensions.");
+
+        let mut mres = unsafe { Self::new_uninitialized_generic(shape.0, shape.1) };
+        let mut jres = unsafe { Self::new_uninitialized_generic(shape.0, shape.1) };
+
+        for i in 0 .. shape.0.value() * shape.1.value() {
+            unsafe {
+                let mj = self.data.get_unchecked_linear(i).meet_join(other.data.get_unchecked_linear(i));
+                *mres.data.get_unchecked_linear_mut(i) = mj.0;
+                *jres.data.get_unchecked_linear_mut(i) = mj.1;
+            }
+        }
+
+        (mres, jres)
+    }
+}

src/core/matrix_array.rs

@@ -0,0 +1,298 @@
+use std::ops::{Deref, DerefMut, Mul};
+use std::fmt::{self, Debug, Formatter};
+use std::hash::{Hash, Hasher};
+
+#[cfg(feature = "serde-serialize")]
+use serde::{Serialize, Serializer, Deserialize, Deserializer};
+#[cfg(feature = "serde-serialize")]
+use serde::ser::SerializeSeq;
+#[cfg(feature = "serde-serialize")]
+use serde::de::{SeqVisitor, Visitor};
+#[cfg(feature = "serde-serialize")]
+use std::mem;
+#[cfg(feature = "serde-serialize")]
+use std::marker::PhantomData;
+
+use typenum::Prod;
+use generic_array::{ArrayLength, GenericArray};
+
+use core::Scalar;
+use core::dimension::{DimName, U1};
+use core::storage::{Storage, StorageMut, Owned, OwnedStorage};
+use core::allocator::Allocator;
+use core::default_allocator::DefaultAllocator;
+
+
+/*
+ *
+ * Static Storage.
+ *
+ */
+/// A array-based statically sized matrix data storage.
+#[repr(C)]
+pub struct MatrixArray<N, R, C>
+where R: DimName,
+      C: DimName,
+      R::Value: Mul<C::Value>,
+      Prod<R::Value, C::Value>: ArrayLength<N> {
+
+    data: GenericArray<N, Prod<R::Value, C::Value>>
+}
+
+
+impl<N, R, C> Hash for MatrixArray<N, R, C>
+where N: Hash,
+      R: DimName,
+      C: DimName,
+      R::Value: Mul<C::Value>,
+      Prod<R::Value, C::Value>: ArrayLength<N> {
+    fn hash<H: Hasher>(&self, state: &mut H) {
+        self.data[..].hash(state)
+    }
+}
+
+impl<N, R, C> Deref for MatrixArray<N, R, C>
+where R: DimName,
+      C: DimName,
+      R::Value: Mul<C::Value>,
+      Prod<R::Value, C::Value>: ArrayLength<N> {
+    type Target = GenericArray<N, Prod<R::Value, C::Value>>;
+
+    #[inline]
+    fn deref(&self) -> &Self::Target {
+        &self.data
+    }
+}
+
+impl<N, R, C> DerefMut for MatrixArray<N, R, C>
+where R: DimName,
+      C: DimName,
+      R::Value: Mul<C::Value>,
+      Prod<R::Value, C::Value>: ArrayLength<N> {
+    #[inline]
+    fn deref_mut(&mut self) -> &mut Self::Target {
+        &mut self.data
+    }
+}
+
+impl<N, R, C> Debug for MatrixArray<N, R, C>
+where N: Debug,
+      R: DimName,
+      C: DimName,
+      R::Value: Mul<C::Value>,
+      Prod<R::Value, C::Value>: ArrayLength<N> {
+    #[inline]
+    fn fmt(&self, fmt: &mut Formatter) -> fmt::Result {
+        self.data.fmt(fmt)
+    }
+}
+
+impl<N, R, C> Copy for MatrixArray<N, R, C>
+    where N: Copy,
+          R: DimName,
+          C: DimName,
+          R::Value: Mul<C::Value>,
+          Prod<R::Value, C::Value>: ArrayLength<N>,
+          GenericArray<N, Prod<R::Value, C::Value>> : Copy
+{ }
+
+impl<N, R, C> Clone for MatrixArray<N, R, C>
+    where N: Clone,
+          R: DimName,
+          C: DimName,
+          R::Value: Mul<C::Value>,
+          Prod<R::Value, C::Value>: ArrayLength<N> {
+    #[inline]
+    fn clone(&self) -> Self {
+        MatrixArray {
+            data: self.data.clone()
+        }
+    }
+}
+
+impl<N, R, C> Eq for MatrixArray<N, R, C>
+    where N: Eq,
+          R: DimName,
+          C: DimName,
+          R::Value: Mul<C::Value>,
+          Prod<R::Value, C::Value>: ArrayLength<N> {
+}
+
+impl<N, R, C> PartialEq for MatrixArray<N, R, C>
+    where N: PartialEq,
+          R: DimName,
+          C: DimName,
+          R::Value: Mul<C::Value>,
+          Prod<R::Value, C::Value>: ArrayLength<N> {
+    #[inline]
+    fn eq(&self, right: &Self) -> bool {
+        self.data  == right.data
+    }
+}
+
+
+unsafe impl<N, R, C> Storage<N, R, C> for MatrixArray<N, R, C>
+    where N: Scalar,
+          R: DimName,
+          C: DimName,
+          R::Value: Mul<C::Value>,
+          Prod<R::Value, C::Value>: ArrayLength<N> {
+    type RStride = U1;
+    type CStride = R;
+    type Alloc   = DefaultAllocator;
+
+    #[inline]
+    fn into_owned(self) -> Owned<N, R, C, Self::Alloc> {
+        self
+    }
+
+    #[inline]
+    fn clone_owned(&self) -> Owned<N, R, C, Self::Alloc> {
+        let it = self.iter().cloned();
+
+        Self::Alloc::allocate_from_iterator(self.shape().0, self.shape().1, it)
+    }
+
+    #[inline]
+    fn ptr(&self) -> *const N {
+        self[..].as_ptr()
+    }
+
+    #[inline]
+    fn shape(&self) -> (R, C) {
+        (R::name(), C::name())
+    }
+
+    #[inline]
+    fn strides(&self) -> (Self::RStride, Self::CStride) {
+        (Self::RStride::name(), Self::CStride::name())
+    }
+}
+
+unsafe impl<N, R, C> StorageMut<N, R, C> for MatrixArray<N, R, C>
+    where N: Scalar,
+          R: DimName,
+          C: DimName,
+          R::Value: Mul<C::Value>,
+          Prod<R::Value, C::Value>: ArrayLength<N> {
+    #[inline]
+    fn ptr_mut(&mut self) -> *mut N {
+        self[..].as_mut_ptr()
+    }
+}
+
+unsafe impl<N, R, C> OwnedStorage<N, R, C> for MatrixArray<N, R, C>
+    where N: Scalar,
+          R: DimName,
+          C: DimName,
+          R::Value: Mul<C::Value>,
+          Prod<R::Value, C::Value>: ArrayLength<N> {
+    #[inline]
+    fn as_slice(&self) -> &[N] {
+        &self[..]
+    }
+
+    #[inline]
+    fn as_mut_slice(&mut self) -> &mut [N] {
+        &mut self[..]
+    }
+}
+
+
+/*
+ *
+ * Allocation-less serde impls.
+ *
+ */
+// XXX: open an issue for GenericArray so that it implements serde traits?
+#[cfg(feature = "serde-serialize")]
+impl<N, R, C> Serialize for MatrixArray<N, R, C>
+where N: Scalar + Serialize,
+      R: DimName,
+      C: DimName,
+      R::Value: Mul<C::Value>,
+      Prod<R::Value, C::Value>: ArrayLength<N> {
+
+
+    fn serialize<S>(&self, serializer: S) -> Result<S::Ok, S::Error>
+        where S: Serializer {
+            let mut serializer = serializer.serialize_seq_fixed_size(R::dim() * C::dim())?;
+
+            for e in self.iter() {
+                serializer.serialize_element(e)?;
+            }
+
+            serializer.end()
+        }
+}
+
+
+#[cfg(feature = "serde-serialize")]
+impl<N, R, C> Deserialize for MatrixArray<N, R, C>
+where N: Scalar + Deserialize,
+      R: DimName,
+      C: DimName,
+      R::Value: Mul<C::Value>,
+      Prod<R::Value, C::Value>: ArrayLength<N> {
+
+
+    fn deserialize<D>(deserializer: D) -> Result<Self, D::Error>
+        where D: Deserializer {
+
+            let len = R::dim() * C::dim();
+            deserializer.deserialize_seq_fixed_size(len, MatrixArrayVisitor::new())
+        }
+}
+
+
+#[cfg(feature = "serde-serialize")]
+/// A visitor that produces a matrix array.
+struct MatrixArrayVisitor<N, R, C> {
+    marker: PhantomData<(N, R, C)>
+}
+
+#[cfg(feature = "serde-serialize")]
+impl<N, R, C> MatrixArrayVisitor<N, R, C>
+where N: Scalar,
+      R: DimName,
+      C: DimName,
+      R::Value: Mul<C::Value>,
+      Prod<R::Value, C::Value>: ArrayLength<N> {
+
+    /// Construct a new sequence visitor.
+    pub fn new() -> Self {
+        MatrixArrayVisitor {
+            marker: PhantomData,
+        }
+    }
+}
+
+#[cfg(feature = "serde-serialize")]
+impl<N, R, C> Visitor for MatrixArrayVisitor<N, R, C>
+where N: Scalar + Deserialize,
+      R: DimName,
+      C: DimName,
+      R::Value: Mul<C::Value>,
+      Prod<R::Value, C::Value>: ArrayLength<N> {
+
+    type Value = MatrixArray<N, R, C>;
+
+    fn expecting(&self, formatter: &mut Formatter) -> fmt::Result {
+        formatter.write_str("a matrix array")
+    }
+
+    #[inline]
+    fn visit_seq<V>(self, mut visitor: V) -> Result<MatrixArray<N, R, C>, V::Error>
+        where V: SeqVisitor {
+
+        let mut out: Self::Value = unsafe { mem::uninitialized() };
+        let mut curr = 0;
+
+        while let Some(value) = try!(visitor.visit()) {
+            out[curr] = value;
+            curr += 1;
+        }
+
+        Ok(out)
+    }
+}

src/core/matrix_slice.rs

@@ -0,0 +1,482 @@
+use std::marker::PhantomData;
+
+use core::{Scalar, Matrix};
+use core::dimension::{Dim, DimName, Dynamic, DimMul, DimProd, U1};
+use core::iter::MatrixIter;
+use core::storage::{Storage, StorageMut, Owned};
+use core::allocator::Allocator;
+
+macro_rules! slice_storage_impl(
+    ($doc: expr; $Storage: ident as $SRef: ty; $T: ident.$get_addr: ident ($Ptr: ty as $Ref: ty)) => {
+        #[doc = $doc]
+        pub struct $T<'a, N: Scalar, R: Dim, C: Dim, RStride: Dim, CStride: Dim, Alloc> {
+            ptr:       $Ptr,
+            shape:     (R, C),
+            strides:   (RStride, CStride),
+            _phantoms: PhantomData<($Ref, Alloc)>,
+        }
+
+        // Dynamic and () are arbitrary. It's just to be able to call the constructors with
+        // `Slice::`
+        impl<'a, N: Scalar, R: Dim, C: Dim> $T<'a, N, R, C, Dynamic, Dynamic, ()> {
+            /// Create a new matrix slice without bound checking.
+            #[inline]
+            pub unsafe fn new_unchecked<RStor, CStor, S>(storage: $SRef, start: (usize, usize), shape: (R, C))
+                -> $T<'a, N, R, C, S::RStride, S::CStride, S::Alloc>
+                where RStor: Dim,
+                      CStor: Dim,
+                      S:     $Storage<N, RStor, CStor> {
+
+                let strides = storage.strides();
+                $T::new_with_strides_unchecked(storage, start, shape, strides)
+            }
+
+            /// Create a new matrix slice without bound checking.
+            #[inline]
+            pub unsafe fn new_with_strides_unchecked<S, RStor, CStor, RStride, CStride>(storage: $SRef,
+                                                                                        start:   (usize, usize),
+                                                                                        shape:   (R, C),
+                                                                                        strides: (RStride, CStride))
+                -> $T<'a, N, R, C, RStride, CStride, S::Alloc>
+                where RStor: Dim,
+                      CStor: Dim,
+                      S: $Storage<N, RStor, CStor>,
+                      RStride: Dim,
+                      CStride: Dim {
+
+                $T {
+                    ptr:       storage.$get_addr(start.0, start.1),
+                    shape:     shape,
+                    strides:   (strides.0, strides.1),
+                    _phantoms: PhantomData
+                }
+            }
+        }
+    }
+);
+
+slice_storage_impl!("A matrix data storage for a matrix slice. Only contains an internal reference \
+                     to another matrix data storage.";
+    Storage as &'a S; SliceStorage.get_address_unchecked(*const N as &'a N));
+
+slice_storage_impl!("A mutable matrix data storage for mutable matrix slice. Only contains an \
+                     internal mutable reference to another matrix data storage.";
+    StorageMut as &'a mut S; SliceStorageMut.get_address_unchecked_mut(*mut N as &'a mut N)
+);
+
+
+impl<'a, N: Scalar, R: Dim, C: Dim, RStride: Dim, CStride: Dim, Alloc> Copy
+for SliceStorage<'a, N, R, C, RStride, CStride, Alloc> { }
+
+impl<'a, N: Scalar, R: Dim, C: Dim, RStride: Dim, CStride: Dim, Alloc> Clone
+for SliceStorage<'a, N, R, C, RStride, CStride, Alloc> {
+    #[inline]
+    fn clone(&self) -> Self {
+        SliceStorage {
+            ptr:       self.ptr,
+            shape:     self.shape,
+            strides:   self.strides,
+            _phantoms: PhantomData,
+        }
+    }
+}
+
+macro_rules! storage_impl(
+    ($($T: ident),* $(,)*) => {$(
+        unsafe impl<'a, N, R: Dim, C: Dim, RStride: Dim, CStride: Dim, Alloc> Storage<N, R, C>
+            for $T<'a, N, R, C, RStride, CStride, Alloc>
+            where N:     Scalar,
+                  Alloc: Allocator<N, R, C> {
+
+            type RStride = RStride;
+            type CStride = CStride;
+            type Alloc   = Alloc;
+
+            #[inline]
+            fn into_owned(self) -> Owned<N, R, C, Self::Alloc> {
+                self.clone_owned()
+            }
+
+            #[inline]
+            fn clone_owned(&self) -> Owned<N, R, C, Self::Alloc> {
+                let (nrows, ncols) = self.shape();
+                let it = MatrixIter::new(self).cloned();
+                Alloc::allocate_from_iterator(nrows, ncols, it)
+            }
+
+            #[inline]
+            fn ptr(&self) -> *const N {
+                self.ptr
+            }
+
+            #[inline]
+            fn shape(&self) -> (R, C) {
+                self.shape
+            }
+
+            #[inline]
+            fn strides(&self) -> (Self::RStride, Self::CStride) {
+                self.strides
+            }
+        }
+    )*}
+);
+
+storage_impl!(SliceStorage, SliceStorageMut);
+
+unsafe impl<'a, N, R: Dim, C: Dim, RStride: Dim, CStride: Dim, Alloc> StorageMut<N, R, C>
+    for SliceStorageMut<'a, N, R, C, RStride, CStride, Alloc>
+    where N: Scalar,
+          Alloc: Allocator<N, R, C> {
+    #[inline]
+    fn ptr_mut(&mut self) -> *mut N {
+        self.ptr
+    }
+}
+
+
+impl<N: Scalar, R: Dim, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S> {
+    #[inline]
+    fn assert_slice_index(&self, start: (usize, usize), shape: (usize, usize), steps: (usize, usize)) {
+        let my_shape = self.shape();
+        assert!(start.0 + (shape.0 - 1) * steps.0 <= my_shape.0, "Matrix slicing out of bounds.");
+        assert!(start.1 + (shape.1 - 1) * steps.1 <= my_shape.1, "Matrix slicing out of bounds.");
+    }
+}
+
+
+macro_rules! matrix_slice_impl(
+    ($me: ident: $Me: ty, $MatrixSlice: ident, $SliceStorage: ident, $Storage: ident, $data: expr;
+     $row: ident,
+     $rows: ident,
+     $rows_with_step: ident,
+     $fixed_rows: ident,
+     $fixed_rows_with_step: ident,
+     $rows_generic: ident,
+     $column: ident,
+     $columns: ident,
+     $columns_with_step: ident,
+     $fixed_columns: ident,
+     $fixed_columns_with_step: ident,
+     $columns_generic: ident,
+     $slice: ident,
+     $slice_with_steps: ident,
+     $fixed_slice: ident,
+     $fixed_slice_with_steps: ident,
+     $generic_slice: ident,
+     $generic_slice_with_steps: ident) => {
+        /// A matrix slice.
+        pub type $MatrixSlice<'a, N, R, C, RStride, CStride, Alloc>
+        = Matrix<N, R, C, $SliceStorage<'a, N, R, C, RStride, CStride, Alloc>>;
+
+        impl<N: Scalar, R: Dim, C: Dim, S: $Storage<N, R, C>> Matrix<N, R, C, S> {
+            /*
+             *
+             * Row slicing.
+             *
+             */
+            /// Returns a slice containing the i-th column of this matrix.
+            #[inline]
+            pub fn $row($me: $Me, i: usize) -> $MatrixSlice<N, U1, C, S::RStride, S::CStride, S::Alloc> {
+                $me.$fixed_rows::<U1>(i)
+            }
+
+            /// Extracts from this matrix a set of consecutive rows.
+            #[inline]
+            pub fn $rows($me: $Me, first_row: usize, nrows: usize)
+                -> $MatrixSlice<N, Dynamic, C, S::RStride, S::CStride, S::Alloc> {
+
+                let my_shape = $me.data.shape();
+                $me.assert_slice_index((first_row, 0), (nrows, my_shape.1.value()), (1, 1));
+                let shape = (Dynamic::new(nrows), my_shape.1);
+
+                unsafe {
+                    let data = $SliceStorage::new_unchecked($data, (first_row, 0), shape);
+                    Matrix::from_data_statically_unchecked(data)
+                }
+            }
+
+            /// Extracts from this matrix a set of consecutive rows regularly spaced by `step` rows.
+            #[inline]
+            pub fn $rows_with_step($me: $Me, first_row: usize, nrows: usize, step: usize)
+                -> $MatrixSlice<N, Dynamic, C, Dynamic, S::CStride, S::Alloc> {
+
+                $me.$rows_generic(first_row, Dynamic::new(nrows), Dynamic::new(step))
+            }
+
+            /// Extracts a compile-time number of consecutive rows from this matrix.
+            #[inline]
+            pub fn $fixed_rows<RSlice>($me: $Me, first_row: usize)
+                -> $MatrixSlice<N, RSlice, C, S::RStride, S::CStride, S::Alloc>
+                where RSlice: DimName {
+
+                let my_shape = $me.data.shape();
+                $me.assert_slice_index((first_row, 0), (RSlice::dim(), my_shape.1.value()), (1, 1));
+                let shape = (RSlice::name(), my_shape.1);
+
+                unsafe {
+                    let data = $SliceStorage::new_unchecked($data, (first_row, 0), shape);
+                    Matrix::from_data_statically_unchecked(data)
+                }
+            }
+
+            /// Extracts from this matrix a compile-time number of rows regularly spaced by `step` rows.
+            #[inline]
+            pub fn $fixed_rows_with_step<RSlice>($me: $Me, first_row: usize, step: usize)
+                -> $MatrixSlice<N, RSlice, C, Dynamic, S::CStride, S::Alloc>
+                where RSlice: DimName {
+
+                $me.$rows_generic(first_row, RSlice::name(), Dynamic::new(step))
+            }
+
+            /// Extracts from this matrix `nrows` rows regularly spaced by `step` rows. Both argument may
+            /// or may not be values known at compile-time.
+            #[inline]
+            pub fn $rows_generic<RSlice, RStep>($me: $Me, row_start: usize, nrows: RSlice, step: RStep)
+                -> $MatrixSlice<N, RSlice, C, DimProd<RStep, S::RStride>, S::CStride, S::Alloc>
+                where RSlice: Dim,
+                      RStep: DimMul<S::RStride> {
+
+                let my_shape   = $me.data.shape();
+                let my_strides = $me.data.strides();
+                $me.assert_slice_index((row_start, 0), (nrows.value(), my_shape.1.value()), (step.value(), 1));
+
+                let strides = (step.mul(my_strides.0), my_strides.1);
+                let shape   = (nrows, my_shape.1);
+
+                unsafe {
+                    let data = $SliceStorage::new_with_strides_unchecked($data, (row_start, 0), shape, strides);
+                    Matrix::from_data_statically_unchecked(data)
+                }
+            }
+
+            /*
+             *
+             * Column slicing.
+             *
+             */
+            /// Returns a slice containing the i-th column of this matrix.
+            #[inline]
+            pub fn $column($me: $Me, i: usize) -> $MatrixSlice<N, R, U1, S::RStride, S::CStride, S::Alloc> {
+                $me.$fixed_columns::<U1>(i)
+            }
+
+            /// Extracts from this matrix a set of consecutive columns.
+            #[inline]
+            pub fn $columns($me: $Me, first_col: usize, ncols: usize)
+                -> $MatrixSlice<N, R, Dynamic, S::RStride, S::CStride, S::Alloc> {
+
+                let my_shape = $me.data.shape();
+                $me.assert_slice_index((0, first_col), (my_shape.0.value(), ncols), (1, 1));
+                let shape = (my_shape.0, Dynamic::new(ncols));
+
+                unsafe {
+                    let data = $SliceStorage::new_unchecked($data, (0, first_col), shape);
+                    Matrix::from_data_statically_unchecked(data)
+                }
+            }
+
+            /// Extracts from this matrix a set of consecutive columns regularly spaced by `step` columns.
+            #[inline]
+            pub fn $columns_with_step($me: $Me, first_col: usize, ncols: usize, step: usize)
+                -> $MatrixSlice<N, R, Dynamic, S::RStride, Dynamic, S::Alloc> {
+
+                $me.$columns_generic(first_col, Dynamic::new(ncols), Dynamic::new(step))
+            }
+
+            /// Extracts a compile-time number of consecutive columns from this matrix.
+            #[inline]
+            pub fn $fixed_columns<CSlice>($me: $Me, first_col: usize)
+                -> $MatrixSlice<N, R, CSlice, S::RStride, S::CStride, S::Alloc>
+                where CSlice: DimName {
+
+                let my_shape = $me.data.shape();
+                $me.assert_slice_index((0, first_col), (my_shape.0.value(), CSlice::dim()), (1, 1));
+                let shape = (my_shape.0, CSlice::name());
+
+                unsafe {
+                    let data = $SliceStorage::new_unchecked($data, (0, first_col), shape);
+                    Matrix::from_data_statically_unchecked(data)
+                }
+            }
+
+            /// Extracts from this matrix a compile-time number of columns regularly spaced by `step`
+            /// columns.
+            #[inline]
+            pub fn $fixed_columns_with_step<CSlice>($me: $Me, first_col: usize, step: usize)
+                -> $MatrixSlice<N, R, CSlice, S::RStride, Dynamic, S::Alloc>
+                where CSlice: DimName {
+
+                $me.$columns_generic(first_col, CSlice::name(), Dynamic::new(step))
+            }
+
+            /// Extracts from this matrix `ncols` columns regularly spaced by `step` columns. Both argument may
+            /// or may not be values known at compile-time.
+            #[inline]
+            pub fn $columns_generic<CSlice, CStep>($me: $Me, first_col: usize, ncols: CSlice, step: CStep)
+                -> $MatrixSlice<N, R, CSlice, S::RStride, DimProd<CStep, S::CStride>, S::Alloc>
+                where CSlice: Dim,
+                      CStep: DimMul<S::CStride> {
+
+                let my_shape   = $me.data.shape();
+                let my_strides = $me.data.strides();
+
+                $me.assert_slice_index((0, first_col), (my_shape.0.value(), ncols.value()), (1, step.value()));
+
+                let strides = (my_strides.0, step.mul(my_strides.1));
+                let shape   = (my_shape.0, ncols);
+
+                unsafe {
+                    let data = $SliceStorage::new_with_strides_unchecked($data, (0, first_col), shape, strides);
+                    Matrix::from_data_statically_unchecked(data)
+                }
+            }
+
+            /*
+             *
+             * General slicing.
+             *
+             */
+            /// Slices this matrix starting at its component `(irow, icol)` and with `(nrows, ncols)`
+            /// consecutive elements.
+            #[inline]
+            pub fn $slice($me: $Me, start: (usize, usize), shape: (usize, usize))
+                -> $MatrixSlice<N, Dynamic, Dynamic, S::RStride, S::CStride, S::Alloc> {
+
+                $me.assert_slice_index(start, shape, (1, 1));
+                let shape = (Dynamic::new(shape.0), Dynamic::new(shape.1));
+
+                unsafe {
+                    let data = $SliceStorage::new_unchecked($data, start, shape);
+                    Matrix::from_data_statically_unchecked(data)
+                }
+            }
+
+
+            /// Slices this matrix starting at its component `(start.0, start.1)` and with
+            /// `(shape.0, shape.1)` components. Each row (resp. column) of the sliced matrix is
+            /// separated by `steps.0` (resp. `steps.1`) ignored rows (resp. columns) of the
+            /// original matrix.
+            #[inline]
+            pub fn $slice_with_steps($me: $Me, start: (usize, usize), shape: (usize, usize), steps: (usize, usize))
+                -> $MatrixSlice<N, Dynamic, Dynamic, Dynamic, Dynamic, S::Alloc> {
+                let shape = (Dynamic::new(shape.0), Dynamic::new(shape.1));
+                let steps = (Dynamic::new(steps.0), Dynamic::new(steps.1));
+
+                $me.$generic_slice_with_steps(start, shape, steps)
+            }
+
+            /// Slices this matrix starting at its component `(irow, icol)` and with `(R::dim(),
+            /// CSlice::dim())` consecutive components.
+            #[inline]
+            pub fn $fixed_slice<RSlice, CSlice>($me: $Me, irow: usize, icol: usize)
+                -> $MatrixSlice<N, RSlice, CSlice, S::RStride, S::CStride, S::Alloc>
+                where RSlice: DimName,
+                      CSlice: DimName {
+
+                $me.assert_slice_index((irow, icol), (RSlice::dim(), CSlice::dim()), (1, 1));
+                let shape = (RSlice::name(), CSlice::name());
+
+                unsafe {
+                    let data = $SliceStorage::new_unchecked($data, (irow, icol), shape);
+                    Matrix::from_data_statically_unchecked(data)
+                }
+            }
+
+            /// Slices this matrix starting at its component `(start.0, start.1)` and with
+            /// `(R::dim(), CSlice::dim())` components. Each row (resp. column) of the sliced
+            /// matrix is separated by `steps.0` (resp. `steps.1`) ignored rows (resp. columns) of
+            /// the original matrix.
+            #[inline]
+            pub fn $fixed_slice_with_steps<RSlice, CSlice>($me: $Me, start: (usize, usize), steps: (usize, usize))
+                -> $MatrixSlice<N, RSlice, CSlice, Dynamic, Dynamic, S::Alloc>
+                where RSlice: DimName,
+                      CSlice: DimName {
+                let shape = (RSlice::name(), CSlice::name());
+                let steps = (Dynamic::new(steps.0), Dynamic::new(steps.1));
+                $me.$generic_slice_with_steps(start, shape, steps)
+            }
+
+            /// Creates a slice that may or may not have a fixed size and stride.
+            #[inline]
+            pub fn $generic_slice<RSlice, CSlice>($me: $Me, start: (usize, usize), shape: (RSlice, CSlice))
+                -> $MatrixSlice<N, RSlice, CSlice, S::RStride, S::CStride, S::Alloc>
+                where RSlice: Dim,
+                      CSlice: Dim {
+
+                $me.assert_slice_index(start, (shape.0.value(), shape.1.value()), (1, 1));
+
+                unsafe {
+                    let data = $SliceStorage::new_unchecked($data, start, shape);
+                    Matrix::from_data_statically_unchecked(data)
+                }
+            }
+
+            /// Creates a slice that may or may not have a fixed size and stride.
+            #[inline]
+            pub fn $generic_slice_with_steps<RSlice, CSlice, RStep, CStep>($me: $Me,
+                                                                           start: (usize, usize),
+                                                                           shape: (RSlice, CSlice),
+                                                                           steps: (RStep, CStep))
+                -> $MatrixSlice<N, RSlice, CSlice, DimProd<RStep, S::RStride>, DimProd<CStep, S::CStride>, S::Alloc>
+                where RSlice: Dim,
+                      CSlice: Dim,
+                      RStep: DimMul<S::RStride>,
+                      CStep: DimMul<S::CStride> {
+
+                $me.assert_slice_index(start, (shape.0.value(), shape.1.value()), (steps.0.value(), steps.1.value()));
+
+                let my_strides = $me.data.strides();
+                let strides    = (steps.0.mul(my_strides.0), steps.1.mul(my_strides.1));
+
+                unsafe {
+                    let data = $SliceStorage::new_with_strides_unchecked($data, start, shape, strides);
+                    Matrix::from_data_statically_unchecked(data)
+                }
+            }
+        }
+    }
+);
+
+matrix_slice_impl!(
+     self: &Self, MatrixSlice, SliceStorage, Storage, &self.data;
+     row,
+     rows,
+     rows_with_step,
+     fixed_rows,
+     fixed_rows_with_step,
+     rows_generic,
+     column,
+     columns,
+     columns_with_step,
+     fixed_columns,
+     fixed_columns_with_step,
+     columns_generic,
+     slice,
+     slice_with_steps,
+     fixed_slice,
+     fixed_slice_with_steps,
+     generic_slice,
+     generic_slice_with_steps);
+
+
+matrix_slice_impl!(
+     self: &mut Self, MatrixSliceMut, SliceStorageMut, StorageMut, &mut self.data;
+     row_mut,
+     rows_mut,
+     rows_with_step_mut,
+     fixed_rows_mut,
+     fixed_rows_with_step_mut,
+     rows_generic_mut,
+     column_mut,
+     columns_mut,
+     columns_with_step_mut,
+     fixed_columns_mut,
+     fixed_columns_with_step_mut,
+     columns_generic_mut,
+     slice_mut,
+     slice_with_steps_mut,
+     fixed_slice_mut,
+     fixed_slice_with_steps_mut,
+     generic_slice_mut,
+     generic_slice_with_steps_mut);

src/core/matrix_vec.rs

@@ -0,0 +1,173 @@
+use std::ops::Deref;
+
+use core::Scalar;
+use core::dimension::{Dim, DimName, Dynamic, U1};
+use core::storage::{Storage, StorageMut, Owned, OwnedStorage};
+use core::default_allocator::DefaultAllocator;
+
+/*
+ *
+ * Storage.
+ *
+ */
+/// A Vec-based matrix data storage. It may be dynamically-sized.
+#[repr(C)]
+#[derive(Eq, Debug, Clone, PartialEq)]
+#[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
+pub struct MatrixVec<N, R: Dim, C: Dim> {
+    data:   Vec<N>,
+    nrows:  R,
+    ncols:  C
+}
+
+impl<N, R: Dim, C: Dim> MatrixVec<N, R, C> {
+    /// Creates a new dynamic matrix data storage from the given vector and shape.
+    #[inline]
+    pub fn new(nrows: R, ncols: C, data: Vec<N>) -> MatrixVec<N, R, C> {
+        assert!(nrows.value() * ncols.value() == data.len(), "Data storage buffer dimension mismatch.");
+        MatrixVec {
+            data:   data,
+            nrows:  nrows,
+            ncols:  ncols
+        }
+    }
+
+    /// The underlying data storage.
+    #[inline]
+    pub fn data(&self) -> &Vec<N> {
+        &self.data
+    }
+
+    /// The underlying mutable data storage.
+    ///
+    /// This is unsafe because this may cause UB if the vector is modified by the user.
+    #[inline]
+    pub unsafe fn data_mut(&mut self) -> &mut Vec<N> {
+        &mut self.data
+    }
+}
+
+impl<N, R: Dim, C: Dim> Deref for MatrixVec<N, R, C> {
+    type Target = Vec<N>;
+
+    #[inline]
+    fn deref(&self) -> &Self::Target {
+        &self.data
+    }
+}
+
+/*
+ *
+ * Dynamic − Static
+ * Dynamic − Dynamic
+ *
+ */
+unsafe impl<N: Scalar, C: Dim> Storage<N, Dynamic, C> for MatrixVec<N, Dynamic, C> {
+    type RStride = U1;
+    type CStride = Dynamic;
+    type Alloc   = DefaultAllocator;
+
+    #[inline]
+    fn into_owned(self) -> Owned<N, Dynamic, C, Self::Alloc> {
+        self
+    }
+
+    #[inline]
+    fn clone_owned(&self) -> Owned<N, Dynamic, C, Self::Alloc> {
+        self.clone()
+    }
+
+    #[inline]
+    fn ptr(&self) -> *const N {
+        self[..].as_ptr()
+    }
+
+    #[inline]
+    fn shape(&self) -> (Dynamic, C) {
+        (self.nrows, self.ncols)
+    }
+
+    #[inline]
+    fn strides(&self) -> (Self::RStride, Self::CStride) {
+        (Self::RStride::name(), self.nrows)
+    }
+}
+
+
+unsafe impl<N: Scalar, R: DimName> Storage<N, R, Dynamic> for MatrixVec<N, R, Dynamic> {
+    type RStride = U1;
+    type CStride = R;
+    type Alloc   = DefaultAllocator;
+
+    #[inline]
+    fn into_owned(self) -> Owned<N, R, Dynamic, Self::Alloc> {
+        self
+    }
+
+    #[inline]
+    fn clone_owned(&self) -> Owned<N, R, Dynamic, Self::Alloc> {
+        self.clone()
+    }
+
+    #[inline]
+    fn ptr(&self) -> *const N {
+        self[..].as_ptr()
+    }
+
+    #[inline]
+    fn shape(&self) -> (R, Dynamic) {
+        (self.nrows, self.ncols)
+    }
+
+    #[inline]
+    fn strides(&self) -> (Self::RStride, Self::CStride) {
+        (Self::RStride::name(), self.nrows)
+    }
+}
+
+
+
+
+/*
+ *
+ * StorageMut, OwnedStorage.
+ *
+ */
+unsafe impl<N: Scalar, C: Dim> StorageMut<N, Dynamic, C> for MatrixVec<N, Dynamic, C> {
+    #[inline]
+    fn ptr_mut(&mut self) -> *mut N {
+        self.as_mut_slice().as_mut_ptr()
+    }
+}
+
+unsafe impl<N: Scalar, C: Dim> OwnedStorage<N, Dynamic, C> for MatrixVec<N, Dynamic, C> {
+    #[inline]
+    fn as_slice(&self) -> &[N] {
+        &self[..]
+    }
+
+    #[inline]
+    fn as_mut_slice(&mut self) -> &mut [N] {
+        &mut self.data[..]
+    }
+}
+
+
+unsafe impl<N: Scalar, R: DimName> StorageMut<N, R, Dynamic> for MatrixVec<N, R, Dynamic> {
+    #[inline]
+    fn ptr_mut(&mut self) -> *mut N {
+        self.as_mut_slice().as_mut_ptr()
+    }
+}
+
+unsafe impl<N: Scalar, R: DimName> OwnedStorage<N, R, Dynamic> for MatrixVec<N, R, Dynamic> {
+    #[inline]
+    fn as_slice(&self) -> &[N] {
+        &self[..]
+    }
+
+    #[inline]
+    fn as_mut_slice(&mut self) -> &mut [N] {
+        &mut self.data[..]
+    }
+}

src/core/mod.rs

@@ -0,0 +1,43 @@
+//! [Reexported at the root of this crate.] Data structures for vector and matrix computations.
+
+pub mod dimension;
+pub mod constraint;
+pub mod allocator;
+pub mod storage;
+pub mod coordinates;
+mod ops;
+pub mod iter;
+pub mod default_allocator;
+
+mod scalar;
+mod matrix;
+mod construction;
+mod properties;
+mod alias;
+mod matrix_alga;
+mod determinant;
+mod inverse;
+mod conversion;
+mod matrix_slice;
+mod matrix_array;
+mod matrix_vec;
+mod cg;
+mod unit;
+mod componentwise;
+
+mod decompositions;
+
+#[doc(hidden)]
+pub mod helper;
+
+pub use self::scalar::*;
+pub use self::matrix::*;
+pub use self::unit::*;
+
+pub use self::dimension::*;
+pub use self::default_allocator::*;
+
+pub use self::alias::*;
+pub use self::matrix_slice::*;
+pub use self::matrix_array::*;
+pub use self::matrix_vec::*;

src/core/ops.rs

@@ -0,0 +1,490 @@
+use std::ops::{Add, AddAssign, Sub, SubAssign, Mul, MulAssign, Div, DivAssign, Neg,
+               Index, IndexMut};
+use num::Zero;
+
+use alga::general::{ClosedMul, ClosedDiv, ClosedAdd, ClosedSub, ClosedNeg};
+
+use core::{Scalar, Matrix, OwnedMatrix, MatrixSum, MatrixMul, MatrixTrMul};
+use core::dimension::Dim;
+use core::constraint::{ShapeConstraint, SameNumberOfRows, SameNumberOfColumns, AreMultipliable};
+use core::storage::{Storage, StorageMut, OwnedStorage};
+use core::allocator::{SameShapeAllocator, Allocator, OwnedAllocator};
+
+/*
+ *
+ * Indexing.
+ *
+ */
+impl<N: Scalar, R: Dim, C: Dim, S: Storage<N, R, C>> Index<usize> for Matrix<N, R, C, S> {
+    type Output = N;
+
+    #[inline]
+    fn index(&self, i: usize) -> &N {
+        let ij = self.vector_to_matrix_index(i);
+        &self[ij]
+    }
+}
+
+
+impl<N, R: Dim, C: Dim, S> Index<(usize, usize)> for Matrix<N, R, C, S>
+    where N: Scalar,
+          S: Storage<N, R, C> {
+    type Output = N;
+
+    #[inline]
+    fn index(&self, ij: (usize, usize)) -> &N {
+        let shape = self.shape();
+        assert!(ij.0 < shape.0 && ij.1 < shape.1, "Matrix index out of bounds.");
+
+        unsafe { self.get_unchecked(ij.0, ij.1) }
+    }
+}
+
+// Mutable versions.
+impl<N: Scalar, R: Dim, C: Dim, S: StorageMut<N, R, C>> IndexMut<usize> for Matrix<N, R, C, S> {
+    #[inline]
+    fn index_mut(&mut self, i: usize) -> &mut N {
+        let ij = self.vector_to_matrix_index(i);
+        &mut self[ij]
+    }
+}
+
+impl<N, R: Dim, C: Dim, S> IndexMut<(usize, usize)> for Matrix<N, R, C, S>
+    where N: Scalar,
+          S: StorageMut<N, R, C> {
+
+    #[inline]
+    fn index_mut(&mut self, ij: (usize, usize)) -> &mut N {
+        let shape = self.shape();
+        assert!(ij.0 < shape.0 && ij.1 < shape.1, "Matrix index out of bounds.");
+
+        unsafe { self.get_unchecked_mut(ij.0, ij.1) }
+    }
+}
+
+/*
+ *
+ * Neg
+ *
+ */
+impl<N, R: Dim, C: Dim, S> Neg for Matrix<N, R, C, S>
+    where N: Scalar + ClosedNeg,
+          S: Storage<N, R, C> {
+    type Output = OwnedMatrix<N, R, C, S::Alloc>;
+
+    #[inline]
+    fn neg(self) -> Self::Output {
+        let mut res = self.into_owned();
+        res.neg_mut();
+        res
+    }
+}
+
+impl<'a, N, R: Dim, C: Dim, S> Neg for &'a Matrix<N, R, C, S>
+    where N: Scalar + ClosedNeg,
+          S: Storage<N, R, C> {
+    type Output = OwnedMatrix<N, R, C, S::Alloc>;
+
+    #[inline]
+    fn neg(self) -> Self::Output {
+        -self.clone_owned()
+    }
+}
+
+impl<N, R: Dim, C: Dim, S> Matrix<N, R, C, S>
+    where N: Scalar + ClosedNeg,
+          S: StorageMut<N, R, C> {
+    /// Negates `self` in-place.
+    #[inline]
+    pub fn neg_mut(&mut self) {
+        for e in self.iter_mut() {
+            *e = -*e
+        }
+    }
+}
+
+/*
+ *
+ * Addition & Substraction
+ *
+ */
+macro_rules! componentwise_binop_impl(
+    ($Trait: ident, $method: ident, $bound: ident;
+     $TraitAssign: ident, $method_assign: ident) => {
+        impl<'b, N, R1, C1, R2, C2, SA, SB> $Trait<&'b Matrix<N, R2, C2, SB>> for Matrix<N, R1, C1, SA>
+            where R1: Dim, C1: Dim, R2: Dim, C2: Dim,
+                  N: Scalar + $bound,
+                  SA: Storage<N, R1, C1>,
+                  SB: Storage<N, R2, C2>,
+                  SA::Alloc: SameShapeAllocator<N, R1, C1, R2, C2, SA>,
+                  ShapeConstraint: SameNumberOfRows<R1, R2> + SameNumberOfColumns<C1, C2> {
+            type Output = MatrixSum<N, R1, C1, R2, C2, SA>;
+
+            #[inline]
+            fn $method(self, right: &'b Matrix<N, R2, C2, SB>) -> Self::Output {
+                assert!(self.shape() == right.shape(), "Matrix addition/subtraction dimensions mismatch.");
+                let mut res = self.into_owned_sum::<R2, C2>();
+
+                // XXX: optimize our iterator!
+                //
+                // Using our own iterator prevents loop unrolling, wich breaks some optimization
+                // (like SIMD). On the other hand, using the slice iterator is 4x faster.
+
+                // for (left, right) in res.iter_mut().zip(right.iter()) {
+                for (left, right) in res.as_mut_slice().iter_mut().zip(right.iter()) {
+                    *left = left.$method(*right)
+                }
+
+                res
+            }
+        }
+
+        impl<'a, N, R1, C1, R2, C2, SA, SB> $Trait<Matrix<N, R2, C2, SB>> for &'a Matrix<N, R1, C1, SA>
+            where R1: Dim, C1: Dim, R2: Dim, C2: Dim,
+                  N: Scalar + $bound,
+                  SA: Storage<N, R1, C1>,
+                  SB: Storage<N, R2, C2>,
+                  SB::Alloc: SameShapeAllocator<N, R2, C2, R1, C1, SB>,
+                  ShapeConstraint: SameNumberOfRows<R2, R1> + SameNumberOfColumns<C2, C1> {
+            type Output = MatrixSum<N, R2, C2, R1, C1, SB>;
+
+            #[inline]
+            fn $method(self, right: Matrix<N, R2, C2, SB>) -> Self::Output {
+                assert!(self.shape() == right.shape(), "Matrix addition/subtraction dimensions mismatch.");
+                let mut res = right.into_owned_sum::<R1, C1>();
+
+                // XXX: optimize our iterator!
+                //
+                // Using our own iterator prevents loop unrolling, wich breaks some optimization
+                // (like SIMD). On the other hand, using the slice iterator is 4x faster.
+
+                // for (left, right) in self.iter().zip(res.iter_mut()) {
+                for (left, right) in self.iter().zip(res.as_mut_slice().iter_mut()) {
+                    *right = left.$method(*right)
+                }
+
+                res
+            }
+        }
+
+        impl<N, R1, C1, R2, C2, SA, SB> $Trait<Matrix<N, R2, C2, SB>> for Matrix<N, R1, C1, SA>
+            where R1: Dim, C1: Dim, R2: Dim, C2: Dim,
+                  N: Scalar + $bound,
+                  SA: Storage<N, R1, C1>,
+                  SB: Storage<N, R2, C2>,
+                  SA::Alloc: SameShapeAllocator<N, R1, C1, R2, C2, SA>,
+                  ShapeConstraint: SameNumberOfRows<R1, R2> + SameNumberOfColumns<C1, C2> {
+            type Output = MatrixSum<N, R1, C1, R2, C2, SA>;
+
+            #[inline]
+            fn $method(self, right: Matrix<N, R2, C2, SB>) -> Self::Output {
+                self.$method(&right)
+            }
+        }
+
+        impl<'a, 'b, N, R1, C1, R2, C2, SA, SB> $Trait<&'b Matrix<N, R2, C2, SB>> for &'a Matrix<N, R1, C1, SA>
+            where R1: Dim, C1: Dim, R2: Dim, C2: Dim,
+                  N: Scalar + $bound,
+                  SA: Storage<N, R1, C1>,
+                  SB: Storage<N, R2, C2>,
+                  SA::Alloc: SameShapeAllocator<N, R1, C1, R2, C2, SA>,
+                  ShapeConstraint: SameNumberOfRows<R1, R2> + SameNumberOfColumns<C1, C2> {
+            type Output = MatrixSum<N, R1, C1, R2, C2, SA>;
+
+            #[inline]
+            fn $method(self, right: &'b Matrix<N, R2, C2, SB>) -> Self::Output {
+                self.clone_owned().$method(right)
+            }
+        }
+
+        impl<'b, N, R1, C1, R2, C2, SA, SB> $TraitAssign<&'b Matrix<N, R2, C2, SB>> for Matrix<N, R1, C1, SA>
+            where R1: Dim, C1: Dim, R2: Dim, C2: Dim,
+                  N: Scalar + $bound,
+                  SA: StorageMut<N, R1, C1>,
+                  SB: Storage<N, R2, C2>,
+                  ShapeConstraint: SameNumberOfRows<R1, R2> + SameNumberOfColumns<C1, C2> {
+
+            #[inline]
+            fn $method_assign(&mut self, right: &'b Matrix<N, R2, C2, SB>) {
+                assert!(self.shape() == right.shape(), "Matrix addition/subtraction dimensions mismatch.");
+                for (left, right) in self.iter_mut().zip(right.iter()) {
+                    left.$method_assign(*right)
+                }
+            }
+        }
+
+        impl<N, R1, C1, R2, C2, SA, SB> $TraitAssign<Matrix<N, R2, C2, SB>> for Matrix<N, R1, C1, SA>
+            where R1: Dim, C1: Dim, R2: Dim, C2: Dim,
+                  N: Scalar + $bound,
+                  SA: StorageMut<N, R1, C1>,
+                  SB: Storage<N, R2, C2>,
+                  ShapeConstraint: SameNumberOfRows<R1, R2> + SameNumberOfColumns<C1, C2> {
+
+            #[inline]
+            fn $method_assign(&mut self, right: Matrix<N, R2, C2, SB>) {
+                self.$method_assign(&right)
+            }
+        }
+    }
+);
+
+componentwise_binop_impl!(Add, add, ClosedAdd; AddAssign, add_assign);
+componentwise_binop_impl!(Sub, sub, ClosedSub; SubAssign, sub_assign);
+
+
+
+/*
+ *
+ * Multiplication
+ *
+ */
+
+// Matrix × Scalar
+// Matrix / Scalar
+macro_rules! componentwise_scalarop_impl(
+    ($Trait: ident, $method: ident, $bound: ident;
+     $TraitAssign: ident, $method_assign: ident) => {
+        impl<N, R: Dim, C: Dim, S> $Trait<N> for Matrix<N, R, C, S>
+            where N: Scalar + $bound,
+                  S: Storage<N, R, C> {
+            type Output = OwnedMatrix<N, R, C, S::Alloc>;
+
+            #[inline]
+            fn $method(self, rhs: N) -> Self::Output {
+                let mut res = self.into_owned();
+
+                // XXX: optimize our iterator!
+                //
+                // Using our own iterator prevents loop unrolling, wich breaks some optimization
+                // (like SIMD). On the other hand, using the slice iterator is 4x faster.
+
+                // for left in res.iter_mut() {
+                for left in res.as_mut_slice().iter_mut() {
+                    *left = left.$method(rhs)
+                }
+
+                res
+            }
+        }
+
+        impl<'a, N, R: Dim, C: Dim, S> $Trait<N> for &'a Matrix<N, R, C, S>
+            where N: Scalar + $bound,
+                  S: Storage<N, R, C> {
+            type Output = OwnedMatrix<N, R, C, S::Alloc>;
+
+            #[inline]
+            fn $method(self, rhs: N) -> Self::Output {
+                self.clone_owned().$method(rhs)
+            }
+        }
+
+        impl<N, R: Dim, C: Dim, S> $TraitAssign<N> for Matrix<N, R, C, S>
+            where N: Scalar + $bound,
+                  S: StorageMut<N, R, C> {
+            #[inline]
+            fn $method_assign(&mut self, right: N) {
+                for left in self.iter_mut() {
+                    left.$method_assign(right)
+                }
+            }
+        }
+    }
+);
+
+componentwise_scalarop_impl!(Mul, mul, ClosedMul; MulAssign, mul_assign);
+componentwise_scalarop_impl!(Div, div, ClosedDiv; DivAssign, div_assign);
+
+macro_rules! left_scalar_mul_impl(
+    ($($T: ty),* $(,)*) => {$(
+        impl<R: Dim, C: Dim, S> Mul<Matrix<$T, R, C, S>> for $T
+            where S: Storage<$T, R, C> {
+            type Output = OwnedMatrix<$T, R, C, S::Alloc>;
+
+            #[inline]
+            fn mul(self, right: Matrix<$T, R, C, S>) -> Self::Output {
+                let mut res = right.into_owned();
+
+                // XXX: optimize our iterator!
+                //
+                // Using our own iterator prevents loop unrolling, wich breaks some optimization
+                // (like SIMD). On the other hand, using the slice iterator is 4x faster.
+
+                // for right in res.iter_mut() {
+                for right in res.as_mut_slice().iter_mut() {
+                    *right = self * *right
+                }
+
+                res
+            }
+        }
+
+        impl<'b, R: Dim, C: Dim, S> Mul<&'b Matrix<$T, R, C, S>> for $T
+            where S: Storage<$T, R, C> {
+            type Output = OwnedMatrix<$T, R, C, S::Alloc>;
+
+            #[inline]
+            fn mul(self, right: &'b Matrix<$T, R, C, S>) -> Self::Output {
+                self * right.clone_owned()
+            }
+        }
+    )*}
+);
+
+left_scalar_mul_impl!(
+    u8, u16, u32, u64, usize,
+    i8, i16, i32, i64, isize,
+    f32, f64
+);
+
+
+
+// Matrix × Matrix
+impl<'a, 'b, N, R1: Dim, C1: Dim, R2: Dim, C2: Dim, SA, SB> Mul<&'b Matrix<N, R2, C2, SB>>
+for &'a Matrix<N, R1, C1, SA>
+    where N:  Scalar + Zero + ClosedAdd + ClosedMul,
+          SB: Storage<N, R2, C2>,
+          SA: Storage<N, R1, C1>,
+          SA::Alloc: Allocator<N, R1, C2>,
+          ShapeConstraint: AreMultipliable<R1, C1, R2, C2> {
+    type Output = MatrixMul<N, R1, C1, C2, SA>;
+
+    #[inline]
+    fn mul(self, right: &'b Matrix<N, R2, C2, SB>) -> Self::Output {
+        let (nrows1, ncols1) = self.shape();
+        let (nrows2, ncols2) = right.shape();
+
+        assert!(ncols1 == nrows2, "Matrix multiplication dimensions mismatch.");
+
+        let mut res: MatrixMul<N, R1, C1, C2, SA> = unsafe {
+            Matrix::new_uninitialized_generic(self.data.shape().0, right.data.shape().1)
+        };
+
+        for i in 0 .. nrows1 {
+            for j in 0 .. ncols2 {
+                let mut acc = N::zero();
+
+                unsafe {
+                    for k in 0 .. ncols1 {
+                        acc = acc + *self.get_unchecked(i, k) * *right.get_unchecked(k, j);
+                    }
+
+                    *res.get_unchecked_mut(i, j) = acc;
+                }
+            }
+        }
+
+        res
+    }
+}
+
+impl<'a, N, R1: Dim, C1: Dim, R2: Dim, C2: Dim, SA, SB> Mul<Matrix<N, R2, C2, SB>>
+for &'a Matrix<N, R1, C1, SA>
+    where N:  Scalar + Zero + ClosedAdd + ClosedMul,
+          SB: Storage<N, R2, C2>,
+          SA: Storage<N, R1, C1>,
+          SA::Alloc: Allocator<N, R1, C2>,
+          ShapeConstraint: AreMultipliable<R1, C1, R2, C2> {
+    type Output = MatrixMul<N, R1, C1, C2, SA>;
+
+    #[inline]
+    fn mul(self, right: Matrix<N, R2, C2, SB>) -> Self::Output {
+        self * &right
+    }
+}
+
+impl<'b, N, R1: Dim, C1: Dim, R2: Dim, C2: Dim, SA, SB> Mul<&'b Matrix<N, R2, C2, SB>>
+for Matrix<N, R1, C1, SA>
+    where N:  Scalar + Zero + ClosedAdd + ClosedMul,
+          SB: Storage<N, R2, C2>,
+          SA: Storage<N, R1, C1>,
+          SA::Alloc: Allocator<N, R1, C2>,
+          ShapeConstraint: AreMultipliable<R1, C1, R2, C2> {
+    type Output = MatrixMul<N, R1, C1, C2, SA>;
+
+    #[inline]
+    fn mul(self, right: &'b Matrix<N, R2, C2, SB>) -> Self::Output {
+        &self * right
+    }
+}
+
+impl<N, R1: Dim, C1: Dim, R2: Dim, C2: Dim, SA, SB> Mul<Matrix<N, R2, C2, SB>>
+for Matrix<N, R1, C1, SA>
+    where N:  Scalar + Zero + ClosedAdd + ClosedMul,
+          SB: Storage<N, R2, C2>,
+          SA: Storage<N, R1, C1>,
+          SA::Alloc: Allocator<N, R1, C2>,
+          ShapeConstraint: AreMultipliable<R1, C1, R2, C2> {
+    type Output = MatrixMul<N, R1, C1, C2, SA>;
+
+    #[inline]
+    fn mul(self, right: Matrix<N, R2, C2, SB>) -> Self::Output {
+        &self * &right
+    }
+}
+
+// FIXME: this is too restrictive:
+//    − we can't use `a *= b` when `a` is a mutable slice.
+//    − we can't use `a *= b` when C2 is not equal to C1.
+impl<N, R1, C1, R2, SA, SB> MulAssign<Matrix<N, R2, C1, SB>> for Matrix<N, R1, C1, SA>
+    where R1: Dim, C1: Dim, R2: Dim,
+          N:  Scalar + Zero + ClosedAdd + ClosedMul,
+          SB: Storage<N, R2, C1>,
+          SA: OwnedStorage<N, R1, C1>,
+          ShapeConstraint: AreMultipliable<R1, C1, R2, C1>,
+          SA::Alloc: OwnedAllocator<N, R1, C1, SA> {
+    #[inline]
+    fn mul_assign(&mut self, right: Matrix<N, R2, C1, SB>) {
+        *self = &*self * right
+    }
+}
+
+impl<'b, N, R1, C1, R2, SA, SB> MulAssign<&'b Matrix<N, R2, C1, SB>> for Matrix<N, R1, C1, SA>
+    where R1: Dim, C1: Dim, R2: Dim,
+          N:  Scalar + Zero + ClosedAdd + ClosedMul,
+          SB: Storage<N, R2, C1>,
+          SA: OwnedStorage<N, R1, C1>,
+          ShapeConstraint: AreMultipliable<R1, C1, R2, C1>,
+          // FIXME: this is too restrictive. See comments for the non-ref version.
+          SA::Alloc: OwnedAllocator<N, R1, C1, SA> {
+    #[inline]
+    fn mul_assign(&mut self, right: &'b Matrix<N, R2, C1, SB>) {
+        *self = &*self * right
+    }
+}
+
+
+impl<N, R1: Dim, C1: Dim, SA> Matrix<N, R1, C1, SA>
+    where N:  Scalar + Zero + ClosedAdd + ClosedMul,
+          SA: Storage<N, R1, C1> {
+    /// Equivalent to `self.transpose() * right`.
+    #[inline]
+    pub fn tr_mul<R2: Dim, C2: Dim, SB>(&self, right: &Matrix<N, R2, C2, SB>) -> MatrixTrMul<N, R1, C1, C2, SA>
+        where SB: Storage<N, R2, C2>,
+              SA::Alloc: Allocator<N, C1, C2>,
+              ShapeConstraint: AreMultipliable<C1, R1, R2, C2> {
+        let (nrows1, ncols1) = self.shape();
+        let (nrows2, ncols2) = right.shape();
+
+        assert!(nrows1 == nrows2, "Matrix multiplication dimensions mismatch.");
+
+        let mut res: MatrixTrMul<N, R1, C1, C2, SA> = unsafe {
+            Matrix::new_uninitialized_generic(self.data.shape().1, right.data.shape().1)
+        };
+
+        for i in 0 .. ncols1 {
+            for j in 0 .. ncols2 {
+                let mut acc = N::zero();
+
+                unsafe {
+                    for k in 0 .. nrows1 {
+                        acc = acc + *self.get_unchecked(k, i) * *right.get_unchecked(k, j);
+                    }
+
+                    *res.get_unchecked_mut(i, j) = acc;
+                }
+            }
+        }
+
+        res
+    }
+}

src/core/properties.rs

@@ -0,0 +1,107 @@
+// Matrix properties checks.
+use num::{Zero, One};
+use approx::ApproxEq;
+
+use alga::general::{ClosedAdd, ClosedMul, ClosedSub, Field};
+
+use core::{Scalar, Matrix, SquareMatrix};
+use core::dimension::Dim;
+use core::storage::Storage;
+
+
+impl<N: Scalar, R: Dim, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S> {
+    /// Indicates if this is a square matrix.
+    #[inline]
+    pub fn is_square(&self) -> bool {
+        let shape = self.shape();
+        shape.0 == shape.1
+    }
+}
+
+impl<N: Scalar, R: Dim, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S>
+    // FIXME: ApproxEq prevents us from using those methods on integer matrices…
+    where N: ApproxEq,
+          N::Epsilon: Copy {
+    /// Indicated if this is the identity matrix within a relative error of `eps`.
+    ///
+    /// If the matrix is diagonal, this checks that diagonal elements (i.e. at coordinates `(i, i)`
+    /// for i from `0` to `min(R, C)`) are equal one; and that all other elements are zero.
+    #[inline]
+    pub fn is_identity(&self, eps: N::Epsilon) -> bool
+        where N: Zero + One {
+        let (nrows, ncols) = self.shape();
+        let d;
+
+        if nrows > ncols {
+            d = ncols;
+
+            for i in d .. nrows {
+                for j in 0 .. ncols {
+                    if !relative_eq!(self[(i, j)], N::zero(), epsilon = eps) {
+                        return false;
+                    }
+                }
+            }
+        }
+        else { // nrows <= ncols
+            d = nrows;
+
+            for i in 0 .. nrows {
+                for j in d .. ncols {
+                    if !relative_eq!(self[(i, j)], N::zero(), epsilon = eps) {
+                        return false;
+                    }
+                }
+            }
+        }
+
+        // Off-diagonal elements of the sub-square matrix.
+        for i in 1 .. d {
+            for j in 0 .. i {
+                // FIXME: use unsafe indexing.
+                if !relative_eq!(self[(i, j)], N::zero(), epsilon = eps) ||
+                   !relative_eq!(self[(j, i)], N::zero(), epsilon = eps) {
+                    return false;
+                }
+            }
+        }
+
+        // Diagonal elements of the sub-square matrix.
+        for i in 0 .. d {
+            if !relative_eq!(self[(i, i)], N::one(), epsilon = eps) {
+                return false;
+            }
+        }
+
+        true
+    }
+}
+
+
+impl<N: Scalar + ApproxEq, D: Dim, S: Storage<N, D, D>> SquareMatrix<N, D, S>
+    where N: Zero + One + ClosedAdd + ClosedMul,
+          N::Epsilon: Copy {
+    /// Checks that this matrix is orthogonal, i.e., that it is square and `M × Mᵀ = Id`.
+    ///
+    /// In this definition `Id` is approximately equal to the identity matrix with a relative error
+    /// equal to `eps`.
+    #[inline]
+    pub fn is_orthogonal(&self, eps: N::Epsilon) -> bool {
+        self.is_square() && (self.tr_mul(self)).is_identity(eps)
+    }
+
+    /// Checks that this matrix is orthogonal and has a determinant equal to 1.
+    #[inline]
+    pub fn is_special_orthogonal(&self, eps: N::Epsilon) -> bool
+        where N: ClosedSub + PartialOrd {
+            self.is_orthogonal(eps) && self.determinant() > N::zero()
+    }
+
+    /// Returns `true` if this matrix is invertible.
+    #[inline]
+    pub fn is_invertible(&self) -> bool
+        where N: Field {
+        // FIXME: improve this?
+        self.clone_owned().try_inverse().is_some()
+    }
+}

src/core/scalar.rs

@@ -0,0 +1,9 @@
+use std::fmt::Debug;
+use std::any::Any;
+
+/// The basic scarar type for all structures of `nalgebra`.
+///
+/// This does not make any assumption on the algebraic properties of `Self`.
+pub trait Scalar: Copy + PartialEq + Debug + Any {
+}
+impl<T: Copy + PartialEq + Debug + Any> Scalar for T { }

src/core/storage.rs

@@ -0,0 +1,184 @@
+//! Abstract definition of a matrix data storage.
+
+use std::mem;
+use std::any::Any;
+
+use core::Scalar;
+use dimension::Dim;
+use allocator::{Allocator, SameShapeR, SameShapeC};
+
+/*
+ * Aliases for sum storage.
+ */
+/// The data storage for the sum of two matrices with dimensions `(R1, C1)` and `(R2, C2)`.
+pub type SumStorage<N, R1, C1, R2, C2, SA> =
+    <<SA as Storage<N, R1, C1>>::Alloc as Allocator<N, SameShapeR<R1, R2>, SameShapeC<C1, C2>>>::Buffer;
+
+/*
+ * Aliases for multiplication storage.
+ */
+/// The data storage for the multiplication of two matrices with dimensions `(R1, C1)` on the left
+/// hand side, and with `C2` columns on the right hand side.
+pub type MulStorage<N, R1, C1, C2, SA> =
+    <<SA as Storage<N, R1, C1>>::Alloc as Allocator<N, R1, C2>>::Buffer;
+
+/// The data storage for the multiplication of two matrices with dimensions `(R1, C1)` on the left
+/// hand side, and with `C2` columns on the right hand side. The first matrix is implicitly
+/// transposed.
+pub type TrMulStorage<N, R1, C1, C2, SA> =
+    <<SA as Storage<N, R1, C1>>::Alloc as Allocator<N, C1, C2>>::Buffer;
+
+/*
+ * Alias for allocation result.
+ */
+/// The owned data storage that can be allocated from `S`.
+pub type Owned<N, R, C, A> =
+    <A as Allocator<N, R, C>>::Buffer;
+
+
+/// The trait shared by all matrix data storage.
+///
+/// FIXME: doc
+///
+/// Note that `Self` must always have a number of elements compatible with the matrix length (given
+/// by `R` and `C` if they are known at compile-time). For example, implementors of this trait
+/// should **not** allow the user to modify the size of the underlying buffer with safe methods
+/// (for example the `MatrixVec::data_mut` method is unsafe because the user could change the
+/// vector's size so that it no longer contains enough elements: this will lead to UB.
+pub unsafe trait Storage<N: Scalar, R: Dim, C: Dim>: Sized {
+    /// The static stride of this storage's rows.
+    type RStride: Dim;
+
+    /// The static stride of this storage's columns.
+    type CStride: Dim;
+
+    /// The allocator for this family of storage.
+    type Alloc: Allocator<N, R, C>;
+
+    /// Builds a matrix data storage that does not contain any reference.
+    fn into_owned(self) -> Owned<N, R, C, Self::Alloc>;
+
+    /// Clones this data storage into one that does not contain any reference.
+    fn clone_owned(&self) -> Owned<N, R, C, Self::Alloc>;
+
+    /// The matrix data pointer.
+    fn ptr(&self) -> *const N;
+
+    /// The dimension of the matrix at run-time. Arr length of zero indicates the additive identity
+    /// element of any dimension. Must be equal to `Self::dimension()` if it is not `None`.
+    fn shape(&self) -> (R, C);
+
+    /// The spacing between concecutive row elements and consecutive column elements.
+    ///
+    /// For example this returns `(1, 5)` for a row-major matrix with 5 columns.
+    fn strides(&self) -> (Self::RStride, Self::CStride);
+
+    /// Compute the index corresponding to the irow-th row and icol-th column of this matrix. The
+    /// index must be such that the following holds:
+    ///
+    /// ```.ignore
+    /// let lindex = self.linear_index(irow, icol);
+    /// assert!(*self.get_unchecked(irow, icol) == *self.get_unchecked_linear(lindex)
+    /// ```
+    #[inline]
+    fn linear_index(&self, irow: usize, icol: usize) -> usize {
+        let (rstride, cstride) = self.strides();
+
+        irow * rstride.value() + icol * cstride.value()
+    }
+
+    /// Gets the address of the i-th matrix component without performing bound-checking.
+    #[inline]
+    unsafe fn get_address_unchecked_linear(&self, i: usize) -> *const N {
+        self.ptr().offset(i as isize)
+    }
+
+    /// Gets the address of the i-th matrix component without performing bound-checking.
+    #[inline]
+    unsafe fn get_address_unchecked(&self, irow: usize, icol: usize) -> *const N {
+        self.get_address_unchecked_linear(self.linear_index(irow, icol))
+    }
+
+    /// Retrieves a reference to the i-th element without bound-checking.
+    #[inline]
+    unsafe fn get_unchecked_linear(&self, i: usize) -> &N {
+        &*self.get_address_unchecked_linear(i)
+    }
+
+    /// Retrieves a reference to the i-th element without bound-checking.
+    #[inline]
+    unsafe fn get_unchecked(&self, irow: usize, icol: usize) -> &N {
+        self.get_unchecked_linear(self.linear_index(irow, icol))
+    }
+}
+
+
+/// Trait implemented by matrix data storage that can provide a mutable access to its elements.
+///
+/// Note that a mutable access does not mean that the matrix owns its data. For example, a mutable
+/// matrix slice can provide mutable access to its elements even if it does not own its data (it
+/// contains only an internal reference to them).
+pub unsafe trait StorageMut<N: Scalar, R: Dim, C: Dim>: Storage<N, R, C> {
+    /// The matrix mutable data pointer.
+    fn ptr_mut(&mut self) -> *mut N;
+
+    /// Gets the mutable address of the i-th matrix component without performing bound-checking.
+    #[inline]
+    unsafe fn get_address_unchecked_linear_mut(&mut self, i: usize) -> *mut N {
+        self.ptr_mut().offset(i as isize)
+    }
+
+    /// Gets the mutable address of the i-th matrix component without performing bound-checking.
+    #[inline]
+    unsafe fn get_address_unchecked_mut(&mut self, irow: usize, icol: usize) -> *mut N {
+        let lid = self.linear_index(irow, icol);
+        self.get_address_unchecked_linear_mut(lid)
+    }
+
+    /// Retrieves a mutable reference to the i-th element without bound-checking.
+    unsafe fn get_unchecked_linear_mut(&mut self, i: usize) -> &mut N {
+        &mut *self.get_address_unchecked_linear_mut(i)
+    }
+
+    /// Retrieves a mutable reference to the element at `(irow, icol)` without bound-checking.
+    #[inline]
+    unsafe fn get_unchecked_mut(&mut self, irow: usize, icol: usize) -> &mut N {
+        &mut *self.get_address_unchecked_mut(irow, icol)
+    }
+
+    /// Swaps two elements using their linear index without bound-checking.
+    #[inline]
+    unsafe fn swap_unchecked_linear(&mut self, i1: usize, i2: usize) {
+        let a = self.get_address_unchecked_linear_mut(i1);
+        let b = self.get_address_unchecked_linear_mut(i2);
+
+        mem::swap(&mut *a, &mut *b);
+    }
+
+    /// Swaps two elements without bound-checking.
+    #[inline]
+    unsafe fn swap_unchecked(&mut self, row_col1: (usize, usize), row_col2: (usize, usize)) {
+        let lid1 = self.linear_index(row_col1.0, row_col1.1);
+        let lid2 = self.linear_index(row_col2.0, row_col2.1);
+
+        self.swap_unchecked_linear(lid1, lid2)
+    }
+}
+
+/// A matrix storage that does not contain any reference and that is stored contiguously in memory.
+///
+/// The storage requirement means that for any value of `i` in `[0, nrows * ncols[`, the value
+/// `.get_unchecked_linear` succeeds. This trait is unsafe because failing to comply to this may
+/// cause Undefined Behaviors.
+pub unsafe trait OwnedStorage<N: Scalar, R: Dim, C: Dim>: StorageMut<N, R, C> + Clone + Any
+    where Self::Alloc: Allocator<N, R, C, Buffer = Self> {
+    // NOTE: We could auto-impl those two methods but we don't to make sure the user is aware that
+    // data must be contiguous.
+    /// Converts this data storage to a slice.
+    #[inline]
+    fn as_slice(&self) -> &[N];
+
+    /// Converts this data storage to a mutable slice.
+    #[inline]
+    fn as_mut_slice(&mut self) -> &mut [N];
+}

src/core/unit.rs

@@ -0,0 +1,176 @@
+use std::mem;
+use std::ops::{Neg, Deref};
+use approx::ApproxEq;
+
+use alga::general::SubsetOf;
+use alga::linear::NormedSpace;
+
+
+/// A wrapper that ensures the undelying algebraic entity has a unit norm.
+///
+/// Use `.as_ref()` or `.unwrap()` to obtain the undelying value by-reference or by-move.
+#[repr(C)]
+#[derive(Eq, PartialEq, Clone, Hash, Debug, Copy)]
+#[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
+pub struct Unit<T> {
+    value: T
+}
+
+impl<T: NormedSpace> Unit<T> {
+    /// Normalize the given value and return it wrapped on a `Unit` structure.
+    #[inline]
+    pub fn new_normalize(value: T) -> Self {
+        Self::new_and_get(value).0
+    }
+
+    /// Attempts to normalize the given value and return it wrapped on a `Unit` structure.
+    ///
+    /// Returns `None` if the norm was smaller or equal to `min_norm`.
+    #[inline]
+    pub fn try_new(value: T, min_norm: T::Field) -> Option<Self> {
+        Self::try_new_and_get(value, min_norm).map(|res| res.0)
+    }
+
+    /// Normalize the given value and return it wrapped on a `Unit` structure and its norm.
+    #[inline]
+    pub fn new_and_get(mut value: T) -> (Self, T::Field) {
+        let n = value.normalize_mut();
+
+        (Unit { value: value }, n)
+    }
+
+    /// Normalize the given value and return it wrapped on a `Unit` structure and its norm.
+    ///
+    /// Returns `None` if the norm was smaller or equal to `min_norm`.
+    #[inline]
+    pub fn try_new_and_get(mut value: T, min_norm: T::Field) -> Option<(Self, T::Field)> {
+        if let Some(n) = value.try_normalize_mut(min_norm) {
+            Some((Unit { value: value }, n))
+        }
+        else {
+            None
+        }
+    }
+
+    /// Normalizes this value again. This is useful when repeated computations
+    /// might cause a drift in the norm because of float inaccuracies.
+    ///
+    /// Returns the norm before re-normalization (should be close to `1.0`).
+    #[inline]
+    pub fn renormalize(&mut self) -> T::Field {
+        self.value.normalize_mut()
+    }
+}
+
+impl<T> Unit<T> {
+    /// Wraps the given value, assuming it is already normalized.
+    ///
+    /// This function is not safe because `value` is not verified to be actually normalized.
+    #[inline]
+    pub fn new_unchecked(value: T) -> Self {
+        Unit { value: value }
+    }
+
+    /// Retrieves the underlying value.
+    #[inline]
+    pub fn unwrap(self) -> T {
+        self.value
+    }
+
+    /// Returns a mutable reference to the underlying value. This is `_unchecked` because modifying
+    /// the underlying value in such a way that it no longer has unit length may lead to unexpected
+    /// results.
+    #[inline]
+    pub fn as_mut_unchecked(&mut self) -> &mut T {
+        &mut self.value
+    }
+}
+
+impl<T> AsRef<T> for Unit<T> {
+    #[inline]
+    fn as_ref(&self) -> &T {
+        &self.value
+    }
+}
+
+/*
+ *
+ * Conversions.
+ *
+ */
+impl<T: NormedSpace> SubsetOf<T> for Unit<T>
+where T::Field: ApproxEq {
+    #[inline]
+    fn to_superset(&self) -> T {
+        self.clone().unwrap()
+    }
+
+    #[inline]
+    fn is_in_subset(value: &T) -> bool {
+        relative_eq!(value.norm_squared(), ::one())
+    }
+
+    #[inline]
+    unsafe fn from_superset_unchecked(value: &T) -> Self {
+        Unit::new_normalize(value.clone()) // We still need to re-normalize because the condition is inexact.
+    }
+}
+
+// impl<T: ApproxEq> ApproxEq for Unit<T> {
+//     type Epsilon = T::Epsilon;
+// 
+//     #[inline]
+//     fn default_epsilon() -> Self::Epsilon {
+//         T::default_epsilon()
+//     }
+// 
+//     #[inline]
+//     fn default_max_relative() -> Self::Epsilon {
+//         T::default_max_relative()
+//     }
+// 
+//     #[inline]
+//     fn default_max_ulps() -> u32 {
+//         T::default_max_ulps()
+//     }
+// 
+//     #[inline]
+//     fn relative_eq(&self, other: &Self, epsilon: Self::Epsilon, max_relative: Self::Epsilon) -> bool {
+//         self.value.relative_eq(&other.value, epsilon, max_relative)
+//     }
+// 
+//     #[inline]
+//     fn ulps_eq(&self, other: &Self, epsilon: Self::Epsilon, max_ulps: u32) -> bool {
+//         self.value.ulps_eq(&other.value, epsilon, max_ulps)
+//     }
+// }
+
+
+// FIXME:re-enable this impl when spacialization is possible.
+// Currently, it is disabled so that we can have a nice output for the `UnitQuaternion` display.
+/*
+impl<T: fmt::Display> fmt::Display for Unit<T> {
+    // XXX: will not always work correctly due to rounding errors.
+    fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
+        self.value.fmt(f)
+    }
+}
+*/
+
+impl<T: Neg> Neg for Unit<T> {
+    type Output = Unit<T::Output>;
+
+    #[inline]
+    fn neg(self) -> Self::Output {
+        Unit::new_unchecked(-self.value)
+    }
+}
+
+impl<T> Deref for Unit<T> {
+    type Target = T;
+
+    #[inline]
+    fn deref(&self) -> &T {
+        unsafe { mem::transmute(self) }
+    }
+}

src/geometry/isometry.rs

@@ -0,0 +1,206 @@
+use std::fmt;
+use std::marker::PhantomData;
+use approx::ApproxEq;
+
+use alga::general::{Real, SubsetOf};
+use alga::linear::Rotation;
+
+use core::{Scalar, OwnedSquareMatrix};
+use core::dimension::{DimName, DimNameSum, DimNameAdd, U1};
+use core::storage::{Storage, OwnedStorage};
+use core::allocator::{Allocator, OwnedAllocator};
+use geometry::{TranslationBase, PointBase};
+
+
+/// An isometry that uses a data storage deduced from the allocator `A`.
+pub type OwnedIsometryBase<N, D, A, R> =
+    IsometryBase<N, D, <A as Allocator<N, D, U1>>::Buffer, R>;
+
+/// A direct isometry, i.e., a rotation followed by a translation.
+#[repr(C)]
+#[derive(Hash, Debug, Clone, Copy)]
+#[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
+pub struct IsometryBase<N: Scalar, D: DimName, S, R> {
+    /// The pure rotational part of this isometry.
+    pub rotation:    R,
+    /// The pure translational part of this isometry.
+    pub translation: TranslationBase<N, D, S>,
+
+
+    // One dummy private field just to prevent explicit construction.
+    #[cfg_attr(feature = "serde-serialize", serde(skip_serializing, skip_deserializing))]
+    _noconstruct: PhantomData<N>
+}
+
+impl<N, D: DimName, S, R> IsometryBase<N, D, S, R>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          R: Rotation<PointBase<N, D, S>>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    /// Creates a new isometry from its rotational and translational parts.
+    #[inline]
+    pub fn from_parts(translation: TranslationBase<N, D, S>, rotation: R) -> IsometryBase<N, D, S, R> {
+        IsometryBase {
+            rotation:     rotation,
+            translation:  translation,
+            _noconstruct: PhantomData
+        }
+    }
+
+    /// Inverts `self`.
+    #[inline]
+    pub fn inverse(&self) -> IsometryBase<N, D, S, R> {
+        let mut res = self.clone();
+        res.inverse_mut();
+        res
+    }
+
+    /// Inverts `self`.
+    #[inline]
+    pub fn inverse_mut(&mut self) {
+        self.rotation.inverse_mut();
+        self.translation.inverse_mut();
+        self.translation.vector = self.rotation.transform_vector(&self.translation.vector);
+    }
+
+    /// Appends to `self` the given translation in-place.
+    #[inline]
+    pub fn append_translation_mut(&mut self, t: &TranslationBase<N, D, S>) {
+        self.translation.vector += &t.vector
+    }
+
+    /// Appends to `self` the given rotation in-place.
+    #[inline]
+    pub fn append_rotation_mut(&mut self, r: &R) {
+        self.rotation           = self.rotation.append_rotation(&r);
+        self.translation.vector = r.transform_vector(&self.translation.vector);
+    }
+
+    /// Appends in-place to `self` a rotation centered at the point `p`, i.e., the rotation that
+    /// lets `p` invariant.
+    #[inline]
+    pub fn append_rotation_wrt_point_mut(&mut self, r: &R, p: &PointBase<N, D, S>) {
+        self.translation.vector -= &p.coords;
+        self.append_rotation_mut(r);
+        self.translation.vector += &p.coords;
+    }
+
+    /// Appends in-place to `self` a rotation centered at the point with coordinates
+    /// `self.translation`.
+    #[inline]
+    pub fn append_rotation_wrt_center_mut(&mut self, r: &R) {
+        let center = PointBase::from_coordinates(self.translation.vector.clone());
+        self.append_rotation_wrt_point_mut(r, &center)
+    }
+}
+
+// NOTE: we don't require `R: Rotation<...>` here because this is not useful for the implementation
+// and makes it hard to use it, e.g., for Transform × Isometry implementation.
+// This is OK since all constructors of the isometry enforce the Rotation bound already (and
+// explicit struct construction is prevented by the dummy ZST field).
+impl<N, D: DimName, S, R> IsometryBase<N, D, S, R>
+    where N: Scalar,
+          S: Storage<N, D, U1> {
+    /// Converts this isometry into its equivalent homogeneous transformation matrix.
+    #[inline]
+    pub fn to_homogeneous(&self) -> OwnedSquareMatrix<N, DimNameSum<D, U1>, S::Alloc>
+        where D: DimNameAdd<U1>,
+              R: SubsetOf<OwnedSquareMatrix<N, DimNameSum<D, U1>, S::Alloc>>,
+              S::Alloc: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1>> {
+        let mut res: OwnedSquareMatrix<N, _, S::Alloc> = ::convert_ref(&self.rotation);
+        res.fixed_slice_mut::<D, U1>(0, D::dim()).copy_from(&self.translation.vector);
+
+        res
+    }
+}
+
+
+impl<N, D: DimName, S, R> Eq for IsometryBase<N, D, S, R>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          R: Rotation<PointBase<N, D, S>> + Eq,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+}
+
+impl<N, D: DimName, S, R> PartialEq for IsometryBase<N, D, S, R>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          R: Rotation<PointBase<N, D, S>> + PartialEq,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    #[inline]
+    fn eq(&self, right: &IsometryBase<N, D, S, R>) -> bool {
+        self.translation == right.translation &&
+        self.rotation    == right.rotation
+    }
+}
+
+impl<N, D: DimName, S, R> ApproxEq for IsometryBase<N, D, S, R>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          R: Rotation<PointBase<N, D, S>> + ApproxEq<Epsilon = N::Epsilon>,
+          S::Alloc: OwnedAllocator<N, D, U1, S>,
+          N::Epsilon: Copy {
+    type Epsilon = N::Epsilon;
+
+    #[inline]
+    fn default_epsilon() -> Self::Epsilon {
+        N::default_epsilon()
+    }
+
+    #[inline]
+    fn default_max_relative() -> Self::Epsilon {
+        N::default_max_relative()
+    }
+
+    #[inline]
+    fn default_max_ulps() -> u32 {
+        N::default_max_ulps()
+    }
+
+    #[inline]
+    fn relative_eq(&self, other: &Self, epsilon: Self::Epsilon, max_relative: Self::Epsilon) -> bool {
+        self.translation.relative_eq(&other.translation, epsilon, max_relative) &&
+        self.rotation.relative_eq(&other.rotation, epsilon, max_relative)
+    }
+
+    #[inline]
+    fn ulps_eq(&self, other: &Self, epsilon: Self::Epsilon, max_ulps: u32) -> bool {
+        self.translation.ulps_eq(&other.translation, epsilon, max_ulps) &&
+        self.rotation.ulps_eq(&other.rotation, epsilon, max_ulps)
+    }
+}
+
+/*
+ *
+ * Display
+ *
+ */
+impl<N, D: DimName, S, R> fmt::Display for IsometryBase<N, D, S, R>
+    where N: Real + fmt::Display,
+          S: OwnedStorage<N, D, U1>,
+          R: fmt::Display,
+          S::Alloc: OwnedAllocator<N, D, U1, S> + Allocator<usize, D, U1> {
+    fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
+        let precision = f.precision().unwrap_or(3);
+
+        try!(writeln!(f, "IsometryBase {{"));
+        try!(write!(f, "{:.*}", precision, self.translation));
+        try!(write!(f, "{:.*}", precision, self.rotation));
+        writeln!(f, "}}")
+    }
+}
+
+
+//         /*
+//          *
+//          * Absolute
+//          *
+//          */
+//         impl<N: Absolute> Absolute for $t<N> {
+//             type AbsoluteValue = $submatrix<N::AbsoluteValue>;
+//
+//             #[inline]
+//             fn abs(m: &$t<N>) -> $submatrix<N::AbsoluteValue> {
+//                 Absolute::abs(&m.submatrix)
+//             }
+//         }

src/geometry/isometry_alga.rs

@@ -0,0 +1,198 @@
+use alga::general::{AbstractMagma, AbstractGroup, AbstractLoop, AbstractMonoid, AbstractQuasigroup,
+                    AbstractSemigroup, Real, Inverse, Multiplicative, Identity, Id};
+use alga::linear::{Transformation, Similarity, AffineTransformation, DirectIsometry, Isometry,
+                   Rotation, ProjectiveTransformation};
+
+use core::ColumnVector;
+use core::dimension::{DimName, U1};
+use core::storage::OwnedStorage;
+use core::allocator::OwnedAllocator;
+
+use geometry::{IsometryBase, TranslationBase, PointBase};
+
+
+/*
+ *
+ * Algebraic structures.
+ *
+ */
+impl<N, D: DimName, S, R> Identity<Multiplicative> for IsometryBase<N, D, S, R>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          R: Rotation<PointBase<N, D, S>>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    #[inline]
+    fn identity() -> Self {
+        Self::identity()
+    }
+}
+
+impl<N, D: DimName, S, R> Inverse<Multiplicative> for IsometryBase<N, D, S, R>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          R: Rotation<PointBase<N, D, S>>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    #[inline]
+    fn inverse(&self) -> Self {
+        self.inverse()
+    }
+
+    #[inline]
+    fn inverse_mut(&mut self) {
+        self.inverse_mut()
+    }
+}
+
+impl<N, D: DimName, S, R> AbstractMagma<Multiplicative> for IsometryBase<N, D, S, R>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          R: Rotation<PointBase<N, D, S>>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    #[inline]
+    fn operate(&self, rhs: &Self) -> Self {
+        self * rhs
+    }
+}
+
+macro_rules! impl_multiplicative_structures(
+    ($($marker: ident<$operator: ident>),* $(,)*) => {$(
+        impl<N, D: DimName, S, R> $marker<$operator> for IsometryBase<N, D, S, R>
+            where N: Real,
+                  S: OwnedStorage<N, D, U1>,
+                  R: Rotation<PointBase<N, D, S>>,
+                  S::Alloc: OwnedAllocator<N, D, U1, S> { }
+    )*}
+);
+
+impl_multiplicative_structures!(
+    AbstractSemigroup<Multiplicative>,
+    AbstractMonoid<Multiplicative>,
+    AbstractQuasigroup<Multiplicative>,
+    AbstractLoop<Multiplicative>,
+    AbstractGroup<Multiplicative>
+);
+
+/*
+ *
+ * Transformation groups.
+ *
+ */
+impl<N, D: DimName, S, R> Transformation<PointBase<N, D, S>> for IsometryBase<N, D, S, R>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          R: Rotation<PointBase<N, D, S>>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    #[inline]
+    fn transform_point(&self, pt: &PointBase<N, D, S>) -> PointBase<N, D, S> {
+        self * pt
+    }
+
+    #[inline]
+    fn transform_vector(&self, v: &ColumnVector<N, D, S>) -> ColumnVector<N, D, S> {
+        self * v
+    }
+}
+
+impl<N, D: DimName, S, R> ProjectiveTransformation<PointBase<N, D, S>> for IsometryBase<N, D, S, R>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          R: Rotation<PointBase<N, D, S>>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    #[inline]
+    fn inverse_transform_point(&self, pt: &PointBase<N, D, S>) -> PointBase<N, D, S> {
+        self.rotation.inverse_transform_point(&(pt - &self.translation.vector))
+    }
+
+    #[inline]
+    fn inverse_transform_vector(&self, v: &ColumnVector<N, D, S>) -> ColumnVector<N, D, S> {
+        self.rotation.inverse_transform_vector(v)
+    }
+}
+
+impl<N, D: DimName, S, R> AffineTransformation<PointBase<N, D, S>> for IsometryBase<N, D, S, R>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          R: Rotation<PointBase<N, D, S>>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    type Rotation          = R;
+    type NonUniformScaling = Id;
+    type Translation       = TranslationBase<N, D, S>;
+
+    #[inline]
+    fn decompose(&self) -> (TranslationBase<N, D, S>, R, Id, R) {
+        (self.translation.clone(), self.rotation.clone(), Id::new(), R::identity())
+    }
+
+    #[inline]
+    fn append_translation(&self, t: &Self::Translation) -> Self {
+        t * self
+    }
+
+    #[inline]
+    fn prepend_translation(&self, t: &Self::Translation) -> Self {
+        self * t
+    }
+
+    #[inline]
+    fn append_rotation(&self, r: &Self::Rotation) -> Self {
+        let shift = r.transform_vector(&self.translation.vector);
+        IsometryBase::from_parts(TranslationBase::from_vector(shift), r.clone() * self.rotation.clone())
+    }
+
+    #[inline]
+    fn prepend_rotation(&self, r: &Self::Rotation) -> Self {
+        self * r
+    }
+
+    #[inline]
+    fn append_scaling(&self, _: &Self::NonUniformScaling) -> Self {
+        self.clone()
+    }
+
+    #[inline]
+    fn prepend_scaling(&self, _: &Self::NonUniformScaling) -> Self {
+        self.clone()
+    }
+
+    #[inline]
+    fn append_rotation_wrt_point(&self, r: &Self::Rotation, p: &PointBase<N, D, S>) -> Option<Self> {
+        let mut res = self.clone();
+        res.append_rotation_wrt_point_mut(r, p);
+        Some(res)
+    }
+}
+
+impl<N, D: DimName, S, R> Similarity<PointBase<N, D, S>> for IsometryBase<N, D, S, R>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          R: Rotation<PointBase<N, D, S>>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    type Scaling = Id;
+
+    #[inline]
+    fn translation(&self) -> TranslationBase<N, D, S> {
+        self.translation.clone()
+    }
+
+    #[inline]
+    fn rotation(&self) -> R {
+        self.rotation.clone()
+    }
+
+    #[inline]
+    fn scaling(&self) -> Id {
+        Id::new()
+    }
+}
+
+macro_rules! marker_impl(
+    ($($Trait: ident),*) => {$(
+        impl<N, D: DimName, S, R> $Trait<PointBase<N, D, S>> for IsometryBase<N, D, S, R>
+        where N: Real,
+              S: OwnedStorage<N, D, U1>,
+              R: Rotation<PointBase<N, D, S>>,
+              S::Alloc: OwnedAllocator<N, D, U1, S> { }
+    )*}
+);
+
+marker_impl!(Isometry, DirectIsometry);

src/geometry/isometry_alias.rs

@@ -0,0 +1,19 @@
+use core::MatrixArray;
+use core::dimension::{U1, U2, U3};
+
+use geometry::{Rotation, IsometryBase, UnitQuaternion, UnitComplex};
+
+/// A D-dimensional isometry.
+pub type Isometry<N, D> = IsometryBase<N, D, MatrixArray<N, D, U1>, Rotation<N, D>>;
+
+/// A 2-dimensional isometry using a unit complex number for its rotational part.
+pub type Isometry2<N> = IsometryBase<N, U2, MatrixArray<N, U2, U1>, UnitComplex<N>>;
+
+/// A 3-dimensional isometry using a unit quaternion for its rotational part.
+pub type Isometry3<N> = IsometryBase<N, U3, MatrixArray<N, U3, U1>, UnitQuaternion<N>>;
+
+/// A 2-dimensional isometry using a rotation matrix for its rotation part.
+pub type IsometryMatrix2<N> = Isometry<N, U2>;
+
+/// A 3-dimensional isometry using a rotation matrix for its rotation part.
+pub type IsometryMatrix3<N> = Isometry<N, U3>;

src/geometry/isometry_construction.rs

@@ -0,0 +1,195 @@
+#[cfg(feature = "arbitrary")]
+use quickcheck::{Arbitrary, Gen};
+
+use num::One;
+use rand::{Rng, Rand};
+
+use alga::general::Real;
+use alga::linear::Rotation as AlgaRotation;
+
+use core::ColumnVector;
+use core::dimension::{DimName, U1, U2, U3, U4};
+use core::allocator::{OwnedAllocator, Allocator};
+use core::storage::OwnedStorage;
+
+use geometry::{PointBase, TranslationBase, RotationBase, IsometryBase, UnitQuaternionBase, UnitComplex};
+
+
+impl<N, D: DimName, S, R> IsometryBase<N, D, S, R>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          R: AlgaRotation<PointBase<N, D, S>>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    /// Creates a new identity isometry.
+    #[inline]
+    pub fn identity() -> Self {
+        Self::from_parts(TranslationBase::identity(), R::identity())
+    }
+}
+
+impl<N, D: DimName, S, R> One for IsometryBase<N, D, S, R>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          R: AlgaRotation<PointBase<N, D, S>>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    /// Creates a new identity isometry.
+    #[inline]
+    fn one() -> Self {
+        Self::identity()
+    }
+}
+
+impl<N, D: DimName, S, R> Rand for IsometryBase<N, D, S, R>
+    where N: Real + Rand,
+          S: OwnedStorage<N, D, U1>,
+          R: AlgaRotation<PointBase<N, D, S>> + Rand,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    #[inline]
+    fn rand<G: Rng>(rng: &mut G) -> Self {
+        Self::from_parts(rng.gen(), rng.gen())
+    }
+}
+
+impl<N, D: DimName, S, R> IsometryBase<N, D, S, R>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          R: AlgaRotation<PointBase<N, D, S>>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    /// The isometry that applies the rotation `r` with its axis passing through the point `p`.
+    /// This effectively lets `p` invariant.
+    #[inline]
+    pub fn rotation_wrt_point(r: R, p: PointBase<N, D, S>) -> Self {
+        let shift = r.transform_vector(&-&p.coords);
+        Self::from_parts(TranslationBase::from_vector(shift + p.coords), r)
+    }
+}
+
+#[cfg(feature = "arbitrary")]
+impl<N, D: DimName, S, R> Arbitrary for IsometryBase<N, D, S, R>
+    where N: Real + Arbitrary + Send,
+          S: OwnedStorage<N, D, U1> + Send,
+          R: AlgaRotation<PointBase<N, D, S>> + Arbitrary + Send,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    #[inline]
+    fn arbitrary<G: Gen>(rng: &mut G) -> Self {
+        Self::from_parts(Arbitrary::arbitrary(rng), Arbitrary::arbitrary(rng))
+    }
+}
+
+/*
+ *
+ * Constructors for various static dimensions.
+ *
+ */
+
+// 2D rotation.
+impl<N, S, SR> IsometryBase<N, U2, S, RotationBase<N, U2, SR>>
+    where N: Real,
+          S:  OwnedStorage<N, U2, U1, Alloc = SR::Alloc>,
+          SR: OwnedStorage<N, U2, U2>,
+          S::Alloc:  OwnedAllocator<N, U2, U1, S>,
+          SR::Alloc: OwnedAllocator<N, U2, U2, SR> {
+    /// Creates a new isometry from a translation and a rotation angle.
+    #[inline]
+    pub fn new(translation: ColumnVector<N, U2, S>, angle: N) -> Self {
+        Self::from_parts(TranslationBase::from_vector(translation), RotationBase::<N, U2, SR>::new(angle))
+    }
+}
+
+impl<N, S> IsometryBase<N, U2, S, UnitComplex<N>>
+    where N: Real,
+          S: OwnedStorage<N, U2, U1>,
+          S::Alloc:  OwnedAllocator<N, U2, U1, S> {
+    /// Creates a new isometry from a translation and a rotation angle.
+    #[inline]
+    pub fn new(translation: ColumnVector<N, U2, S>, angle: N) -> Self {
+        Self::from_parts(TranslationBase::from_vector(translation), UnitComplex::from_angle(angle))
+    }
+}
+
+// 3D rotation.
+macro_rules! isometry_construction_impl(
+    ($RotId: ident < $($RotParams: ident),*>, $RRDim: ty, $RCDim: ty) => {
+        impl<N, S, SR> IsometryBase<N, U3, S, $RotId<$($RotParams),*>>
+            where N: Real,
+                  S:  OwnedStorage<N, U3, U1, Alloc = SR::Alloc>,
+                  SR: OwnedStorage<N, $RRDim, $RCDim>,
+                  S::Alloc:  OwnedAllocator<N, U3, U1, S>,
+                  SR::Alloc: OwnedAllocator<N, $RRDim, $RCDim, SR> +
+                             Allocator<N, U3, U3> {
+            /// Creates a new isometry from a translation and a rotation axis-angle.
+            #[inline]
+            pub fn new(translation: ColumnVector<N, U3, S>, axisangle: ColumnVector<N, U3, S>) -> Self {
+                Self::from_parts(
+                    TranslationBase::from_vector(translation),
+                    $RotId::<$($RotParams),*>::from_scaled_axis(axisangle))
+            }
+
+            /// Creates an isometry that corresponds to the local frame of an observer standing at the
+            /// point `eye` and looking toward `target`.
+            ///
+            /// It maps the view direction `target - eye` to the positive `z` axis and the origin to the
+            /// `eye`.
+            ///
+            /// # Arguments
+            ///   * eye - The observer position.
+            ///   * target - The target position.
+            ///   * up - Vertical direction. The only requirement of this parameter is to not be collinear
+            ///   to `eye - at`. Non-collinearity is not checked.
+            #[inline]
+            pub fn new_observer_frame(eye:    &PointBase<N, U3, S>,
+                                      target: &PointBase<N, U3, S>,
+                                      up:     &ColumnVector<N, U3, S>)
+                                      -> Self {
+                Self::from_parts(
+                    TranslationBase::from_vector(eye.coords.clone()),
+                    $RotId::new_observer_frame(&(target - eye), up))
+            }
+
+            /// Builds a right-handed look-at view matrix.
+            ///
+            /// This conforms to the common notion of right handed look-at matrix from the computer
+            /// graphics community.
+            ///
+            /// # Arguments
+            ///   * eye - The eye position.
+            ///   * target - The target position.
+            ///   * up - A vector approximately aligned with required the vertical axis. The only
+            ///   requirement of this parameter is to not be collinear to `target - eye`.
+            #[inline]
+            pub fn look_at_rh(eye:    &PointBase<N, U3, S>,
+                              target: &PointBase<N, U3, S>,
+                              up:     &ColumnVector<N, U3, S>)
+                              -> Self {
+                let rotation = $RotId::look_at_rh(&(target - eye), up);
+                let trans    = &rotation * (-eye);
+
+                Self::from_parts(TranslationBase::from_vector(trans.coords), rotation)
+            }
+
+            /// Builds a left-handed look-at view matrix.
+            ///
+            /// This conforms to the common notion of left handed look-at matrix from the computer
+            /// graphics community.
+            ///
+            /// # Arguments
+            ///   * eye - The eye position.
+            ///   * target - The target position.
+            ///   * up - A vector approximately aligned with required the vertical axis. The only
+            ///   requirement of this parameter is to not be collinear to `target - eye`.
+            #[inline]
+            pub fn look_at_lh(eye:    &PointBase<N, U3, S>,
+                              target: &PointBase<N, U3, S>,
+                              up:     &ColumnVector<N, U3, S>)
+                              -> Self {
+                let rotation = $RotId::look_at_lh(&(target - eye), up);
+                let trans    = &rotation * (-eye);
+
+                Self::from_parts(TranslationBase::from_vector(trans.coords), rotation)
+            }
+        }
+    }
+);
+
+isometry_construction_impl!(RotationBase<N, U3, SR>, U3, U3);
+isometry_construction_impl!(UnitQuaternionBase<N, SR>, U4, U1);

src/geometry/isometry_conversion.rs

@@ -0,0 +1,163 @@
+use alga::general::{Real, SubsetOf, SupersetOf};
+use alga::linear::Rotation;
+
+use core::{SquareMatrix, OwnedSquareMatrix};
+use core::dimension::{DimName, DimNameAdd, DimNameSum, U1};
+use core::storage::OwnedStorage;
+use core::allocator::{Allocator, OwnedAllocator};
+
+use geometry::{PointBase, TranslationBase, IsometryBase, SimilarityBase, TransformBase, SuperTCategoryOf, TAffine};
+
+/*
+ * This file provides the following conversions:
+ * =============================================
+ *
+ * IsometryBase -> IsometryBase
+ * IsometryBase -> SimilarityBase
+ * IsometryBase -> TransformBase
+ * IsometryBase -> Matrix (homogeneous)
+ */
+
+
+impl<N1, N2, D: DimName, SA, SB, R1, R2> SubsetOf<IsometryBase<N2, D, SB, R2>> for IsometryBase<N1, D, SA, R1>
+    where N1: Real,
+          N2: Real + SupersetOf<N1>,
+          R1: Rotation<PointBase<N1, D, SA>> + SubsetOf<R2>,
+          R2: Rotation<PointBase<N2, D, SB>>,
+          SA: OwnedStorage<N1, D, U1>,
+          SB: OwnedStorage<N2, D, U1>,
+          SA::Alloc: OwnedAllocator<N1, D, U1, SA>,
+          SB::Alloc: OwnedAllocator<N2, D, U1, SB> {
+    #[inline]
+    fn to_superset(&self) -> IsometryBase<N2, D, SB, R2> {
+        IsometryBase::from_parts(
+            self.translation.to_superset(),
+            self.rotation.to_superset()
+        )
+    }
+
+    #[inline]
+    fn is_in_subset(iso: &IsometryBase<N2, D, SB, R2>) -> bool {
+        ::is_convertible::<_, TranslationBase<N1, D, SA>>(&iso.translation) &&
+        ::is_convertible::<_, R1>(&iso.rotation)
+    }
+
+    #[inline]
+    unsafe fn from_superset_unchecked(iso: &IsometryBase<N2, D, SB, R2>) -> Self {
+        IsometryBase::from_parts(
+            iso.translation.to_subset_unchecked(),
+            iso.rotation.to_subset_unchecked()
+        )
+    }
+}
+
+
+impl<N1, N2, D: DimName, SA, SB, R1, R2> SubsetOf<SimilarityBase<N2, D, SB, R2>> for IsometryBase<N1, D, SA, R1>
+    where N1: Real,
+          N2: Real + SupersetOf<N1>,
+          R1: Rotation<PointBase<N1, D, SA>> + SubsetOf<R2>,
+          R2: Rotation<PointBase<N2, D, SB>>,
+          SA: OwnedStorage<N1, D, U1>,
+          SB: OwnedStorage<N2, D, U1>,
+          SA::Alloc: OwnedAllocator<N1, D, U1, SA>,
+          SB::Alloc: OwnedAllocator<N2, D, U1, SB> {
+    #[inline]
+    fn to_superset(&self) -> SimilarityBase<N2, D, SB, R2> {
+        SimilarityBase::from_isometry(
+            self.to_superset(),
+            N2::one()
+        )
+    }
+
+    #[inline]
+    fn is_in_subset(sim: &SimilarityBase<N2, D, SB, R2>) -> bool {
+        ::is_convertible::<_, IsometryBase<N1, D, SA, R1>>(&sim.isometry) &&
+        sim.scaling() == N2::one()
+    }
+
+    #[inline]
+    unsafe fn from_superset_unchecked(sim: &SimilarityBase<N2, D, SB, R2>) -> Self {
+        ::convert_ref_unchecked(&sim.isometry)
+    }
+}
+
+
+impl<N1, N2, D, SA, SB, R, C> SubsetOf<TransformBase<N2, D, SB, C>> for IsometryBase<N1, D, SA, R>
+    where N1: Real,
+          N2: Real + SupersetOf<N1>,
+          SA: OwnedStorage<N1, D, U1>,
+          SB: OwnedStorage<N2, DimNameSum<D, U1>, DimNameSum<D, U1>>,
+          C:  SuperTCategoryOf<TAffine>,
+          R: Rotation<PointBase<N1, D, SA>> +
+             SubsetOf<OwnedSquareMatrix<N1, DimNameSum<D, U1>, SA::Alloc>> + // needed by: .to_homogeneous()
+             SubsetOf<SquareMatrix<N2, DimNameSum<D, U1>, SB>>,              // needed by: ::convert_unchecked(mm)
+          D:  DimNameAdd<U1>,
+          SA::Alloc: OwnedAllocator<N1, D, U1, SA> +
+                     Allocator<N1, D, D> +                                 // needed by R
+                     Allocator<N1, DimNameSum<D, U1>, DimNameSum<D, U1>> + // needed by: .to_homogeneous()
+                     Allocator<N2, DimNameSum<D, U1>, DimNameSum<D, U1>>,  // needed by R
+          SB::Alloc: OwnedAllocator<N2, DimNameSum<D, U1>, DimNameSum<D, U1>, SB> +
+                     Allocator<N2, D, D> +  // needed by: mm.fixed_slice_mut
+                     Allocator<N2, D, U1> + // needed by: m.fixed_slice
+                     Allocator<N2, U1, D> { // needed by: m.fixed_slice
+    #[inline]
+    fn to_superset(&self) -> TransformBase<N2, D, SB, C> {
+        TransformBase::from_matrix_unchecked(self.to_homogeneous().to_superset())
+    }
+
+    #[inline]
+    fn is_in_subset(t: &TransformBase<N2, D, SB, C>) -> bool {
+        <Self as SubsetOf<_>>::is_in_subset(t.matrix())
+    }
+
+    #[inline]
+    unsafe fn from_superset_unchecked(t: &TransformBase<N2, D, SB, C>) -> Self {
+        Self::from_superset_unchecked(t.matrix())
+    }
+}
+
+
+impl<N1, N2, D, SA, SB, R> SubsetOf<SquareMatrix<N2, DimNameSum<D, U1>, SB>> for IsometryBase<N1, D, SA, R>
+    where N1: Real,
+          N2: Real + SupersetOf<N1>,
+          SA: OwnedStorage<N1, D, U1>,
+          SB: OwnedStorage<N2, DimNameSum<D, U1>, DimNameSum<D, U1>>,
+          R: Rotation<PointBase<N1, D, SA>> +
+             SubsetOf<OwnedSquareMatrix<N1, DimNameSum<D, U1>, SA::Alloc>> + // needed by: .to_homogeneous()
+             SubsetOf<SquareMatrix<N2, DimNameSum<D, U1>, SB>>,              // needed by: ::convert_unchecked(mm)
+          D:  DimNameAdd<U1>,
+          SA::Alloc: OwnedAllocator<N1, D, U1, SA> +
+                     Allocator<N1, D, D> +                                 // needed by R
+                     Allocator<N1, DimNameSum<D, U1>, DimNameSum<D, U1>> + // needed by: .to_homogeneous()
+                     Allocator<N2, DimNameSum<D, U1>, DimNameSum<D, U1>>,  // needed by R
+          SB::Alloc: OwnedAllocator<N2, DimNameSum<D, U1>, DimNameSum<D, U1>, SB> +
+                     Allocator<N2, D, D> +  // needed by: mm.fixed_slice_mut
+                     Allocator<N2, D, U1> + // needed by: m.fixed_slice
+                     Allocator<N2, U1, D> { // needed by: m.fixed_slice
+    #[inline]
+    fn to_superset(&self) -> SquareMatrix<N2, DimNameSum<D, U1>, SB> {
+        self.to_homogeneous().to_superset()
+    }
+
+    #[inline]
+    fn is_in_subset(m: &SquareMatrix<N2, DimNameSum<D, U1>, SB>) -> bool {
+        let rot    = m.fixed_slice::<D, D>(0, 0);
+        let bottom = m.fixed_slice::<U1, D>(D::dim(), 0);
+
+        // Scalar types agree.
+        m.iter().all(|e| SupersetOf::<N1>::is_in_subset(e)) &&
+        // The block part is a rotation.
+        rot.is_special_orthogonal(N2::default_epsilon() * ::convert(100.0)) &&
+        // The bottom row is (0, 0, ..., 1)
+        bottom.iter().all(|e| e.is_zero()) &&
+        m[(D::dim(), D::dim())] == N2::one()
+    }
+
+    #[inline]
+    unsafe fn from_superset_unchecked(m: &SquareMatrix<N2, DimNameSum<D, U1>, SB>) -> Self {
+        let t = m.fixed_slice::<D, U1>(0, D::dim()).into_owned();
+        let t = TranslationBase::from_vector(::convert_unchecked(t));
+
+        Self::from_parts(t, ::convert_unchecked(m.clone_owned()))
+    }
+}

src/geometry/isometry_ops.rs

@@ -0,0 +1,435 @@
+use std::ops::{Mul, MulAssign, Div, DivAssign};
+
+use alga::general::Real;
+use alga::linear::Rotation;
+
+use core::ColumnVector;
+use core::dimension::{DimName, U1, U3, U4};
+use core::storage::OwnedStorage;
+use core::allocator::OwnedAllocator;
+
+use geometry::{PointBase, RotationBase, IsometryBase, TranslationBase, UnitQuaternionBase};
+
+// FIXME: there are several cloning of rotations that we could probably get rid of (but we didn't
+// yet because that would require to add a bound like `where for<'a, 'b> &'a R: Mul<&'b R, Output = R>`
+// which is quite ugly.
+
+/*
+ *
+ * In this file, we provide:
+ * =========================
+ *
+ *
+ * (Operators)
+ *
+ * IsometryBase × IsometryBase
+ * IsometryBase × R
+ *
+ *
+ * IsometryBase ÷ IsometryBase
+ * IsometryBase ÷ R
+ *
+ * IsometryBase × PointBase
+ * IsometryBase × ColumnVector
+ *
+ *
+ * IsometryBase    × TranslationBase
+ * TranslationBase × IsometryBase
+ * TranslationBase × R           -> IsometryBase<R>
+ *
+ * NOTE: The following are provided explicitly because we can't have R × IsometryBase.
+ * RotationBase   × IsometryBase<RotationBase>
+ * UnitQuaternion × IsometryBase<UnitQuaternion>
+ *
+ * RotationBase   ÷ IsometryBase<RotationBase>
+ * UnitQuaternion ÷ IsometryBase<UnitQuaternion>
+ *
+ * RotationBase   × TranslationBase -> IsometryBase<RotationBase>
+ * UnitQuaternion × TranslationBase -> IsometryBase<UnitQuaternion>
+ *
+ *
+ * (Assignment Operators)
+ *
+ * IsometryBase ×= TranslationBase
+ *
+ * IsometryBase ×= IsometryBase
+ * IsometryBase ×= R
+ *
+ * IsometryBase ÷= IsometryBase
+ * IsometryBase ÷= R
+ *
+ */
+
+
+macro_rules! isometry_binop_impl(
+    ($Op: ident, $op: ident;
+     $lhs: ident: $Lhs: ty, $rhs: ident: $Rhs: ty, Output = $Output: ty;
+     $action: expr; $($lives: tt),*) => {
+        impl<$($lives ,)* N, D: DimName, S, R> $Op<$Rhs> for $Lhs
+            where N: Real,
+                  S: OwnedStorage<N, D, U1>,
+                  R: Rotation<PointBase<N, D, S>>,
+                  S::Alloc: OwnedAllocator<N, D, U1, S> {
+            type Output = $Output;
+
+            #[inline]
+            fn $op($lhs, $rhs: $Rhs) -> Self::Output {
+                $action
+            }
+        }
+    }
+);
+
+macro_rules! isometry_binop_impl_all(
+    ($Op: ident, $op: ident;
+     $lhs: ident: $Lhs: ty, $rhs: ident: $Rhs: ty, Output = $Output: ty;
+     [val val] => $action_val_val: expr;
+     [ref val] => $action_ref_val: expr;
+     [val ref] => $action_val_ref: expr;
+     [ref ref] => $action_ref_ref: expr;) => {
+        isometry_binop_impl!(
+            $Op, $op;
+            $lhs: $Lhs, $rhs: $Rhs, Output = $Output;
+            $action_val_val; );
+
+        isometry_binop_impl!(
+            $Op, $op;
+            $lhs: &'a $Lhs, $rhs: $Rhs, Output = $Output;
+            $action_ref_val; 'a);
+
+        isometry_binop_impl!(
+            $Op, $op;
+            $lhs: $Lhs, $rhs: &'b $Rhs, Output = $Output;
+            $action_val_ref; 'b);
+
+        isometry_binop_impl!(
+            $Op, $op;
+            $lhs: &'a $Lhs, $rhs: &'b $Rhs, Output = $Output;
+            $action_ref_ref; 'a, 'b);
+    }
+);
+
+macro_rules! isometry_binop_assign_impl_all(
+    ($OpAssign: ident, $op_assign: ident;
+     $lhs: ident: $Lhs: ty, $rhs: ident: $Rhs: ty;
+     [val] => $action_val: expr;
+     [ref] => $action_ref: expr;) => {
+        impl<N, D: DimName, S, R> $OpAssign<$Rhs> for $Lhs
+            where N: Real,
+                  S: OwnedStorage<N, D, U1>,
+                  R: Rotation<PointBase<N, D, S>>,
+                  S::Alloc: OwnedAllocator<N, D, U1, S> {
+            #[inline]
+            fn $op_assign(&mut $lhs, $rhs: $Rhs) {
+                $action_val
+            }
+        }
+
+        impl<'b, N, D: DimName, S, R> $OpAssign<&'b $Rhs> for $Lhs
+            where N: Real,
+                  S: OwnedStorage<N, D, U1>,
+                  R: Rotation<PointBase<N, D, S>>,
+                  S::Alloc: OwnedAllocator<N, D, U1, S> {
+            #[inline]
+            fn $op_assign(&mut $lhs, $rhs: &'b $Rhs) {
+                $action_ref
+            }
+        }
+    }
+);
+
+// IsometryBase × IsometryBase
+// IsometryBase ÷ IsometryBase
+isometry_binop_impl_all!(
+    Mul, mul;
+    self: IsometryBase<N, D, S, R>, rhs: IsometryBase<N, D, S, R>, Output = IsometryBase<N, D, S, R>;
+    [val val] => &self * &rhs;
+    [ref val] => self * &rhs;
+    [val ref] => &self * rhs;
+    [ref ref] => {
+        let shift = self.rotation.transform_vector(&rhs.translation.vector);
+
+        IsometryBase::from_parts(TranslationBase::from_vector(&self.translation.vector + shift),
+                             self.rotation.clone() * rhs.rotation.clone()) // FIXME: too bad we have to clone.
+    };
+);
+
+
+isometry_binop_impl_all!(
+    Div, div;
+    self: IsometryBase<N, D, S, R>, rhs: IsometryBase<N, D, S, R>, Output = IsometryBase<N, D, S, R>;
+    [val val] => self * rhs.inverse();
+    [ref val] => self * rhs.inverse();
+    [val ref] => self * rhs.inverse();
+    [ref ref] => self * rhs.inverse();
+);
+
+
+// IsometryBase ×= TranslationBase
+isometry_binop_assign_impl_all!(
+    MulAssign, mul_assign;
+    self: IsometryBase<N, D, S, R>, rhs: TranslationBase<N, D, S>;
+    [val] => *self *= &rhs;
+    [ref] => {
+        let shift = self.rotation.transform_vector(&rhs.vector);
+        self.translation.vector += shift;
+    };
+);
+
+// IsometryBase ×= IsometryBase
+// IsometryBase ÷= IsometryBase
+isometry_binop_assign_impl_all!(
+    MulAssign, mul_assign;
+    self: IsometryBase<N, D, S, R>, rhs: IsometryBase<N, D, S, R>;
+    [val] => *self *= &rhs;
+    [ref] => {
+        let shift = self.rotation.transform_vector(&rhs.translation.vector);
+        self.translation.vector += shift;
+        self.rotation *= rhs.rotation.clone();
+    };
+);
+
+isometry_binop_assign_impl_all!(
+    DivAssign, div_assign;
+    self: IsometryBase<N, D, S, R>, rhs: IsometryBase<N, D, S, R>;
+    [val] => *self /= &rhs;
+    [ref] => *self *= rhs.inverse();
+);
+
+// IsometryBase ×= R
+// IsometryBase ÷= R
+isometry_binop_assign_impl_all!(
+    MulAssign, mul_assign;
+    self: IsometryBase<N, D, S, R>, rhs: R;
+    [val] => self.rotation *= rhs;
+    [ref] => self.rotation *= rhs.clone();
+);
+
+isometry_binop_assign_impl_all!(
+    DivAssign, div_assign;
+    self: IsometryBase<N, D, S, R>, rhs: R;
+    // FIXME: don't invert explicitly?
+    [val] => *self *= rhs.inverse();
+    [ref] => *self *= rhs.inverse();
+);
+
+
+// IsometryBase × R
+// IsometryBase ÷ R
+isometry_binop_impl_all!(
+    Mul, mul;
+    self: IsometryBase<N, D, S, R>, rhs: R, Output = IsometryBase<N, D, S, R>;
+    [val val] => IsometryBase::from_parts(self.translation, self.rotation * rhs);
+    [ref val] => IsometryBase::from_parts(self.translation.clone(), self.rotation.clone() * rhs); // FIXME: do not clone.
+    [val ref] => IsometryBase::from_parts(self.translation, self.rotation * rhs.clone());
+    [ref ref] => IsometryBase::from_parts(self.translation.clone(), self.rotation.clone() * rhs.clone());
+);
+
+
+isometry_binop_impl_all!(
+    Div, div;
+    self: IsometryBase<N, D, S, R>, rhs: R, Output = IsometryBase<N, D, S, R>;
+    [val val] => IsometryBase::from_parts(self.translation, self.rotation / rhs);
+    [ref val] => IsometryBase::from_parts(self.translation.clone(), self.rotation.clone() / rhs);
+    [val ref] => IsometryBase::from_parts(self.translation, self.rotation / rhs.clone());
+    [ref ref] => IsometryBase::from_parts(self.translation.clone(), self.rotation.clone() / rhs.clone());
+);
+
+
+// IsometryBase × PointBase
+isometry_binop_impl_all!(
+    Mul, mul;
+    self: IsometryBase<N, D, S, R>, right: PointBase<N, D, S>, Output = PointBase<N, D, S>;
+    [val val] => self.translation * self.rotation.transform_point(&right);
+    [ref val] => &self.translation * self.rotation.transform_point(&right);
+    [val ref] => self.translation * self.rotation.transform_point(right);
+    [ref ref] => &self.translation * self.rotation.transform_point(right);
+);
+
+
+// IsometryBase × Vector
+isometry_binop_impl_all!(
+    Mul, mul;
+    self: IsometryBase<N, D, S, R>, right: ColumnVector<N, D, S>, Output = ColumnVector<N, D, S>;
+    [val val] => self.rotation.transform_vector(&right);
+    [ref val] => self.rotation.transform_vector(&right);
+    [val ref] => self.rotation.transform_vector(right);
+    [ref ref] => self.rotation.transform_vector(right);
+);
+
+
+// IsometryBase × TranslationBase
+isometry_binop_impl_all!(
+    Mul, mul;
+    self: IsometryBase<N, D, S, R>, right: TranslationBase<N, D, S>, Output = IsometryBase<N, D, S, R>;
+    [val val] => &self * &right;
+    [ref val] => self * &right;
+    [val ref] => &self * right;
+    [ref ref] => {
+        let new_tr = &self.translation.vector + self.rotation.transform_vector(&right.vector);
+        IsometryBase::from_parts(TranslationBase::from_vector(new_tr), self.rotation.clone())
+    };
+);
+
+// TranslationBase × IsometryBase
+isometry_binop_impl_all!(
+    Mul, mul;
+    self: TranslationBase<N, D, S>, right: IsometryBase<N, D, S, R>, Output = IsometryBase<N, D, S, R>;
+    [val val] => IsometryBase::from_parts(self * right.translation, right.rotation);
+    [ref val] => IsometryBase::from_parts(self * &right.translation, right.rotation);
+    [val ref] => IsometryBase::from_parts(self * &right.translation, right.rotation.clone());
+    [ref ref] => IsometryBase::from_parts(self * &right.translation, right.rotation.clone());
+);
+
+
+// TranslationBase × R
+isometry_binop_impl_all!(
+    Mul, mul;
+    self: TranslationBase<N, D, S>, right: R, Output = IsometryBase<N, D, S, R>;
+    [val val] => IsometryBase::from_parts(self, right);
+    [ref val] => IsometryBase::from_parts(self.clone(), right);
+    [val ref] => IsometryBase::from_parts(self, right.clone());
+    [ref ref] => IsometryBase::from_parts(self.clone(), right.clone());
+);
+
+
+
+
+macro_rules! isometry_from_composition_impl(
+    ($Op: ident, $op: ident;
+     ($R1: ty, $C1: ty),($R2: ty, $C2: ty) $(for $Dims: ident: $DimsBound: ident),*;
+     $lhs: ident: $Lhs: ty, $rhs: ident: $Rhs: ty, Output = $Output: ty;
+     $action: expr; $($lives: tt),*) => {
+        impl<$($lives ,)* N $(, $Dims: $DimsBound)*, SA, SB> $Op<$Rhs> for $Lhs
+            where N: Real,
+                  SA: OwnedStorage<N, $R1, $C1>,
+                  SB: OwnedStorage<N, $R2, $C2, Alloc = SA::Alloc>,
+                  SA::Alloc: OwnedAllocator<N, $R1, $C1, SA>,
+                  SB::Alloc: OwnedAllocator<N, $R2, $C2, SB> {
+            type Output = $Output;
+
+            #[inline]
+            fn $op($lhs, $rhs: $Rhs) -> Self::Output {
+                $action
+            }
+        }
+    }
+);
+
+macro_rules! isometry_from_composition_impl_all(
+    ($Op: ident, $op: ident;
+     ($R1: ty, $C1: ty),($R2: ty, $C2: ty) $(for $Dims: ident: $DimsBound: ident),*;
+     $lhs: ident: $Lhs: ty, $rhs: ident: $Rhs: ty, Output = $Output: ty;
+     [val val] => $action_val_val: expr;
+     [ref val] => $action_ref_val: expr;
+     [val ref] => $action_val_ref: expr;
+     [ref ref] => $action_ref_ref: expr;) => {
+
+        isometry_from_composition_impl!(
+            $Op, $op;
+            ($R1, $C1),($R2, $C2) $(for $Dims: $DimsBound),*;
+            $lhs: $Lhs, $rhs: $Rhs, Output = $Output;
+            $action_val_val; );
+
+        isometry_from_composition_impl!(
+            $Op, $op;
+            ($R1, $C1),($R2, $C2) $(for $Dims: $DimsBound),*;
+            $lhs: &'a $Lhs, $rhs: $Rhs, Output = $Output;
+            $action_ref_val; 'a);
+
+        isometry_from_composition_impl!(
+            $Op, $op;
+            ($R1, $C1),($R2, $C2) $(for $Dims: $DimsBound),*;
+            $lhs: $Lhs, $rhs: &'b $Rhs, Output = $Output;
+            $action_val_ref; 'b);
+
+        isometry_from_composition_impl!(
+            $Op, $op;
+            ($R1, $C1),($R2, $C2) $(for $Dims: $DimsBound),*;
+            $lhs: &'a $Lhs, $rhs: &'b $Rhs, Output = $Output;
+            $action_ref_ref; 'a, 'b);
+    }
+);
+
+
+// RotationBase × TranslationBase
+isometry_from_composition_impl_all!(
+    Mul, mul;
+    (D, D), (D, U1) for D: DimName;
+    self: RotationBase<N, D, SA>, right: TranslationBase<N, D, SB>, Output = IsometryBase<N, D, SB, RotationBase<N, D, SA>>;
+    [val val] => IsometryBase::from_parts(TranslationBase::from_vector(&self * right.vector),  self);
+    [ref val] => IsometryBase::from_parts(TranslationBase::from_vector(self * right.vector),   self.clone());
+    [val ref] => IsometryBase::from_parts(TranslationBase::from_vector(&self * &right.vector), self);
+    [ref ref] => IsometryBase::from_parts(TranslationBase::from_vector(self * &right.vector),  self.clone());
+);
+
+
+// UnitQuaternionBase × TranslationBase
+isometry_from_composition_impl_all!(
+    Mul, mul;
+    (U4, U1), (U3, U1);
+    self: UnitQuaternionBase<N, SA>, right: TranslationBase<N, U3, SB>,
+    Output = IsometryBase<N, U3, SB, UnitQuaternionBase<N, SA>>;
+    [val val] => IsometryBase::from_parts(TranslationBase::from_vector(&self *  right.vector), self);
+    [ref val] => IsometryBase::from_parts(TranslationBase::from_vector( self *  right.vector), self.clone());
+    [val ref] => IsometryBase::from_parts(TranslationBase::from_vector(&self * &right.vector), self);
+    [ref ref] => IsometryBase::from_parts(TranslationBase::from_vector( self * &right.vector), self.clone());
+);
+
+// RotationBase × IsometryBase
+isometry_from_composition_impl_all!(
+    Mul, mul;
+    (D, D), (D, U1) for D: DimName;
+    self: RotationBase<N, D, SA>, right: IsometryBase<N, D, SB, RotationBase<N, D, SA>>,
+    Output = IsometryBase<N, D, SB, RotationBase<N, D, SA>>;
+    [val val] => &self * &right;
+    [ref val] =>  self * &right;
+    [val ref] => &self * right;
+    [ref ref] => {
+        let shift = self * &right.translation.vector;
+        IsometryBase::from_parts(TranslationBase::from_vector(shift), self * &right.rotation)
+    };
+);
+
+// RotationBase ÷ IsometryBase
+isometry_from_composition_impl_all!(
+    Div, div;
+    (D, D), (D, U1) for D: DimName;
+    self: RotationBase<N, D, SA>, right: IsometryBase<N, D, SB, RotationBase<N, D, SA>>,
+    Output = IsometryBase<N, D, SB, RotationBase<N, D, SA>>;
+    // FIXME: don't call iverse explicitly?
+    [val val] => self * right.inverse();
+    [ref val] => self * right.inverse();
+    [val ref] => self * right.inverse();
+    [ref ref] => self * right.inverse();
+);
+
+
+// UnitQuaternion × IsometryBase
+isometry_from_composition_impl_all!(
+    Mul, mul;
+    (U4, U1), (U3, U1);
+    self: UnitQuaternionBase<N, SA>, right: IsometryBase<N, U3, SB, UnitQuaternionBase<N, SA>>,
+    Output = IsometryBase<N, U3, SB, UnitQuaternionBase<N, SA>>;
+    [val val] => &self * &right;
+    [ref val] =>  self * &right;
+    [val ref] => &self * right;
+    [ref ref] => {
+        let shift = self * &right.translation.vector;
+        IsometryBase::from_parts(TranslationBase::from_vector(shift), self * &right.rotation)
+    };
+);
+
+
+// UnitQuaternion ÷ IsometryBase
+isometry_from_composition_impl_all!(
+    Div, div;
+    (U4, U1), (U3, U1);
+    self: UnitQuaternionBase<N, SA>, right: IsometryBase<N, U3, SB, UnitQuaternionBase<N, SA>>,
+    Output = IsometryBase<N, U3, SB, UnitQuaternionBase<N, SA>>;
+    // FIXME: don't call inverse explicitly?
+    [val val] => self * right.inverse();
+    [ref val] => self * right.inverse();
+    [val ref] => self * right.inverse();
+    [ref ref] => self * right.inverse();
+);

src/geometry/mod.rs

@@ -0,0 +1,91 @@
+//! [Reexported at the root of this crate.] Data structures for points and usual transformations
+//! (rotations, isometries, etc.)
+
+mod op_macros;
+
+mod point;
+mod point_construction;
+mod point_alias;
+mod point_ops;
+mod point_alga;
+mod point_conversion;
+mod point_coordinates;
+
+mod rotation;
+mod rotation_construction;
+mod rotation_ops;
+mod rotation_alga; // FIXME: implement RotationBase methods.
+mod rotation_conversion;
+mod rotation_alias;
+mod rotation_specialization;
+
+mod quaternion;
+mod quaternion_construction;
+mod quaternion_ops;
+mod quaternion_alga;
+mod quaternion_alias;
+mod quaternion_coordinates;
+mod quaternion_conversion;
+
+mod unit_complex;
+mod unit_complex_construction;
+mod unit_complex_ops;
+mod unit_complex_alga;
+mod unit_complex_conversion;
+
+mod translation;
+mod translation_construction;
+mod translation_ops;
+mod translation_alga;
+mod translation_conversion;
+mod translation_alias;
+
+mod isometry;
+mod isometry_construction;
+mod isometry_ops;
+mod isometry_alga;
+mod isometry_conversion;
+mod isometry_alias;
+
+mod similarity;
+mod similarity_construction;
+mod similarity_ops;
+mod similarity_alga;
+mod similarity_conversion;
+mod similarity_alias;
+
+mod transform;
+mod transform_construction;
+mod transform_ops;
+mod transform_alga;
+mod transform_conversion;
+mod transform_alias;
+
+mod orthographic;
+mod perspective;
+
+pub use self::point::*;
+pub use self::point_alias::*;
+
+pub use self::rotation::*;
+pub use self::rotation_alias::*;
+
+pub use self::quaternion::*;
+pub use self::quaternion_alias::*;
+
+pub use self::unit_complex::*;
+
+pub use self::translation::*;
+pub use self::translation_alias::*;
+
+pub use self::isometry::*;
+pub use self::isometry_alias::*;
+
+pub use self::similarity::*;
+pub use self::similarity_alias::*;
+
+pub use self::transform::*;
+pub use self::transform_alias::*;
+
+pub use self::orthographic::{OrthographicBase, Orthographic3};
+pub use self::perspective::{PerspectiveBase, Perspective3};

src/geometry/op_macros.rs

@@ -0,0 +1,188 @@
+#![macro_use]
+
+
+// FIXME: merge with `md_impl`.
+/// Macro for the implementation of multiplication and division.
+macro_rules! md_impl(
+    (
+    // Operator, operator method, and calar bounds.
+     $Op: ident, $op: ident $(where N: $($ScalarBounds: ident),*)*;
+     // Storage dimensions, and dimension bounds.
+     ($R1: ty, $C1: ty),($R2: ty, $C2: ty) for $($Dims: ident: $DimsBound: ident $(<$BoundParam: ty>)*),+
+     // [Optional] Extra allocator bounds.
+     $(where $ConstraintType: ty: $ConstraintBound: ident<$($ConstraintBoundParams: ty $( = $EqBound: ty )*),*> )*;
+     // Argument identifiers and types + output.
+     $lhs: ident: $Lhs: ty, $rhs: ident: $Rhs: ty, Output = $Result: ty;
+     // Operator actual mplementation.
+     $action: expr;
+     // Lifetime.
+     $($lives: tt),*) => {
+        impl<$($lives ,)* N $(, $Dims: $DimsBound $(<$BoundParam>)*)*, SA, SB> $Op<$Rhs> for $Lhs
+            where N:  Scalar + Zero + ClosedAdd + ClosedMul $($(+ $ScalarBounds)*)*,
+                  SA: Storage<N, $R1, $C1>,
+                  SB: Storage<N, $R2, $C2>,
+                  $( $ConstraintType: $ConstraintBound<$( $ConstraintBoundParams $( = $EqBound )*),*> ),*
+                   {
+            type Output = $Result;
+
+            #[inline]
+            fn $op($lhs, $rhs: $Rhs) -> Self::Output {
+                $action
+            }
+        }
+    }
+);
+
+
+/// Macro for the implementation of multiplication and division.
+/// Implements all the argument reference combinations.
+macro_rules! md_impl_all(
+    (
+     // Operator, operator method, and calar bounds.
+     $Op: ident, $op: ident $(where N: $($ScalarBounds: ident),*)*;
+     // Storage dimensions, and dimension bounds.
+     ($R1: ty, $C1: ty),($R2: ty, $C2: ty) for $($Dims: ident: $DimsBound: ident $(<$BoundParam: ty>)*),+
+     // [Optional] Extra allocator bounds.
+     $(where $ConstraintType: ty: $ConstraintBound: ident<$($ConstraintBoundParams: ty $( = $EqBound: ty )*),*> )*;
+     // Argument identifiers and types + output.
+     $lhs: ident: $Lhs: ty, $rhs: ident: $Rhs: ty, Output = $Result: ty;
+     // Operators actual implementations.
+     [val val] => $action_val_val: expr;
+     [ref val] => $action_ref_val: expr;
+     [val ref] => $action_val_ref: expr;
+     [ref ref] => $action_ref_ref: expr;) => {
+
+        md_impl!(
+            $Op, $op $(where N: $($ScalarBounds),*)*;
+            ($R1, $C1),($R2, $C2) for $($Dims: $DimsBound $(<$BoundParam>)*),+
+            $(where $ConstraintType: $ConstraintBound<$($ConstraintBoundParams $( = $EqBound )*),*>)*;
+            $lhs: $Lhs, $rhs: $Rhs, Output = $Result;
+            $action_val_val; );
+
+        md_impl!(
+            $Op, $op $(where N: $($ScalarBounds),*)*;
+            ($R1, $C1),($R2, $C2) for $($Dims: $DimsBound $(<$BoundParam>)*),+
+            $(where $ConstraintType: $ConstraintBound<$($ConstraintBoundParams $( = $EqBound )*),*>)*;
+            $lhs: &'a $Lhs, $rhs: $Rhs, Output = $Result;
+            $action_ref_val; 'a);
+
+        md_impl!(
+            $Op, $op $(where N: $($ScalarBounds),*)*;
+            ($R1, $C1),($R2, $C2) for $($Dims: $DimsBound $(<$BoundParam>)*),+
+            $(where $ConstraintType: $ConstraintBound<$($ConstraintBoundParams $( = $EqBound )*),*>)*;
+            $lhs: $Lhs, $rhs: &'b $Rhs, Output = $Result;
+            $action_val_ref; 'b);
+
+        md_impl!(
+            $Op, $op $(where N: $($ScalarBounds),*)*;
+            ($R1, $C1),($R2, $C2) for $($Dims: $DimsBound $(<$BoundParam>)*),+
+            $(where $ConstraintType: $ConstraintBound<$($ConstraintBoundParams $( = $EqBound )*),*>)*;
+            $lhs: &'a $Lhs, $rhs: &'b $Rhs, Output = $Result;
+            $action_ref_ref; 'a, 'b);
+    }
+);
+
+
+/// Macro for the implementation of assignement-multiplication and assignement-division.
+macro_rules! md_assign_impl(
+    (
+     // Operator, operator method, and scalar bounds.
+     $Op: ident, $op: ident $(where N: $($ScalarBounds: ident),*)*;
+     // Storage dimensions, and dimension bounds.
+     ($R1: ty, $C1: ty),($R2: ty, $C2: ty) for $($Dims: ident: $DimsBound: ident $(<$BoundParam: ty>)*),+
+     // [Optional] Extra allocator bounds.
+     $(where $ConstraintType: ty: $ConstraintBound: ident<$($ConstraintBoundParams: ty $( = $EqBound: ty )*),*> )*;
+     // Argument identifiers and types.
+     $lhs: ident: $Lhs: ty, $rhs: ident: $Rhs: ty;
+     // Actual implementation and lifetimes.
+     $action: expr; $($lives: tt),*) => {
+        impl<$($lives ,)* N $(, $Dims: $DimsBound $(<$BoundParam>)*)*, SA, SB> $Op<$Rhs> for $Lhs
+            where N:  Scalar + Zero + ClosedAdd + ClosedMul $($(+ $ScalarBounds)*)*,
+                  SA: OwnedStorage<N, $R1, $C1>, // FIXME: this is too restrictive.
+                  SB: Storage<N, $R2, $C2>,
+                  SA::Alloc: OwnedAllocator<N, $R1, $C1, SA>,
+                  $( $ConstraintType: $ConstraintBound<$( $ConstraintBoundParams $( = $EqBound )*),*> ),*
+        {
+            #[inline]
+            fn $op(&mut $lhs, $rhs: $Rhs) {
+                $action
+            }
+        }
+    }
+);
+
+/// Macro for the implementation of assignement-multiplication and assignement-division with and
+/// without reference to the right-hand-side.
+macro_rules! md_assign_impl_all(
+    (
+     // Operator, operator method, and scalar bounds.
+     $Op: ident, $op: ident $(where N: $($ScalarBounds: ident),*)*;
+     // Storage dimensions, and dimension bounds.
+     ($R1: ty, $C1: ty),($R2: ty, $C2: ty) for $($Dims: ident: $DimsBound: ident $(<$BoundParam: ty>)*),+
+     // [Optional] Extra allocator bounds.
+     $(where $ConstraintType: ty: $ConstraintBound: ident<$($ConstraintBoundParams: ty $( = $EqBound: ty )*),*> )*;
+     // Argument identifiers and types.
+     $lhs: ident: $Lhs: ty, $rhs: ident: $Rhs: ty;
+     // Actual implementation and lifetimes.
+     [val] => $action_val: expr;
+     [ref] => $action_ref: expr;) => {
+        md_assign_impl!(
+            $Op, $op $(where N: $($ScalarBounds),*)*;
+            ($R1, $C1),($R2, $C2) for $($Dims: $DimsBound $(<$BoundParam>)*),+
+            $(where $ConstraintType: $ConstraintBound<$($ConstraintBoundParams $( = $EqBound )*),*>)*;
+            $lhs: $Lhs, $rhs: $Rhs;
+            $action_val; );
+
+        md_assign_impl!(
+            $Op, $op $(where N: $($ScalarBounds),*)*;
+            ($R1, $C1),($R2, $C2) for $($Dims: $DimsBound $(<$BoundParam>)*),+
+            $(where $ConstraintType: $ConstraintBound<$($ConstraintBoundParams $( = $EqBound )*),*>)*;
+            $lhs: $Lhs, $rhs: &'b $Rhs;
+            $action_ref; 'b);
+    }
+);
+
+// FIXME: merge with `as_impl`.
+/// Macro for the implementation of addition and subtraction.
+macro_rules! add_sub_impl(
+    ($Op: ident, $op: ident, $bound: ident;
+     ($R1: ty, $C1: ty),($R2: ty, $C2: ty) $(-> ($RRes: ty))* for $($Dims: ident: $DimsBound: ident),+;
+     $lhs: ident: $Lhs: ty, $rhs: ident: $Rhs: ty, Output = $Result: ty;
+     $action: expr; $($lives: tt),*) => {
+        impl<$($lives ,)* N $(, $Dims: $DimsBound)*, SA, SB> $Op<$Rhs> for $Lhs
+            where N:  Scalar + $bound,
+                  SA: Storage<N, $R1, $C1>,
+                  SB: Storage<N, $R2, $C2>,
+                  SA::Alloc: SameShapeAllocator<N, $R1, $C1, $R2, $C2, SA>,
+                  ShapeConstraint: SameNumberOfRows<$R1, $R2 $(, Representative = $RRes)*> +
+                                   SameNumberOfColumns<$C1, $C2> {
+            type Output = $Result;
+
+            #[inline]
+            fn $op($lhs, $rhs: $Rhs) -> Self::Output {
+                $action
+            }
+        }
+    }
+);
+
+// FIXME: merge with `md_assign_impl`.
+/// Macro for the implementation of assignement-addition and assignement-subtraction.
+macro_rules! add_sub_assign_impl(
+    ($Op: ident, $op: ident, $bound: ident;
+     ($R1: ty, $C1: ty),($R2: ty, $C2: ty) for $($Dims: ident: $DimsBound: ident),+;
+     $lhs: ident: $Lhs: ty, $rhs: ident: $Rhs: ty;
+     $action: expr; $($lives: tt),*) => {
+        impl<$($lives ,)* N $(, $Dims: $DimsBound)*, SA, SB> $Op<$Rhs> for $Lhs
+            where N:  Scalar + $bound,
+                  SA: OwnedStorage<N, $R1, $C1>, // FIXME: this is too restrictive.
+                  SB: Storage<N, $R2, $C2>,
+                  SA::Alloc: OwnedAllocator<N, $R1, $C1, SA>,
+                  ShapeConstraint: SameNumberOfRows<$R1, $R2> + SameNumberOfColumns<$C1, $C2> {
+            #[inline]
+            fn $op(&mut $lhs, $rhs: $Rhs) {
+                $action
+            }
+        }
+    }
+);

src/geometry/orthographic.rs

@@ -0,0 +1,295 @@
+#[cfg(feature="arbitrary")]
+use quickcheck::{Arbitrary, Gen};
+use rand::{Rand, Rng};
+
+use alga::general::Real;
+
+use core::{Scalar, SquareMatrix, OwnedSquareMatrix, ColumnVector, OwnedColumnVector, MatrixArray};
+use core::dimension::{U1, U3, U4};
+use core::storage::{OwnedStorage, Storage, StorageMut};
+use core::allocator::OwnedAllocator;
+use core::helper;
+
+use geometry::{PointBase, OwnedPoint};
+
+/// A 3D orthographic projection stored as an homogeneous 4x4 matrix.
+#[derive(Debug, Clone, Copy)] // FIXME: Hash
+#[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
+pub struct OrthographicBase<N: Scalar, S: Storage<N, U4, U4>> {
+    matrix: SquareMatrix<N, U4, S>
+}
+
+/// A 3D orthographic projection stored as a static homogeneous 4x4 matrix.
+pub type Orthographic3<N> = OrthographicBase<N, MatrixArray<N, U4, U4>>;
+
+impl<N, S> Eq for OrthographicBase<N, S>
+    where N: Scalar + Eq,
+          S: Storage<N, U4, U4> { }
+
+impl<N: Scalar, S: Storage<N, U4, U4>> PartialEq for OrthographicBase<N, S> {
+    #[inline]
+    fn eq(&self, right: &Self) -> bool {
+        self.matrix == right.matrix
+    }
+}
+
+impl<N, S> OrthographicBase<N, S>
+    where N: Real,
+          S: OwnedStorage<N, U4, U4>,
+          S::Alloc: OwnedAllocator<N, U4, U4, S> {
+    /// Creates a new orthographic projection matrix.
+    #[inline]
+    pub fn new(left: N, right: N, bottom: N, top: N, znear: N, zfar: N) -> Self {
+        assert!(left < right, "The left corner must be farther than the right corner.");
+        assert!(bottom < top, "The top corner must be higher than the bottom corner.");
+        assert!(znear < zfar, "The far plane must be farther than the near plane.");
+
+        let matrix = SquareMatrix::<N, U4, S>::identity();
+        let mut res = Self::from_matrix_unchecked(matrix);
+
+        res.set_left_and_right(left, right);
+        res.set_bottom_and_top(bottom, top);
+        res.set_znear_and_zfar(znear, zfar);
+
+        res
+    }
+
+    /// Wraps the given matrix to interpret it as a 3D orthographic matrix.
+    ///
+    /// It is not checked whether or not the given matrix actually represents an orthographic
+    /// projection.
+    #[inline]
+    pub fn from_matrix_unchecked(matrix: SquareMatrix<N, U4, S>) -> Self {
+        OrthographicBase {
+            matrix: matrix
+        }
+    }
+
+    /// Creates a new orthographic projection matrix from an aspect ratio and the vertical field of view.
+    #[inline]
+    pub fn from_fov(aspect: N, vfov: N, znear: N, zfar: N) -> Self {
+        assert!(znear < zfar, "The far plane must be farther than the near plane.");
+        assert!(!relative_eq!(aspect, N::zero()), "The apsect ratio must not be zero.");
+
+        let half: N = ::convert(0.5);
+        let width  = zfar * (vfov * half).tan();
+        let height = width / aspect;
+
+        Self::new(-width * half, width * half, -height * half, height * half, znear, zfar)
+    }
+}
+
+impl<N: Real, S: Storage<N, U4, U4>> OrthographicBase<N, S> {
+    /// A reference to the underlying homogeneous transformation matrix.
+    #[inline]
+    pub fn as_matrix(&self) -> &SquareMatrix<N, U4, S> {
+        &self.matrix
+    }
+
+    /// Retrieves the underlying homogeneous matrix.
+    #[inline]
+    pub fn unwrap(self) -> SquareMatrix<N, U4, S> {
+        self.matrix
+    }
+
+    /// Retrieves the inverse of the underlying homogeneous matrix.
+    #[inline]
+    pub fn inverse(&self) -> OwnedSquareMatrix<N, U4, S::Alloc> {
+        let mut res = self.to_homogeneous();
+
+        let inv_m11 = N::one() / self.matrix[(0, 0)];
+        let inv_m22 = N::one() / self.matrix[(1, 1)];
+        let inv_m33 = N::one() / self.matrix[(2, 2)];
+
+        res[(0, 0)] = inv_m11;
+        res[(1, 1)] = inv_m22;
+        res[(2, 2)] = inv_m33;
+
+        res[(0, 3)] = -self.matrix[(0, 3)] * inv_m11; 
+        res[(1, 3)] = -self.matrix[(1, 3)] * inv_m22; 
+        res[(2, 3)] = -self.matrix[(2, 3)] * inv_m33; 
+
+        res
+    }
+
+    /// Computes the corresponding homogeneous matrix.
+    #[inline]
+    pub fn to_homogeneous(&self) -> OwnedSquareMatrix<N, U4, S::Alloc> {
+        self.matrix.clone_owned()
+    }
+
+    /// The smallest x-coordinate of the view cuboid.
+    #[inline]
+    pub fn left(&self) -> N {
+        (-N::one() - self.matrix[(0, 3)]) / self.matrix[(0, 0)]
+    }
+
+    /// The largest x-coordinate of the view cuboid.
+    #[inline]
+    pub fn right(&self) -> N {
+        (N::one() - self.matrix[(0, 3)]) / self.matrix[(0, 0)]
+    }
+
+    /// The smallest y-coordinate of the view cuboid.
+    #[inline]
+    pub fn bottom(&self) -> N {
+        (-N::one() - self.matrix[(1, 3)]) / self.matrix[(1, 1)]
+    }
+
+    /// The largest y-coordinate of the view cuboid.
+    #[inline]
+    pub fn top(&self) -> N {
+        (N::one() - self.matrix[(1, 3)]) / self.matrix[(1, 1)]
+    }
+
+    /// The near plane offset of the view cuboid.
+    #[inline]
+    pub fn znear(&self) -> N {
+        (N::one() + self.matrix[(2, 3)]) / self.matrix[(2, 2)]
+    }
+
+    /// The far plane offset of the view cuboid.
+    #[inline]
+    pub fn zfar(&self) -> N {
+        (-N::one() + self.matrix[(2, 3)]) / self.matrix[(2, 2)]
+    }
+
+    // FIXME: when we get specialization, specialize the Mul impl instead.
+    /// Projects a point. Faster than matrix multiplication.
+    #[inline]
+    pub fn project_point<SB>(&self, p: &PointBase<N, U3, SB>) -> OwnedPoint<N, U3, SB::Alloc>
+        where SB: Storage<N, U3, U1> {
+
+        OwnedPoint::<N, U3, SB::Alloc>::new(
+            self.matrix[(0, 0)] * p[0] + self.matrix[(0, 3)],
+            self.matrix[(1, 1)] * p[1] + self.matrix[(1, 3)],
+            self.matrix[(2, 2)] * p[2] + self.matrix[(2, 3)]
+        )
+    }
+
+    /// Un-projects a point. Faster than multiplication by the underlying matrix inverse.
+    #[inline]
+    pub fn unproject_point<SB>(&self, p: &PointBase<N, U3, SB>) -> OwnedPoint<N, U3, SB::Alloc>
+        where SB: Storage<N, U3, U1> {
+
+        OwnedPoint::<N, U3, SB::Alloc>::new(
+            (p[0] - self.matrix[(0, 3)]) / self.matrix[(0, 0)],
+            (p[1] - self.matrix[(1, 3)]) / self.matrix[(1, 1)],
+            (p[2] - self.matrix[(2, 3)]) / self.matrix[(2, 2)]
+        )
+    }
+
+    // FIXME: when we get specialization, specialize the Mul impl instead.
+    /// Projects a vector. Faster than matrix multiplication.
+    #[inline]
+    pub fn project_vector<SB>(&self, p: &ColumnVector<N, U3, SB>) -> OwnedColumnVector<N, U3, SB::Alloc>
+        where SB: Storage<N, U3, U1> {
+
+        OwnedColumnVector::<N, U3, SB::Alloc>::new(
+            self.matrix[(0, 0)] * p[0],
+            self.matrix[(1, 1)] * p[1],
+            self.matrix[(2, 2)] * p[2]
+        )
+    }
+}
+
+impl<N: Real, S: StorageMut<N, U4, U4>> OrthographicBase<N, S> {
+    /// Sets the smallest x-coordinate of the view cuboid.
+    #[inline]
+    pub fn set_left(&mut self, left: N) {
+        let right = self.right();
+        self.set_left_and_right(left, right);
+    }
+
+    /// Sets the largest x-coordinate of the view cuboid.
+    #[inline]
+    pub fn set_right(&mut self, right: N) {
+        let left = self.left();
+        self.set_left_and_right(left, right);
+    }
+
+    /// Sets the smallest y-coordinate of the view cuboid.
+    #[inline]
+    pub fn set_bottom(&mut self, bottom: N) {
+        let top = self.top();
+        self.set_bottom_and_top(bottom, top);
+    }
+
+    /// Sets the largest y-coordinate of the view cuboid.
+    #[inline]
+    pub fn set_top(&mut self, top: N) {
+        let bottom = self.bottom();
+        self.set_bottom_and_top(bottom, top);
+    }
+
+    /// Sets the near plane offset of the view cuboid.
+    #[inline]
+    pub fn set_znear(&mut self, znear: N) {
+        let zfar = self.zfar();
+        self.set_znear_and_zfar(znear, zfar);
+    }
+
+    /// Sets the far plane offset of the view cuboid.
+    #[inline]
+    pub fn set_zfar(&mut self, zfar: N) {
+        let znear = self.znear();
+        self.set_znear_and_zfar(znear, zfar);
+    }
+
+    /// Sets the view cuboid coordinates along the `x` axis.
+    #[inline]
+    pub fn set_left_and_right(&mut self, left: N, right: N) {
+        assert!(left < right, "The left corner must be farther than the right corner.");
+        self.matrix[(0, 0)] = ::convert::<_, N>(2.0) / (right - left);
+        self.matrix[(0, 3)] = -(right + left) / (right - left);
+    }
+
+    /// Sets the view cuboid coordinates along the `y` axis.
+    #[inline]
+    pub fn set_bottom_and_top(&mut self, bottom: N, top: N) {
+        assert!(bottom < top, "The top corner must be higher than the bottom corner.");
+        self.matrix[(1, 1)] = ::convert::<_, N>(2.0) / (top - bottom);
+        self.matrix[(1, 3)] = -(top + bottom) / (top - bottom);
+    }
+
+    /// Sets the near and far plane offsets of the view cuboid.
+    #[inline]
+    pub fn set_znear_and_zfar(&mut self, znear: N, zfar: N) {
+        assert!(!relative_eq!(zfar - znear, N::zero()), "The near-plane and far-plane must not be superimposed.");
+        self.matrix[(2, 2)] = -::convert::<_, N>(2.0) / (zfar - znear);
+        self.matrix[(2, 3)] = -(zfar + znear) / (zfar - znear);
+    }
+}
+
+impl<N, S> Rand for OrthographicBase<N, S>
+    where N: Real + Rand,
+          S: OwnedStorage<N, U4, U4>,
+          S::Alloc: OwnedAllocator<N, U4, U4, S> {
+    fn rand<R: Rng>(r: &mut R) -> Self {
+        let left   = Rand::rand(r);
+        let right  = helper::reject_rand(r, |x: &N| *x > left);
+        let bottom = Rand::rand(r);
+        let top    = helper::reject_rand(r, |x: &N| *x > bottom);
+        let znear  = Rand::rand(r);
+        let zfar   = helper::reject_rand(r, |x: &N| *x > znear);
+
+        Self::new(left, right, bottom, top, znear, zfar)
+    }
+}
+
+#[cfg(feature="arbitrary")]
+impl<N, S> Arbitrary for OrthographicBase<N, S>
+    where N: Real + Arbitrary,
+          S: OwnedStorage<N, U4, U4> + Send,
+          S::Alloc: OwnedAllocator<N, U4, U4, S> {
+    fn arbitrary<G: Gen>(g: &mut G) -> Self {
+        let left   = Arbitrary::arbitrary(g);
+        let right  = helper::reject(g, |x: &N| *x > left);
+        let bottom = Arbitrary::arbitrary(g);
+        let top    = helper::reject(g, |x: &N| *x > bottom);
+        let znear  = Arbitrary::arbitrary(g);
+        let zfar   = helper::reject(g, |x: &N| *x > znear);
+
+        Self::new(left, right, bottom, top, znear, zfar)
+    }
+}

src/geometry/perspective.rs

@@ -0,0 +1,256 @@
+#[cfg(feature="arbitrary")]
+use quickcheck::{Arbitrary, Gen};
+use rand::{Rand, Rng};
+
+use alga::general::Real;
+
+use core::{Scalar, SquareMatrix, OwnedSquareMatrix, ColumnVector, OwnedColumnVector, MatrixArray};
+use core::dimension::{U1, U3, U4};
+use core::storage::{OwnedStorage, Storage, StorageMut};
+use core::allocator::OwnedAllocator;
+use core::helper;
+
+use geometry::{PointBase, OwnedPoint};
+
+/// A 3D perspective projection stored as an homogeneous 4x4 matrix.
+#[derive(Debug, Clone, Copy)] // FIXME: Hash
+#[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
+pub struct PerspectiveBase<N: Scalar, S: Storage<N, U4, U4>> {
+    matrix: SquareMatrix<N, U4, S>
+}
+
+/// A 3D perspective projection stored as a static homogeneous 4x4 matrix.
+pub type Perspective3<N> = PerspectiveBase<N, MatrixArray<N, U4, U4>>;
+
+impl<N, S> Eq for PerspectiveBase<N, S>
+    where N: Scalar + Eq,
+          S: Storage<N, U4, U4> { }
+
+impl<N, S> PartialEq for PerspectiveBase<N, S>
+    where N: Scalar,
+          S: Storage<N, U4, U4> {
+    #[inline]
+    fn eq(&self, right: &Self) -> bool {
+        self.matrix == right.matrix
+    }
+}
+
+impl<N, S> PerspectiveBase<N, S>
+    where N: Real,
+          S: OwnedStorage<N, U4, U4>,
+          S::Alloc: OwnedAllocator<N, U4, U4, S> {
+    /// Creates a new perspective matrix from the aspect ratio, y field of view, and near/far planes.
+    pub fn new(aspect: N, fovy: N, znear: N, zfar: N) -> Self {
+        assert!(!relative_eq!(zfar - znear, N::zero()), "The near-plane and far-plane must not be superimposed.");
+        assert!(!relative_eq!(aspect, N::zero()), "The apsect ratio must not be zero.");
+
+        let matrix = SquareMatrix::<N, U4, S>::identity();
+        let mut res = PerspectiveBase::from_matrix_unchecked(matrix);
+
+        res.set_fovy(fovy);
+        res.set_aspect(aspect);
+        res.set_znear_and_zfar(znear, zfar);
+
+        res.matrix[(3, 3)] = N::zero();
+        res.matrix[(3, 2)] = -N::one();
+
+        res
+    }
+
+
+    /// Wraps the given matrix to interpret it as a 3D perspective matrix.
+    ///
+    /// It is not checked whether or not the given matrix actually represents an orthographic
+    /// projection.
+    #[inline]
+    pub fn from_matrix_unchecked(matrix: SquareMatrix<N, U4, S>) -> Self {
+        PerspectiveBase {
+            matrix: matrix
+        }
+    }
+}
+
+impl<N, S> PerspectiveBase<N, S>
+    where N: Real,
+          S: Storage<N, U4, U4> {
+
+    /// A reference to the underlying homogeneous transformation matrix.
+    #[inline]
+    pub fn as_matrix(&self) -> &SquareMatrix<N, U4, S> {
+        &self.matrix
+    }
+
+    /// Retrieves the underlying homogeneous matrix.
+    #[inline]
+    pub fn unwrap(self) -> SquareMatrix<N, U4, S> {
+        self.matrix
+    }
+
+    /// Retrieves the inverse of the underlying homogeneous matrix.
+    #[inline]
+    pub fn inverse(&self) -> OwnedSquareMatrix<N, U4, S::Alloc> {
+        let mut res = self.to_homogeneous();
+
+        res[(0, 0)] = N::one() / self.matrix[(0, 0)];
+        res[(1, 1)] = N::one() / self.matrix[(1, 1)];
+        res[(2, 2)] = N::zero();
+
+        let m23 = self.matrix[(2, 3)];
+        let m32 = self.matrix[(3, 2)];
+
+        res[(2, 3)] = N::one() / m32;
+        res[(3, 2)] = N::one() / m23;
+        res[(3, 3)] = -self.matrix[(2, 2)] / (m23 * m32);
+
+        res
+    }
+
+    /// Computes the corresponding homogeneous matrix.
+    #[inline]
+    pub fn to_homogeneous(&self) -> OwnedSquareMatrix<N, U4, S::Alloc> {
+        self.matrix.clone_owned()
+    }
+
+    /// Gets the `width / height` aspect ratio of the view frustrum.
+    #[inline]
+    pub fn aspect(&self) -> N {
+        self.matrix[(1, 1)] / self.matrix[(0, 0)]
+    }
+
+    /// Gets the y field of view of the view frustrum.
+    #[inline]
+    pub fn fovy(&self) -> N {
+        (N::one() / self.matrix[(1, 1)]).atan() * ::convert(2.0)
+    }
+
+    /// Gets the near plane offset of the view frustrum.
+    #[inline]
+    pub fn znear(&self) -> N {
+        let ratio = (-self.matrix[(2, 2)] + N::one()) / (-self.matrix[(2, 2)] - N::one());
+
+        self.matrix[(2, 3)] / (ratio * ::convert(2.0)) - self.matrix[(2, 3)] / ::convert(2.0)
+    }
+
+    /// Gets the far plane offset of the view frustrum.
+    #[inline]
+    pub fn zfar(&self) -> N {
+        let ratio = (-self.matrix[(2, 2)] + N::one()) / (-self.matrix[(2, 2)] - N::one());
+
+        (self.matrix[(2, 3)] - ratio * self.matrix[(2, 3)]) / ::convert(2.0)
+    }
+
+    // FIXME: add a method to retrieve znear and zfar simultaneously?
+
+
+
+    // FIXME: when we get specialization, specialize the Mul impl instead.
+    /// Projects a point. Faster than matrix multiplication.
+    #[inline]
+    pub fn project_point<SB>(&self, p: &PointBase<N, U3, SB>) -> OwnedPoint<N, U3, SB::Alloc>
+        where SB: Storage<N, U3, U1> {
+
+        let inverse_denom = -N::one() / p[2];
+        OwnedPoint::<N, U3, SB::Alloc>::new(
+             self.matrix[(0, 0)] * p[0] * inverse_denom,
+             self.matrix[(1, 1)] * p[1] * inverse_denom,
+            (self.matrix[(2, 2)] * p[2] + self.matrix[(2, 3)]) * inverse_denom
+        )
+    }
+
+    /// Un-projects a point. Faster than multiplication by the matrix inverse.
+    #[inline]
+    pub fn unproject_point<SB>(&self, p: &PointBase<N, U3, SB>) -> OwnedPoint<N, U3, SB::Alloc>
+        where SB: Storage<N, U3, U1> {
+
+        let inverse_denom = self.matrix[(2, 3)] / (p[2] + self.matrix[(2, 2)]);
+
+        OwnedPoint::<N, U3, SB::Alloc>::new(
+            p[0] * inverse_denom / self.matrix[(0, 0)],
+            p[1] * inverse_denom / self.matrix[(1, 1)],
+            -inverse_denom
+        )
+    }
+
+    // FIXME: when we get specialization, specialize the Mul impl instead.
+    /// Projects a vector. Faster than matrix multiplication.
+    #[inline]
+    pub fn project_vector<SB>(&self, p: &ColumnVector<N, U3, SB>) -> OwnedColumnVector<N, U3, SB::Alloc>
+        where SB: Storage<N, U3, U1> {
+
+        let inverse_denom = -N::one() / p[2];
+        OwnedColumnVector::<N, U3, SB::Alloc>::new(
+            self.matrix[(0, 0)] * p[0] * inverse_denom,
+            self.matrix[(1, 1)] * p[1] * inverse_denom,
+            self.matrix[(2, 2)]
+        )
+    }
+}
+
+
+impl<N, S> PerspectiveBase<N, S>
+    where N: Real,
+          S: StorageMut<N, U4, U4> {
+    /// Updates this perspective matrix with a new `width / height` aspect ratio of the view
+    /// frustrum.
+    #[inline]
+    pub fn set_aspect(&mut self, aspect: N) {
+        assert!(!relative_eq!(aspect, N::zero()), "The aspect ratio must not be zero.");
+        self.matrix[(0, 0)] = self.matrix[(1, 1)] / aspect;
+    }
+
+    /// Updates this perspective with a new y field of view of the view frustrum.
+    #[inline]
+    pub fn set_fovy(&mut self, fovy: N) {
+        let old_m22  = self.matrix[(1, 1)];
+        self.matrix[(1, 1)] = N::one() / (fovy / ::convert(2.0)).tan();
+        self.matrix[(0, 0)] = self.matrix[(0, 0)] * (self.matrix[(1, 1)] / old_m22);
+    }
+
+    /// Updates this perspective matrix with a new near plane offset of the view frustrum.
+    #[inline]
+    pub fn set_znear(&mut self, znear: N) {
+        let zfar = self.zfar();
+        self.set_znear_and_zfar(znear, zfar);
+    }
+
+    /// Updates this perspective matrix with a new far plane offset of the view frustrum.
+    #[inline]
+    pub fn set_zfar(&mut self, zfar: N) {
+        let znear = self.znear();
+        self.set_znear_and_zfar(znear, zfar);
+    }
+
+    /// Updates this perspective matrix with new near and far plane offsets of the view frustrum.
+    #[inline]
+    pub fn set_znear_and_zfar(&mut self, znear: N, zfar: N) {
+        self.matrix[(2, 2)] = (zfar + znear) / (znear - zfar);
+        self.matrix[(2, 3)] = zfar * znear * ::convert(2.0) / (znear - zfar);
+    }
+}
+
+impl<N, S> Rand for PerspectiveBase<N, S>
+    where N: Real + Rand,
+          S: OwnedStorage<N, U4, U4>,
+          S::Alloc: OwnedAllocator<N, U4, U4, S> {
+    fn rand<R: Rng>(r: &mut R) -> Self {
+        let znear  = Rand::rand(r);
+        let zfar   = helper::reject_rand(r, |&x: &N| !(x - znear).is_zero());
+        let aspect = helper::reject_rand(r, |&x: &N| !x.is_zero());
+
+        Self::new(aspect, Rand::rand(r), znear, zfar)
+    }
+}
+
+#[cfg(feature="arbitrary")]
+impl<N, S> Arbitrary for PerspectiveBase<N, S>
+    where N: Real + Arbitrary,
+          S: OwnedStorage<N, U4, U4> + Send,
+          S::Alloc: OwnedAllocator<N, U4, U4, S> {
+    fn arbitrary<G: Gen>(g: &mut G) -> Self {
+        let znear  = Arbitrary::arbitrary(g);
+        let zfar   = helper::reject(g, |&x: &N| !(x - znear).is_zero());
+        let aspect = helper::reject(g, |&x: &N| !x.is_zero());
+
+        Self::new(aspect, Arbitrary::arbitrary(g), znear, zfar)
+    }
+}

src/geometry/point.rs

@@ -0,0 +1,228 @@
+use num::One;
+use std::fmt;
+use std::cmp::Ordering;
+use approx::ApproxEq;
+
+use core::{Scalar, ColumnVector, OwnedColumnVector};
+use core::iter::{MatrixIter, MatrixIterMut};
+use core::dimension::{DimName, DimNameSum, DimNameAdd, U1};
+use core::storage::{Storage, StorageMut, MulStorage};
+use core::allocator::{Allocator, SameShapeR};
+
+// XXX Bad name: we can't even add points…
+/// The type of the result of the sum of a point with a vector.
+pub type PointSum<N, D1, D2, SA> =
+    PointBase<N, SameShapeR<D1, D2>,
+    <<SA as Storage<N, D1, U1>>::Alloc as Allocator<N, SameShapeR<D1, D2>, U1>>::Buffer>;
+
+/// The type of the result of the multiplication of a point by a matrix.
+pub type PointMul<N, R1, C1, SA> = PointBase<N, R1, MulStorage<N, R1, C1, U1, SA>>;
+
+/// A point with an owned storage.
+pub type OwnedPoint<N, D, A> = PointBase<N, D, <A as Allocator<N, D, U1>>::Buffer>;
+
+/// A point in a n-dimensional euclidean space.
+#[repr(C)]
+#[derive(Hash, Debug)]
+#[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
+pub struct PointBase<N: Scalar, D: DimName, S: Storage<N, D, U1>> {
+    /// The coordinates of this point, i.e., the shift from the origin.
+    pub coords: ColumnVector<N, D, S>
+}
+
+impl<N, D, S> Copy for PointBase<N, D, S>
+    where N: Scalar,
+          D: DimName,
+          S: Storage<N, D, U1> + Copy { }
+
+impl<N, D, S> Clone for PointBase<N, D, S>
+    where N: Scalar,
+          D: DimName,
+          S: Storage<N, D, U1> + Clone {
+    #[inline]
+    fn clone(&self) -> Self {
+        PointBase::from_coordinates(self.coords.clone())
+    }
+}
+
+impl<N: Scalar, D: DimName, S: Storage<N, D, U1>> PointBase<N, D, S> {
+    /// Creates a new point with the given coordinates.
+    #[inline]
+    pub fn from_coordinates(coords: ColumnVector<N, D, S>) -> PointBase<N, D, S> {
+        PointBase {
+            coords: coords
+        }
+    }
+}
+
+impl<N: Scalar, D: DimName, S: Storage<N, D, U1>> PointBase<N, D, S> {
+    /// Moves this point into one that owns its data.
+    #[inline]
+    pub fn into_owned(self) -> OwnedPoint<N, D, S::Alloc> {
+        PointBase::from_coordinates(self.coords.into_owned())
+    }
+
+    /// Clones this point into one that owns its data.
+    #[inline]
+    pub fn clone_owned(&self) -> OwnedPoint<N, D, S::Alloc> {
+        PointBase::from_coordinates(self.coords.clone_owned())
+    }
+
+    /// The dimension of this point.
+    #[inline]
+    pub fn len(&self) -> usize {
+        self.coords.len()
+    }
+
+    /// The stride of this point. This is the number of buffer element separating each component of
+    /// this point.
+    #[inline]
+    pub fn stride(&self) -> usize {
+        self.coords.strides().0
+    }
+
+    /// Iterates through this point coordinates.
+    #[inline]
+    pub fn iter(&self) -> MatrixIter<N, D, U1, S> {
+        self.coords.iter()
+    }
+
+    /// Gets a reference to i-th element of this point without bound-checking.
+    #[inline]
+    pub unsafe fn get_unchecked(&self, i: usize) -> &N {
+        self.coords.get_unchecked(i, 0)
+    }
+
+
+    /// Converts this point into a vector in homogeneous coordinates, i.e., appends a `1` at the
+    /// end of it.
+    #[inline]
+    pub fn to_homogeneous(&self) -> OwnedColumnVector<N, DimNameSum<D, U1>, S::Alloc>
+        where N: One,
+              D: DimNameAdd<U1>,
+              S::Alloc: Allocator<N, DimNameSum<D, U1>, U1> {
+
+        let mut res = unsafe { OwnedColumnVector::<N, _, S::Alloc>::new_uninitialized() };
+        res.fixed_slice_mut::<D, U1>(0, 0).copy_from(&self.coords);
+        res[(D::dim(), 0)] = N::one();
+
+        res
+    }
+}
+
+impl<N: Scalar, D: DimName, S: StorageMut<N, D, U1>> PointBase<N, D, S> {
+    /// Mutably iterates through this point coordinates.
+    #[inline]
+    pub fn iter_mut(&mut self) -> MatrixIterMut<N, D, U1, S> {
+        self.coords.iter_mut()
+    }
+
+    /// Gets a mutable reference to i-th element of this point without bound-checking.
+    #[inline]
+    pub unsafe fn get_unchecked_mut(&mut self, i: usize) -> &mut N {
+        self.coords.get_unchecked_mut(i, 0)
+    }
+
+    /// Swaps two entries without bound-checking.
+    #[inline]
+    pub unsafe fn swap_unchecked(&mut self, i1: usize, i2: usize) {
+        self.coords.swap_unchecked((i1, 0), (i2, 0))
+    }
+}
+
+impl<N, D: DimName, S> ApproxEq for PointBase<N, D, S>
+    where N: Scalar + ApproxEq,
+          S: Storage<N, D, U1>,
+          N::Epsilon: Copy {
+    type Epsilon = N::Epsilon;
+
+    #[inline]
+    fn default_epsilon() -> Self::Epsilon {
+        N::default_epsilon()
+    }
+
+    #[inline]
+    fn default_max_relative() -> Self::Epsilon {
+        N::default_max_relative()
+    }
+
+    #[inline]
+    fn default_max_ulps() -> u32 {
+        N::default_max_ulps()
+    }
+
+    #[inline]
+    fn relative_eq(&self, other: &Self, epsilon: Self::Epsilon, max_relative: Self::Epsilon) -> bool {
+        self.coords.relative_eq(&other.coords, epsilon, max_relative)
+    }
+
+    #[inline]
+    fn ulps_eq(&self, other: &Self, epsilon: Self::Epsilon, max_ulps: u32) -> bool {
+        self.coords.ulps_eq(&other.coords, epsilon, max_ulps)
+    }
+}
+
+impl<N, D: DimName, S> Eq for PointBase<N, D, S>
+    where N: Scalar + Eq,
+          S: Storage<N, D, U1> { }
+
+impl<N, D: DimName, S> PartialEq for PointBase<N, D, S>
+    where N: Scalar,
+          S: Storage<N, D, U1> {
+    #[inline]
+    fn eq(&self, right: &Self) -> bool {
+        self.coords == right.coords
+    }
+}
+
+impl<N, D: DimName, S> PartialOrd for PointBase<N, D, S>
+    where N: Scalar + PartialOrd,
+          S: Storage<N, D, U1> {
+    #[inline]
+    fn partial_cmp(&self, other: &Self) -> Option<Ordering> {
+        self.coords.partial_cmp(&other.coords)
+    }
+
+    #[inline]
+    fn lt(&self, right: &Self) -> bool {
+        self.coords.lt(&right.coords)
+    }
+
+    #[inline]
+    fn le(&self, right: &Self) -> bool {
+        self.coords.le(&right.coords)
+    }
+
+    #[inline]
+    fn gt(&self, right: &Self) -> bool {
+        self.coords.gt(&right.coords)
+    }
+
+    #[inline]
+    fn ge(&self, right: &Self) -> bool {
+        self.coords.ge(&right.coords)
+    }
+}
+
+/*
+ *
+ * Display
+ *
+ */
+impl<N, D: DimName, S> fmt::Display for PointBase<N, D, S>
+    where N: Scalar + fmt::Display,
+          S: Storage<N, D, U1> {
+    fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
+        try!(write!(f, "{{"));
+
+        let mut it = self.coords.iter();
+
+        try!(write!(f, "{}", *it.next().unwrap()));
+
+        for comp in it {
+            try!(write!(f, ", {}", *comp));
+        }
+
+        write!(f, "}}")
+    }
+}

src/geometry/point_alga.rs

@@ -0,0 +1,83 @@
+use alga::general::{Field, Real, MeetSemilattice, JoinSemilattice, Lattice};
+use alga::linear::{AffineSpace, EuclideanSpace};
+
+use core::{ColumnVector, Scalar};
+use core::dimension::{DimName, U1};
+use core::storage::OwnedStorage;
+use core::allocator::OwnedAllocator;
+
+use geometry::PointBase;
+
+
+impl<N, D: DimName, S> AffineSpace for PointBase<N, D, S>
+    where N: Scalar + Field,
+          S: OwnedStorage<N, D, U1>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    type Translation = ColumnVector<N, D, S>;
+}
+
+impl<N, D: DimName, S> EuclideanSpace for PointBase<N, D, S>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    type Coordinates = ColumnVector<N, D, S>;
+    type Real        = N;
+
+    #[inline]
+    fn origin() -> Self {
+        Self::origin()
+    }
+
+    #[inline]
+    fn coordinates(&self) -> Self::Coordinates {
+        self.coords.clone()
+    }
+
+    #[inline]
+    fn from_coordinates(coords: Self::Coordinates) -> Self {
+        Self::from_coordinates(coords)
+    }
+
+    #[inline]
+    fn scale_by(&self, n: N) -> Self {
+        self * n
+    }
+}
+
+/*
+ *
+ * Ordering
+ *
+ */
+impl<N, D: DimName, S> MeetSemilattice for PointBase<N, D, S>
+    where N: Scalar + MeetSemilattice,
+          S: OwnedStorage<N, D, U1>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    #[inline]
+    fn meet(&self, other: &Self) -> Self {
+        PointBase::from_coordinates(self.coords.meet(&other.coords))
+    }
+}
+
+impl<N, D: DimName, S> JoinSemilattice for PointBase<N, D, S>
+    where N: Scalar + JoinSemilattice,
+          S: OwnedStorage<N, D, U1>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    #[inline]
+    fn join(&self, other: &Self) -> Self {
+        PointBase::from_coordinates(self.coords.join(&other.coords))
+    }
+}
+
+
+impl<N, D: DimName, S> Lattice for PointBase<N, D, S>
+    where N: Scalar + Lattice,
+          S: OwnedStorage<N, D, U1>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    #[inline]
+    fn meet_join(&self, other: &Self) -> (Self, Self) {
+        let (meet, join) = self.coords.meet_join(&other.coords);
+
+        (PointBase::from_coordinates(meet), PointBase::from_coordinates(join))
+    }
+}

src/geometry/point_alias.rs

@@ -0,0 +1,20 @@
+use core::MatrixArray;
+use core::dimension::{U1, U2, U3, U4, U5, U6};
+
+use geometry::PointBase;
+
+/// A statically sized D-dimensional column point.
+pub type Point<N, D> = PointBase<N, D, MatrixArray<N, D, U1>>;
+
+/// A statically sized 1-dimensional column point.
+pub type Point1<N> = Point<N, U1>;
+/// A statically sized 2-dimensional column point.
+pub type Point2<N> = Point<N, U2>;
+/// A statically sized 3-dimensional column point.
+pub type Point3<N> = Point<N, U3>;
+/// A statically sized 4-dimensional column point.
+pub type Point4<N> = Point<N, U4>;
+/// A statically sized 5-dimensional column point.
+pub type Point5<N> = Point<N, U5>;
+/// A statically sized 6-dimensional column point.
+pub type Point6<N> = Point<N, U6>;

src/geometry/point_construction.rs

@@ -0,0 +1,127 @@
+#[cfg(feature = "arbitrary")]
+use quickcheck::{Arbitrary, Gen};
+
+use rand::{Rand, Rng};
+use num::{Zero, One, Bounded};
+
+use alga::general::ClosedDiv;
+use core::{Scalar, ColumnVector};
+use core::storage::{Storage, OwnedStorage};
+use core::allocator::{Allocator, OwnedAllocator};
+use core::dimension::{DimName, DimNameAdd, DimNameSum, U1, U2, U3, U4, U5, U6};
+
+use geometry::PointBase;
+
+impl<N, D: DimName, S> PointBase<N, D, S>
+    where N: Scalar,
+          S: OwnedStorage<N, D, U1>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    /// Creates a new point with uninitialized coordinates.
+    #[inline]
+    pub unsafe fn new_uninitialized() -> Self {
+        Self::from_coordinates(ColumnVector::<_, D, _>::new_uninitialized())
+    }
+
+    /// Creates a new point with all coordinates equal to zero.
+    #[inline]
+    pub fn origin() -> Self
+        where N: Zero {
+        Self::from_coordinates(ColumnVector::<_, D, _>::from_element(N::zero()))
+    }
+
+    /// Creates a new point from its homogeneous vector representation.
+    ///
+    /// In practice, this builds a D-dimensional points with the same first D component as `v`
+    /// divided by the last component of `v`. Returns `None` if this divisor is zero.
+    #[inline]
+    pub fn from_homogeneous<SB>(v: ColumnVector<N, DimNameSum<D, U1>, SB>) -> Option<Self>
+        where N:  Scalar + Zero + One + ClosedDiv,
+              D:  DimNameAdd<U1>,
+              SB: Storage<N, DimNameSum<D, U1>, U1, Alloc = S::Alloc>,
+              S::Alloc: Allocator<N, DimNameSum<D, U1>, U1> {
+
+        if !v[D::dim()].is_zero() {
+            let coords =  v.fixed_slice::<D, U1>(0, 0) / v[D::dim()];
+            Some(Self::from_coordinates(coords))
+        }
+        else {
+            None
+        }
+    }
+}
+
+
+/*
+ *
+ * Traits that buid points.
+ *
+ */
+impl<N, D: DimName, S> Bounded for PointBase<N, D, S>
+    where N: Scalar + Bounded,
+          S: OwnedStorage<N, D, U1>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    #[inline]
+    fn max_value() -> Self {
+        Self::from_coordinates(ColumnVector::max_value())
+    }
+
+    #[inline]
+    fn min_value() -> Self {
+        Self::from_coordinates(ColumnVector::min_value())
+    }
+}
+
+impl<N, D: DimName, S> Rand for PointBase<N, D, S>
+    where N: Scalar + Rand,
+          S: OwnedStorage<N, D, U1>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    #[inline]
+    fn rand<G: Rng>(rng: &mut G) -> Self {
+        PointBase::from_coordinates(rng.gen())
+    }
+}
+
+#[cfg(feature="arbitrary")]
+impl<N, D: DimName, S> Arbitrary for PointBase<N, D, S>
+    where N: Scalar + Arbitrary + Send,
+          S: OwnedStorage<N, D, U1> + Send,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    #[inline]
+    fn arbitrary<G: Gen>(g: &mut G) -> Self {
+        PointBase::from_coordinates(ColumnVector::arbitrary(g))
+    }
+}
+
+/*
+ *
+ * Small points construction from components.
+ *
+ */
+macro_rules! componentwise_constructors_impl(
+    ($($D: ty, $($args: ident:$irow: expr),*);* $(;)*) => {$(
+        impl<N, S> PointBase<N, $D, S>
+            where N: Scalar,
+                  S: OwnedStorage<N, $D, U1>,
+                  S::Alloc: OwnedAllocator<N, $D, U1, S> {
+            /// Initializes this matrix from its components.
+            #[inline]
+            pub fn new($($args: N),*) -> PointBase<N, $D, S> {
+                unsafe {
+                    let mut res = Self::new_uninitialized();
+                    $( *res.get_unchecked_mut($irow) = $args; )*
+
+                    res
+                }
+            }
+        }
+    )*}
+);
+
+componentwise_constructors_impl!(
+    U1, x:0;
+    U2, x:0, y:1;
+    U3, x:0, y:1, z:2;
+    U4, x:0, y:1, z:2, w:3;
+    U5, x:0, y:1, z:2, w:3, a:4;
+    U6, x:0, y:1, z:2, w:3, a:4, b:5;
+);

src/geometry/point_conversion.rs

@@ -0,0 +1,73 @@
+use num::{One, Zero};
+use alga::general::{SubsetOf, SupersetOf, ClosedDiv};
+
+use core::{Scalar, Matrix, ColumnVector, OwnedColumnVector};
+use core::dimension::{DimName, DimNameSum, DimNameAdd, U1};
+use core::storage::OwnedStorage;
+use core::allocator::{Allocator, OwnedAllocator};
+
+use geometry::{PointBase, OwnedPoint};
+
+/*
+ * This file provides the following conversions:
+ * =============================================
+ *
+ * PointBase -> PointBase
+ * PointBase -> ColumnVector (homogeneous)
+ */
+
+impl<N1, N2, D, SA, SB> SubsetOf<PointBase<N2, D, SB>> for PointBase<N1, D, SA>
+    where D: DimName,
+          N1: Scalar,
+          N2: Scalar + SupersetOf<N1>,
+          SA: OwnedStorage<N1, D, U1>,
+          SB: OwnedStorage<N2, D, U1>,
+          SB::Alloc: OwnedAllocator<N2, D, U1, SB>,
+          SA::Alloc: OwnedAllocator<N1, D, U1, SA> {
+    #[inline]
+    fn to_superset(&self) -> PointBase<N2, D, SB> {
+        PointBase::from_coordinates(self.coords.to_superset())
+    }
+
+    #[inline]
+    fn is_in_subset(m: &PointBase<N2, D, SB>) -> bool {
+        // FIXME: is there a way to reuse the `.is_in_subset` from the matrix implementation of
+        // SubsetOf?
+        m.iter().all(|e| e.is_in_subset())
+    }
+
+    #[inline]
+    unsafe fn from_superset_unchecked(m: &PointBase<N2, D, SB>) -> Self {
+        PointBase::from_coordinates(Matrix::from_superset_unchecked(&m.coords))
+    }
+}
+
+
+impl<N1, N2, D, SA, SB> SubsetOf<ColumnVector<N2, DimNameSum<D, U1>, SB>> for PointBase<N1, D, SA>
+    where D:  DimNameAdd<U1>,
+          N1: Scalar,
+          N2: Scalar + Zero + One + ClosedDiv + SupersetOf<N1>,
+          SA: OwnedStorage<N1, D, U1>,
+          SB: OwnedStorage<N2, DimNameSum<D, U1>, U1>,
+          SA::Alloc: OwnedAllocator<N1, D, U1, SA> +
+                     Allocator<N1, DimNameSum<D, U1>, U1>,
+          SB::Alloc: OwnedAllocator<N2, DimNameSum<D, U1>, U1, SB> +
+                     Allocator<N2, D, U1> {
+    #[inline]
+    fn to_superset(&self) -> ColumnVector<N2, DimNameSum<D, U1>, SB> {
+        let p: OwnedPoint<N2, D, SB::Alloc> = self.to_superset();
+        p.to_homogeneous()
+    }
+
+    #[inline]
+    fn is_in_subset(v: &ColumnVector<N2, DimNameSum<D, U1>, SB>) -> bool {
+        ::is_convertible::<_, OwnedColumnVector<N1, DimNameSum<D, U1>, SA::Alloc>>(v) &&
+        !v[D::dim()].is_zero()
+    }
+
+    #[inline]
+    unsafe fn from_superset_unchecked(v: &ColumnVector<N2, DimNameSum<D, U1>, SB>) -> Self {
+        let coords =  v.fixed_slice::<D, U1>(0, 0) / v[D::dim()];
+        Self::from_coordinates(::convert_unchecked(coords))
+    }
+}

src/geometry/point_coordinates.rs

@@ -0,0 +1,47 @@
+use std::mem;
+use std::ops::{Deref, DerefMut};
+
+use core::Scalar;
+use core::dimension::{U1, U2, U3, U4, U5, U6};
+use core::coordinates::{X, XY, XYZ, XYZW, XYZWA, XYZWAB};
+use core::allocator::OwnedAllocator;
+use core::storage::OwnedStorage;
+
+use geometry::PointBase;
+
+/*
+ *
+ * Give coordinates to PointBase{1 .. 6}
+ *
+ */
+
+macro_rules! deref_impl(
+    ($D: ty, $Target: ident $(, $comps: ident)*) => {
+        impl<N: Scalar, S> Deref for PointBase<N, $D, S>
+            where S: OwnedStorage<N, $D, U1>,
+                  S::Alloc: OwnedAllocator<N, $D, U1, S> {
+            type Target = $Target<N>;
+
+            #[inline]
+            fn deref(&self) -> &Self::Target {
+                unsafe { mem::transmute(self) }
+            }
+        }
+
+        impl<N: Scalar, S> DerefMut for PointBase<N, $D, S>
+            where S: OwnedStorage<N, $D, U1>,
+                  S::Alloc: OwnedAllocator<N, $D, U1, S> {
+            #[inline]
+            fn deref_mut(&mut self) -> &mut Self::Target {
+                unsafe { mem::transmute(self) }
+            }
+        }
+    }
+);
+
+deref_impl!(U1, X, x);
+deref_impl!(U2, XY, x, y);
+deref_impl!(U3, XYZ, x, y, z);
+deref_impl!(U4, XYZW, x, y, z, w);
+deref_impl!(U5, XYZWA, x, y, z, w, a);
+deref_impl!(U6, XYZWAB, x, y, z, w, a, b);

src/geometry/point_ops.rs

@@ -0,0 +1,263 @@
+use std::ops::{Neg, Add, AddAssign, Sub, SubAssign, Mul, MulAssign, Div, DivAssign, Index, IndexMut};
+use num::Zero;
+
+use alga::general::{ClosedNeg, ClosedAdd, ClosedSub, ClosedMul, ClosedDiv};
+
+use core::{Scalar, ColumnVector, Matrix, ColumnVectorSum};
+use core::dimension::{Dim, DimName, U1};
+use core::constraint::{ShapeConstraint, SameNumberOfRows, SameNumberOfColumns, AreMultipliable};
+use core::storage::{Storage, StorageMut};
+use core::allocator::{SameShapeAllocator, Allocator};
+
+use geometry::{PointBase, OwnedPoint, PointMul};
+
+
+/*
+ *
+ * Indexing.
+ *
+ */
+impl<N, D: DimName, S> Index<usize> for PointBase<N, D, S>
+    where N: Scalar,
+          S: Storage<N, D, U1> {
+    type Output = N;
+
+    #[inline]
+    fn index(&self, i: usize) -> &Self::Output {
+        &self.coords[i]
+    }
+}
+
+impl<N, D: DimName, S> IndexMut<usize> for PointBase<N, D, S>
+    where N: Scalar,
+          S: StorageMut<N, D, U1> {
+    #[inline]
+    fn index_mut(&mut self, i: usize) -> &mut Self::Output {
+        &mut self.coords[i]
+    }
+}
+
+/*
+ * Neg.
+ *
+ */
+impl<N, D: DimName, S> Neg for PointBase<N, D, S>
+    where N: Scalar + ClosedNeg,
+          S: Storage<N, D, U1> {
+    type Output = OwnedPoint<N, D, S::Alloc>;
+
+    #[inline]
+    fn neg(self) -> Self::Output {
+        PointBase::from_coordinates(-self.coords)
+    }
+}
+
+impl<'a, N, D: DimName, S> Neg for &'a PointBase<N, D, S>
+    where N: Scalar + ClosedNeg,
+          S: Storage<N, D, U1> {
+    type Output = OwnedPoint<N, D, S::Alloc>;
+
+    #[inline]
+    fn neg(self) -> Self::Output {
+        PointBase::from_coordinates(-&self.coords)
+    }
+}
+
+/*
+ *
+ * Subtraction & Addition.
+ *
+ */
+
+// PointBase - PointBase
+add_sub_impl!(Sub, sub, ClosedSub;
+    (D, U1), (D, U1) for D: DimName;
+    self: &'a PointBase<N, D, SA>, right: &'b PointBase<N, D, SB>, Output = ColumnVectorSum<N, D, D, SA>;
+    &self.coords - &right.coords; 'a, 'b);
+
+add_sub_impl!(Sub, sub, ClosedSub;
+    (D, U1), (D, U1) for D: DimName;
+    self: &'a PointBase<N, D, SB>, right: PointBase<N, D, SA>, Output = ColumnVectorSum<N, D, D, SA>;
+    &self.coords - right.coords; 'a);
+
+add_sub_impl!(Sub, sub, ClosedSub;
+    (D, U1), (D, U1) for D: DimName;
+    self: PointBase<N, D, SA>, right: &'b PointBase<N, D, SB>, Output = ColumnVectorSum<N, D, D, SA>;
+    self.coords - &right.coords; 'b);
+
+add_sub_impl!(Sub, sub, ClosedSub;
+    (D, U1), (D, U1) for D: DimName;
+    self: PointBase<N, D, SA>, right: PointBase<N, D, SB>, Output = ColumnVectorSum<N, D, D, SA>;
+    self.coords - right.coords; );
+
+// PointBase - Vector
+add_sub_impl!(Sub, sub, ClosedSub;
+    (D1, U1), (D2, U1) -> (D1) for D1: DimName, D2: Dim;
+    self: &'a PointBase<N, D1, SA>, right: &'b ColumnVector<N, D2, SB>, Output = OwnedPoint<N, D1, SA::Alloc>;
+    Self::Output::from_coordinates(&self.coords - right); 'a, 'b);
+
+add_sub_impl!(Sub, sub, ClosedSub;
+    (D1, U1), (D2, U1) -> (D1) for D1: DimName, D2: Dim;
+    self: &'a PointBase<N, D1, SA>, right: ColumnVector<N, D2, SB>, Output = OwnedPoint<N, D1, SA::Alloc>;
+    Self::Output::from_coordinates(&self.coords - &right); 'a); // FIXME: should not be a ref to `right`.
+
+add_sub_impl!(Sub, sub, ClosedSub;
+    (D1, U1), (D2, U1) -> (D1) for D1: DimName, D2: Dim;
+    self: PointBase<N, D1, SA>, right: &'b ColumnVector<N, D2, SB>, Output = OwnedPoint<N, D1, SA::Alloc>;
+    Self::Output::from_coordinates(self.coords - right); 'b);
+
+add_sub_impl!(Sub, sub, ClosedSub;
+    (D1, U1), (D2, U1) -> (D1) for D1: DimName, D2: Dim;
+    self: PointBase<N, D1, SA>, right: ColumnVector<N, D2, SB>, Output = OwnedPoint<N, D1, SA::Alloc>;
+    Self::Output::from_coordinates(self.coords - right); );
+
+
+// PointBase + Vector
+add_sub_impl!(Add, add, ClosedAdd;
+    (D1, U1), (D2, U1) -> (D1) for D1: DimName, D2: Dim;
+    self: &'a PointBase<N, D1, SA>, right: &'b ColumnVector<N, D2, SB>, Output = OwnedPoint<N, D1, SA::Alloc>;
+    Self::Output::from_coordinates(&self.coords + right); 'a, 'b);
+
+add_sub_impl!(Add, add, ClosedAdd;
+    (D1, U1), (D2, U1) -> (D1) for D1: DimName, D2: Dim;
+    self: &'a PointBase<N, D1, SA>, right: ColumnVector<N, D2, SB>, Output = OwnedPoint<N, D1, SA::Alloc>;
+    Self::Output::from_coordinates(&self.coords + &right); 'a); // FIXME: should not be a ref to `right`.
+
+add_sub_impl!(Add, add, ClosedAdd;
+    (D1, U1), (D2, U1) -> (D1) for D1: DimName, D2: Dim;
+    self: PointBase<N, D1, SA>, right: &'b ColumnVector<N, D2, SB>, Output = OwnedPoint<N, D1, SA::Alloc>;
+    Self::Output::from_coordinates(self.coords + right); 'b);
+
+add_sub_impl!(Add, add, ClosedAdd;
+    (D1, U1), (D2, U1) -> (D1) for D1: DimName, D2: Dim;
+    self: PointBase<N, D1, SA>, right: ColumnVector<N, D2, SB>, Output = OwnedPoint<N, D1, SA::Alloc>;
+    Self::Output::from_coordinates(self.coords + right); );
+
+
+// XXX: replace by the shared macro: add_sub_assign_impl
+macro_rules! op_assign_impl(
+    ($($TraitAssign: ident, $method_assign: ident, $bound: ident);* $(;)*) => {$(
+        impl<'b, N, D1: DimName, D2: Dim, SA, SB> $TraitAssign<&'b ColumnVector<N, D2, SB>> for PointBase<N, D1, SA>
+            where N: Scalar + $bound,
+                  SA: StorageMut<N, D1, U1>,
+                  SB: Storage<N, D2, U1>,
+                  ShapeConstraint: SameNumberOfRows<D1, D2> {
+
+            #[inline]
+            fn $method_assign(&mut self, right: &'b ColumnVector<N, D2, SB>) {
+                self.coords.$method_assign(right)
+            }
+        }
+
+        impl<N, D1: DimName, D2: Dim, SA, SB> $TraitAssign<ColumnVector<N, D2, SB>> for PointBase<N, D1, SA>
+            where N: Scalar + $bound,
+                  SA: StorageMut<N, D1, U1>,
+                  SB: Storage<N, D2, U1>,
+                  ShapeConstraint: SameNumberOfRows<D1, D2> {
+
+            #[inline]
+            fn $method_assign(&mut self, right: ColumnVector<N, D2, SB>) {
+                self.coords.$method_assign(right)
+            }
+        }
+    )*}
+);
+
+op_assign_impl!(
+    AddAssign, add_assign, ClosedAdd;
+    SubAssign, sub_assign, ClosedSub;
+);
+
+
+/*
+ *
+ * Matrix × PointBase
+ *
+ */
+md_impl_all!(
+    Mul, mul;
+    (R1, C1), (D2, U1) for R1: DimName, C1: Dim, D2: DimName
+    where SA::Alloc: Allocator<N, R1, U1>
+    where ShapeConstraint: AreMultipliable<R1, C1, D2, U1>;
+    self: Matrix<N, R1, C1, SA>, right: PointBase<N, D2, SB>, Output = PointMul<N, R1, C1, SA>;
+    [val val] => PointBase::from_coordinates(self * right.coords);
+    [ref val] => PointBase::from_coordinates(self * right.coords);
+    [val ref] => PointBase::from_coordinates(self * &right.coords);
+    [ref ref] => PointBase::from_coordinates(self * &right.coords);
+);
+
+
+
+/*
+ *
+ * PointBase ×/÷ Scalar
+ *
+ */
+macro_rules! componentwise_scalarop_impl(
+    ($Trait: ident, $method: ident, $bound: ident;
+     $TraitAssign: ident, $method_assign: ident) => {
+        impl<N, D: DimName, S> $Trait<N> for PointBase<N, D, S>
+            where N: Scalar + $bound,
+                  S: Storage<N, D, U1> {
+            type Output = OwnedPoint<N, D, S::Alloc>;
+
+            #[inline]
+            fn $method(self, right: N) -> Self::Output {
+                PointBase::from_coordinates(self.coords.$method(right))
+            }
+        }
+
+        impl<'a, N, D: DimName, S> $Trait<N> for &'a PointBase<N, D, S>
+            where N: Scalar + $bound,
+                  S: Storage<N, D, U1> {
+            type Output = OwnedPoint<N, D, S::Alloc>;
+
+            #[inline]
+            fn $method(self, right: N) -> Self::Output {
+                PointBase::from_coordinates((&self.coords).$method(right))
+            }
+        }
+
+        impl<N, D: DimName, S> $TraitAssign<N> for PointBase<N, D, S>
+            where N: Scalar + $bound,
+                  S: StorageMut<N, D, U1> {
+            #[inline]
+            fn $method_assign(&mut self, right: N) {
+                self.coords.$method_assign(right)
+            }
+        }
+    }
+);
+
+componentwise_scalarop_impl!(Mul, mul, ClosedMul; MulAssign, mul_assign);
+componentwise_scalarop_impl!(Div, div, ClosedDiv; DivAssign, div_assign);
+
+macro_rules! left_scalar_mul_impl(
+    ($($T: ty),* $(,)*) => {$(
+        impl<D: DimName, S> Mul<PointBase<$T, D, S>> for $T
+            where S: Storage<$T, D, U1> {
+            type Output = OwnedPoint<$T, D, S::Alloc>;
+
+            #[inline]
+            fn mul(self, right: PointBase<$T, D, S>) -> Self::Output {
+                PointBase::from_coordinates(self * right.coords)
+            }
+        }
+
+        impl<'b, D: DimName, S> Mul<&'b PointBase<$T, D, S>> for $T
+            where S: Storage<$T, D, U1> {
+            type Output = OwnedPoint<$T, D, S::Alloc>;
+
+            #[inline]
+            fn mul(self, right: &'b PointBase<$T, D, S>) -> Self::Output {
+                PointBase::from_coordinates(self * &right.coords)
+            }
+        }
+    )*}
+);
+
+left_scalar_mul_impl!(
+    u8, u16, u32, u64, usize,
+    i8, i16, i32, i64, isize,
+    f32, f64
+);

src/geometry/quaternion.rs

@@ -0,0 +1,703 @@
+use std::fmt;
+use num::Zero;
+use approx::ApproxEq;
+
+use alga::general::Real;
+
+use core::{Unit, ColumnVector, OwnedColumnVector, MatrixSlice, MatrixSliceMut, SquareMatrix,
+           OwnedSquareMatrix};
+use core::storage::{Storage, StorageMut};
+use core::allocator::Allocator;
+use core::dimension::{U1, U3, U4};
+
+use geometry::{RotationBase, OwnedRotation};
+
+/// A quaternion with an owned storage allocated by `A`.
+pub type OwnedQuaternionBase<N, A> = QuaternionBase<N, <A as Allocator<N, U4, U1>>::Buffer>;
+
+/// A unit quaternion with an owned storage allocated by `A`.
+pub type OwnedUnitQuaternionBase<N, A> = UnitQuaternionBase<N, <A as Allocator<N, U4, U1>>::Buffer>;
+
+/// A quaternion. See the type alias `UnitQuaternionBase = Unit<QuaternionBase>` for a quaternion
+/// that may be used as a rotation.
+#[repr(C)]
+#[derive(Hash, Debug, Copy, Clone)]
+#[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
+pub struct QuaternionBase<N: Real, S: Storage<N, U4, U1>> {
+    /// This quaternion as a 4D vector of coordinates in the `[ x, y, z, w ]` storage order.
+    pub coords: ColumnVector<N, U4, S>
+}
+
+impl<N, S> Eq for QuaternionBase<N, S>
+    where N: Real + Eq,
+          S: Storage<N, U4, U1> {
+}
+
+impl<N, S> PartialEq for QuaternionBase<N, S>
+    where N: Real,
+          S: Storage<N, U4, U1> {
+    fn eq(&self, rhs: &Self) -> bool {
+        self.coords == rhs.coords ||
+        // Account for the double-covering of S², i.e. q = -q
+        self.as_vector().iter().zip(rhs.as_vector().iter()).all(|(a, b)| *a == -*b)
+    }
+}
+
+impl<N, S> QuaternionBase<N, S>
+    where N: Real,
+          S: Storage<N, U4, U1> {
+    /// Moves this quaternion into one that owns its data.
+    #[inline]
+    pub fn into_owned(self) -> OwnedQuaternionBase<N, S::Alloc> {
+        QuaternionBase::from_vector(self.coords.into_owned())
+    }
+
+    /// Clones this quaternion into one that owns its data.
+    #[inline]
+    pub fn clone_owned(&self) -> OwnedQuaternionBase<N, S::Alloc> {
+        QuaternionBase::from_vector(self.coords.clone_owned())
+    }
+
+    /// The vector part `(i, j, k)` of this quaternion.
+    #[inline]
+    pub fn vector(&self) -> MatrixSlice<N, U3, U1, S::RStride, S::CStride, S::Alloc> {
+        self.coords.fixed_rows::<U3>(0)
+    }
+
+    /// The scalar part `w` of this quaternion.
+    #[inline]
+    pub fn scalar(&self) -> N {
+        self.coords[3]
+    }
+
+    /// Reinterprets this quaternion as a 4D vector.
+    #[inline]
+    pub fn as_vector(&self) -> &ColumnVector<N, U4, S> {
+        &self.coords
+    }
+
+    /// The norm of this quaternion.
+    #[inline]
+    pub fn norm(&self) -> N {
+        self.coords.norm()
+    }
+
+    /// The squared norm of this quaternion.
+    #[inline]
+    pub fn norm_squared(&self) -> N {
+        self.coords.norm_squared()
+    }
+
+    /// Normalizes this quaternion.
+    #[inline]
+    pub fn normalize(&self) -> OwnedQuaternionBase<N, S::Alloc> {
+        QuaternionBase::from_vector(self.coords.normalize())
+    }
+
+    /// Compute the conjugate of this quaternion.
+    #[inline]
+    pub fn conjugate(&self) -> OwnedQuaternionBase<N, S::Alloc> {
+        let v = OwnedColumnVector::<N, U4, S::Alloc>::new(-self.coords[0],
+                                                          -self.coords[1],
+                                                          -self.coords[2],
+                                                          self.coords[3]);
+        QuaternionBase::from_vector(v)
+    }
+
+    /// Inverts this quaternion if it is not zero.
+    #[inline]
+    pub fn try_inverse(&self) -> Option<OwnedQuaternionBase<N, S::Alloc>> {
+        let mut res = QuaternionBase::from_vector(self.coords.clone_owned());
+
+        if res.try_inverse_mut() {
+            Some(res)
+        }
+        else {
+            None
+        }
+    }
+
+    /// Linear interpolation between two quaternion.
+    #[inline]
+    pub fn lerp<S2>(&self, other: &QuaternionBase<N, S2>, t: N) -> OwnedQuaternionBase<N, S::Alloc>
+        where S2: Storage<N, U4, U1> {
+        self * (N::one() - t) + other * t
+    }
+}
+
+
+impl<N, S> QuaternionBase<N, S>
+    where N: Real,
+          S: Storage<N, U4, U1>,
+          S::Alloc: Allocator<N, U3, U1> {
+    /// The polar decomposition of this quaternion.
+    ///
+    /// Returns, from left to right: the quaternion norm, the half rotation angle, the rotation
+    /// axis. If the rotation angle is zero, the rotation axis is set to `None`.
+    pub fn polar_decomposition(&self) -> (N, N, Option<Unit<OwnedColumnVector<N, U3, S::Alloc>>>) {
+        if let Some((q, n)) = Unit::try_new_and_get(self.clone_owned(), N::zero()) {
+            if let Some(axis) = Unit::try_new(self.vector().clone_owned(), N::zero()) {
+                let angle = q.angle() / ::convert(2.0f64);
+
+                (n, angle, Some(axis))
+            }
+            else {
+                (n, N::zero(), None)
+            }
+        }
+        else {
+            (N::zero(), N::zero(), None)
+        }
+    }
+
+    /// Compute the exponential of a quaternion.
+    #[inline]
+    pub fn exp(&self) -> OwnedQuaternionBase<N, S::Alloc> {
+        let v  = self.vector();
+        let nn = v.norm_squared();
+
+        if relative_eq!(nn, N::zero()) {
+            QuaternionBase::identity()
+        }
+        else {
+            let w_exp = self.scalar().exp();
+            let n  = nn.sqrt();
+            let nv = v * (w_exp * n.sin() / n);
+
+            QuaternionBase::from_parts(n.cos(), nv)
+        }
+    }
+
+    /// Compute the natural logarithm of a quaternion.
+    #[inline]
+    pub fn ln(&self) -> OwnedQuaternionBase<N, S::Alloc> {
+        let n = self.norm();
+        let v = self.vector();
+        let s = self.scalar();
+
+        QuaternionBase::from_parts(n.ln(), v.normalize() *  (s / n).acos())
+    }
+
+    /// Raise the quaternion to a given floating power.
+    #[inline]
+    pub fn powf(&self, n: N) -> OwnedQuaternionBase<N, S::Alloc> {
+        (self.ln() * n).exp()
+    }
+}
+
+impl<N, S> QuaternionBase<N, S>
+    where N: Real,
+          S: StorageMut<N, U4, U1> {
+    /// Transforms this quaternion into its 4D vector form (Vector part, Scalar part).
+    #[inline]
+    pub fn as_vector_mut(&mut self) -> &mut ColumnVector<N, U4, S> {
+        &mut self.coords
+    }
+
+    /// The mutable vector part `(i, j, k)` of this quaternion.
+    #[inline]
+    pub fn vector_mut(&mut self) -> MatrixSliceMut<N, U3, U1, S::RStride, S::CStride, S::Alloc> {
+        self.coords.fixed_rows_mut::<U3>(0)
+    }
+
+    /// Replaces this quaternion by its conjugate.
+    #[inline]
+    pub fn conjugate_mut(&mut self) {
+        self.coords[0] = -self.coords[0];
+        self.coords[1] = -self.coords[1];
+        self.coords[2] = -self.coords[2];
+    }
+
+    /// Inverts this quaternion in-place if it is not zero.
+    #[inline]
+    pub fn try_inverse_mut(&mut self) -> bool {
+        let norm_squared = self.norm_squared();
+
+        if relative_eq!(&norm_squared, &N::zero()) {
+            false
+        }
+        else {
+            self.conjugate_mut();
+            self.coords /= norm_squared;
+
+            true
+        }
+    }
+
+    /// Normalizes this quaternion.
+    #[inline]
+    pub fn normalize_mut(&mut self) -> N {
+        self.coords.normalize_mut()
+    }
+}
+
+impl<N, S> ApproxEq for QuaternionBase<N, S>
+    where N: Real + ApproxEq<Epsilon = N>,
+          S: Storage<N, U4, U1> {
+    type Epsilon = N;
+
+    #[inline]
+    fn default_epsilon() -> Self::Epsilon {
+        N::default_epsilon()
+    }
+
+    #[inline]
+    fn default_max_relative() -> Self::Epsilon {
+        N::default_max_relative()
+    }
+
+    #[inline]
+    fn default_max_ulps() -> u32 {
+        N::default_max_ulps()
+    }
+
+    #[inline]
+    fn relative_eq(&self, other: &Self, epsilon: Self::Epsilon, max_relative: Self::Epsilon) -> bool {
+        self.as_vector().relative_eq(other.as_vector(), epsilon, max_relative) ||
+        // Account for the double-covering of S², i.e. q = -q
+       self.as_vector().iter().zip(other.as_vector().iter()).all(|(a, b)| a.relative_eq(&-*b, epsilon, max_relative))
+    }
+
+    #[inline]
+    fn ulps_eq(&self, other: &Self, epsilon: Self::Epsilon, max_ulps: u32) -> bool {
+        self.as_vector().ulps_eq(other.as_vector(), epsilon, max_ulps) ||
+        // Account for the double-covering of S², i.e. q = -q.
+       self.as_vector().iter().zip(other.as_vector().iter()).all(|(a, b)| a.ulps_eq(&-*b, epsilon, max_ulps))
+    }
+}
+
+
+impl<N, S> fmt::Display for QuaternionBase<N, S>
+    where N: Real + fmt::Display,
+          S: Storage<N, U4, U1> {
+    fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
+        write!(f, "Quaternion {} − ({}, {}, {})", self[3], self[0], self[1], self[2])
+    }
+}
+
+/// A unit quaternions. May be used to represent a rotation.
+///
+///
+/// <center>
+/// <big><b>
+/// Due to a [bug](https://github.com/rust-lang/rust/issues/32077) in rustdoc, the documentation
+/// below has been written manually lists only method signatures.<br>
+/// Trait implementations are not listed either.
+/// </b></big>
+/// </center>
+///
+/// Please refer directly to the documentation written above each function definition on the source
+/// code for more details.
+///
+/// <h2 id="methods">Methods</h2>
+///
+///
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">angle</a>(&self) -> N</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">angle_to</a>(&self, other: &UnitQuaternionBase) -> N</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">axis</a>(&self) -> Option&lt;Unit&lt;OwnedColumnVector&gt;&gt;</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">clone_owned</a>(&self) -> OwnedUnitQuaternionBase</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">conjugate_mut</a>(&mut self)</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">conjugate</a>(&self) -> OwnedUnitQuaternionBase</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">exp</a>(&self) -> OwnedQuaternionBase</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">from_axis_angle</a>(axis: &Unit&lt;ColumnVector&gt;, angle: N) -> Self</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">from_euler_angles</a>(roll: N, pitch: N, yaw: N) -> Self</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">from_quaternion</a>(q: QuaternionBase) -> Self</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">from_rotation_matrix</a>(rotmat: &RotationBase) -> Self</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">from_scaled_axis</a>(axisangle: ColumnVector) -> Self</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">identity</a>() -> Self</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">into_owned</a>(self) -> OwnedUnitQuaternionBase</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">inverse_mut</a>(&mut self)</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">inverse</a>(&self) -> OwnedUnitQuaternionBase</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">lerp</a>(&self, other: &UnitQuaternionBase, t: N) -> OwnedQuaternionBase</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">ln</a>(&self) -> OwnedQuaternionBase</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">look_at_lh</a>(dir: &ColumnVector, up: &ColumnVector) -> Self</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">look_at_rh</a>(dir: &ColumnVector, up: &ColumnVector) -> Self</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">new</a>(axisangle: ColumnVector) -> Self</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">new_observer_frame</a>(dir: &ColumnVector, up: &ColumnVector) -> Self</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">nlerp</a>(&self, other: &UnitQuaternionBase, t: N) -> OwnedUnitQuaternionBase</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">powf</a>(&self, n: N) -> OwnedUnitQuaternionBase</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">quaternion</a>(&self) -> &QuaternionBase</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">rotation_between</a>(a: &ColumnVector, b: &ColumnVector) -> Option&lt;Self&gt;</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">rotation_to</a>(&self, other: &UnitQuaternionBase) -> OwnedUnitQuaternionBase</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">scaled_axis</a>(&self) -> OwnedColumnVector</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">scaled_rotation_between</a>(a: &ColumnVector, b: &ColumnVector, s: N) -> Option&lt;Self&gt;</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">slerp</a>(&self, other: &UnitQuaternionBase, t: N) -> OwnedUnitQuaternionBase</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">to_homogeneous</a>(&self) -> OwnedSquareMatrix</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">to_rotation_matrix</a>(&self) -> OwnedRotation</code>
+/// </h4>
+/// <h4 class="method"><span class="invisible">
+/// <code>fn <a class="fnname">try_slerp</a>(&self, other: &UnitQuaternionBase, t: N, epsilon: N) -> Option&lt;OwnedUnitQuaternionBase&gt;</code>
+/// </h4>
+pub type UnitQuaternionBase<N, S> = Unit<QuaternionBase<N, S>>;
+
+
+
+impl<N, S> UnitQuaternionBase<N, S>
+    where N: Real,
+          S: Storage<N, U4, U1> {
+    /// Moves this unit quaternion into one that owns its data.
+    #[inline]
+    pub fn into_owned(self) -> OwnedUnitQuaternionBase<N, S::Alloc> {
+        UnitQuaternionBase::new_unchecked(self.unwrap().into_owned())
+    }
+
+    /// Clones this unit quaternion into one that owns its data.
+    #[inline]
+    pub fn clone_owned(&self) -> OwnedUnitQuaternionBase<N, S::Alloc> {
+        UnitQuaternionBase::new_unchecked(self.as_ref().clone_owned())
+    }
+
+    /// The rotation angle in [0; pi] of this unit quaternion.
+    #[inline]
+    pub fn angle(&self) -> N {
+        let w = self.quaternion().scalar().abs();
+
+        // Handle innacuracies that make break `.acos`.
+        if w >= N::one() {
+            N::zero()
+        }
+        else {
+            w.acos() * ::convert(2.0f64)
+        }
+    }
+
+    /// The underlying quaternion.
+    ///
+    /// Same as `self.as_ref()`.
+    #[inline]
+    pub fn quaternion(&self) -> &QuaternionBase<N, S> {
+        self.as_ref()
+    }
+
+    /// Compute the conjugate of this unit quaternion.
+    #[inline]
+    pub fn conjugate(&self) -> OwnedUnitQuaternionBase<N, S::Alloc> {
+        UnitQuaternionBase::new_unchecked(self.as_ref().conjugate())
+    }
+
+    /// Inverts this quaternion if it is not zero.
+    #[inline]
+    pub fn inverse(&self) -> OwnedUnitQuaternionBase<N, S::Alloc> {
+        self.conjugate()
+    }
+
+    /// The rotation angle needed to make `self` and `other` coincide.
+    #[inline]
+    pub fn angle_to<S2>(&self, other: &UnitQuaternionBase<N, S2>) -> N
+        where S2: Storage<N, U4, U1> {
+        let delta = self.rotation_to(other);
+        delta.angle()
+    }
+
+    /// The unit quaternion needed to make `self` and `other` coincide.
+    ///
+    /// The result is such that: `self.rotation_to(other) * self == other`.
+    #[inline]
+    pub fn rotation_to<S2>(&self, other: &UnitQuaternionBase<N, S2>) -> OwnedUnitQuaternionBase<N, S2::Alloc>
+        where S2: Storage<N, U4, U1> {
+        other / self
+    }
+
+    /// Linear interpolation between two unit quaternions.
+    ///
+    /// The result is not normalized.
+    #[inline]
+    pub fn lerp<S2>(&self, other: &UnitQuaternionBase<N, S2>, t: N) -> OwnedQuaternionBase<N, S::Alloc>
+        where S2: Storage<N, U4, U1> {
+        self.as_ref().lerp(other.as_ref(), t)
+    }
+
+    /// Normalized linear interpolation between two unit quaternions.
+    #[inline]
+    pub fn nlerp<S2>(&self, other: &UnitQuaternionBase<N, S2>, t: N) -> OwnedUnitQuaternionBase<N, S::Alloc>
+        where S2: Storage<N, U4, U1> {
+        let mut res = self.lerp(other, t);
+        let _ = res.normalize_mut();
+
+        UnitQuaternionBase::new_unchecked(res)
+    }
+
+    /// Spherical linear interpolation between two unit quaternions.
+    ///
+    /// Panics if the angle between both quaternion is 180 degrees (in which case the interpolation
+    /// is not well-defined).
+    #[inline]
+    pub fn slerp<S2>(&self, other: &UnitQuaternionBase<N, S2>, t: N) -> OwnedUnitQuaternionBase<N, S::Alloc>
+        where S2: Storage<N, U4, U1, Alloc = S::Alloc> {
+        self.try_slerp(other, t, N::zero()).expect(
+            "Unable to perform a spherical quaternion interpolation when they \
+             are 180 degree apart (the result is not unique).")
+    }
+
+    /// Computes the spherical linear interpolation between two unit quaternions or returns `None`
+    /// if both quaternions are approximately 180 degrees apart (in which case the interpolation is
+    /// not well-defined).
+    ///
+    /// # Arguments
+    /// * `self`: the first quaternion to interpolate from.
+    /// * `other`: the second quaternion to interpolate toward.
+    /// * `t`: the interpolation parameter. Should be between 0 and 1.
+    /// * `epsilon`: the value bellow which the sinus of the angle separating both quaternion
+    /// must be to return `None`.
+    #[inline]
+    pub fn try_slerp<S2>(&self, other: &UnitQuaternionBase<N, S2>, t: N, epsilon: N)
+                         -> Option<OwnedUnitQuaternionBase<N, S::Alloc>>
+        where S2: Storage<N, U4, U1, Alloc = S::Alloc> {
+
+        let c_hang = self.coords.dot(&other.coords);
+
+        // self == other
+        if c_hang.abs() >= N::one() {
+            return Some(self.clone_owned())
+        }
+
+        let hang   = c_hang.acos();
+        let s_hang = (N::one() - c_hang * c_hang).sqrt();
+
+        // FIXME: what if s_hang is 0.0 ? The result is not well-defined.
+        if relative_eq!(s_hang, N::zero(), epsilon = epsilon) {
+            None
+        }
+        else {
+            let ta = ((N::one() - t) * hang).sin() / s_hang;
+            let tb = (t * hang).sin() / s_hang; 
+            let res = self.as_ref() * ta + other.as_ref() * tb;
+
+            Some(UnitQuaternionBase::new_unchecked(res))
+        }
+    }
+}
+
+impl<N, S> UnitQuaternionBase<N, S>
+    where N: Real,
+          S: StorageMut<N, U4, U1> {
+    /// Compute the conjugate of this unit quaternion in-place.
+    #[inline]
+    pub fn conjugate_mut(&mut self) {
+        self.as_mut_unchecked().conjugate_mut()
+    }
+
+    /// Inverts this quaternion if it is not zero.
+    #[inline]
+    pub fn inverse_mut(&mut self) {
+        self.as_mut_unchecked().conjugate_mut()
+    }
+}
+
+impl<N, S> UnitQuaternionBase<N, S>
+    where N: Real,
+          S: Storage<N, U4, U1>,
+          S::Alloc: Allocator<N, U3, U1> {
+    /// The rotation axis of this unit quaternion or `None` if the rotation is zero.
+    #[inline]
+    pub fn axis(&self) -> Option<Unit<OwnedColumnVector<N, U3, S::Alloc>>> {
+        let v =
+            if self.quaternion().scalar() >= N::zero() {
+                self.as_ref().vector().clone_owned()
+            }
+            else {
+                -self.as_ref().vector()
+            };
+
+        Unit::try_new(v, N::zero())
+    }
+
+
+    /// The rotation axis of this unit quaternion multiplied by the rotation agle.
+    #[inline]
+    pub fn scaled_axis(&self) -> OwnedColumnVector<N, U3, S::Alloc> {
+        if let Some(axis) = self.axis() {
+            axis.unwrap() * self.angle()
+        }
+        else {
+            ColumnVector::zero()
+        }
+    }
+
+    /// Compute the exponential of a quaternion.
+    ///
+    /// Note that this function yields a `QuaternionBase<N>` because it looses the unit property.
+    #[inline]
+    pub fn exp(&self) -> OwnedQuaternionBase<N, S::Alloc> {
+        self.as_ref().exp()
+    }
+
+    /// Compute the natural logarithm of a quaternion.
+    ///
+    /// Note that this function yields a `QuaternionBase<N>` because it looses the unit property.
+    /// The vector part of the return value corresponds to the axis-angle representation (divided
+    /// by 2.0) of this unit quaternion.
+    #[inline]
+    pub fn ln(&self) -> OwnedQuaternionBase<N, S::Alloc> {
+        if let Some(v) = self.axis() {
+            QuaternionBase::from_parts(N::zero(), v.unwrap() * self.angle())
+        }
+        else {
+            QuaternionBase::zero()
+        }
+    }
+
+    /// Raise the quaternion to a given floating power.
+    ///
+    /// This returns the unit quaternion that identifies a rotation with axis `self.axis()` and
+    /// angle `self.angle() × n`.
+    #[inline]
+    pub fn powf(&self, n: N) -> OwnedUnitQuaternionBase<N, S::Alloc> {
+        if let Some(v) = self.axis() {
+            UnitQuaternionBase::from_axis_angle(&v, self.angle() * n)
+        }
+        else {
+            UnitQuaternionBase::identity()
+        }
+    }
+}
+
+impl<N, S> UnitQuaternionBase<N, S>
+    where N: Real,
+          S: Storage<N, U4, U1>,
+          S::Alloc: Allocator<N, U3, U3> {
+    /// Builds a rotation matrix from this unit quaternion.
+    #[inline]
+    pub fn to_rotation_matrix(&self) -> OwnedRotation<N, U3, S::Alloc> {
+        let i = self.as_ref()[0];
+        let j = self.as_ref()[1];
+        let k = self.as_ref()[2];
+        let w = self.as_ref()[3];
+
+        let ww = w * w;
+        let ii = i * i;
+        let jj = j * j;
+        let kk = k * k;
+        let ij = i * j * ::convert(2.0f64);
+        let wk = w * k * ::convert(2.0f64);
+        let wj = w * j * ::convert(2.0f64);
+        let ik = i * k * ::convert(2.0f64);
+        let jk = j * k * ::convert(2.0f64);
+        let wi = w * i * ::convert(2.0f64);
+
+        RotationBase::from_matrix_unchecked(
+            SquareMatrix::<_, U3, _>::new(
+                ww + ii - jj - kk, ij - wk,           wj + ik,
+                wk + ij,           ww - ii + jj - kk, jk - wi,
+                ik - wj,           wi + jk,           ww - ii - jj + kk
+            )
+        )
+    }
+
+    /// Converts this unit quaternion into its equivalent homogeneous transformation matrix.
+    #[inline]
+    pub fn to_homogeneous(&self) -> OwnedSquareMatrix<N, U4, S::Alloc>
+        where S::Alloc: Allocator<N, U4, U4> {
+        self.to_rotation_matrix().to_homogeneous()
+    }
+}
+
+
+impl<N, S> fmt::Display for UnitQuaternionBase<N, S>
+    where N: Real + fmt::Display,
+          S: Storage<N, U4, U1>,
+          S::Alloc: Allocator<N, U3, U1> {
+    fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
+        if let Some(axis) = self.axis() {
+            let axis = axis.unwrap();
+            write!(f, "UnitQuaternion angle: {} − axis: ({}, {}, {})", self.angle(), axis[0], axis[1], axis[2])
+        }
+        else {
+            write!(f, "UnitQuaternion angle: {} − axis: (undefined)", self.angle())
+        }
+    }
+}
+
+impl<N, S> ApproxEq for UnitQuaternionBase<N, S>
+    where N: Real + ApproxEq<Epsilon = N>,
+          S: Storage<N, U4, U1> {
+    type Epsilon = N;
+
+    #[inline]
+    fn default_epsilon() -> Self::Epsilon {
+        N::default_epsilon()
+    }
+
+    #[inline]
+    fn default_max_relative() -> Self::Epsilon {
+        N::default_max_relative()
+    }
+
+    #[inline]
+    fn default_max_ulps() -> u32 {
+        N::default_max_ulps()
+    }
+
+    #[inline]
+    fn relative_eq(&self, other: &Self, epsilon: Self::Epsilon, max_relative: Self::Epsilon) -> bool {
+        self.as_ref().relative_eq(other.as_ref(), epsilon, max_relative)
+    }
+
+    #[inline]
+    fn ulps_eq(&self, other: &Self, epsilon: Self::Epsilon, max_ulps: u32) -> bool {
+        self.as_ref().ulps_eq(other.as_ref(), epsilon, max_ulps)
+    }
+}

src/geometry/quaternion_alga.rs

@@ -0,0 +1,382 @@
+use num::Zero;
+
+use alga::general::{AbstractMagma, AbstractGroup, AbstractGroupAbelian, AbstractLoop,
+                    AbstractMonoid, AbstractQuasigroup, AbstractSemigroup, AbstractModule,
+                    Module, Real, Inverse, Multiplicative, Additive, Identity, Id};
+use alga::linear::{Transformation, AffineTransformation, Similarity, Isometry, DirectIsometry,
+                   OrthogonalTransformation, VectorSpace, FiniteDimVectorSpace, NormedSpace,
+                   Rotation, ProjectiveTransformation};
+
+use core::ColumnVector;
+use core::storage::OwnedStorage;
+use core::allocator::{Allocator, OwnedAllocator};
+use core::dimension::{U1, U3, U4};
+use geometry::{PointBase, QuaternionBase, UnitQuaternionBase};
+
+
+impl<N, S> Identity<Multiplicative> for QuaternionBase<N, S>
+    where N: Real,
+          S: OwnedStorage<N, U4, U1>,
+          S::Alloc: OwnedAllocator<N, U4, U1, S> {
+    #[inline]
+    fn identity() -> Self {
+        Self::identity()
+    }
+}
+
+impl<N, S> Identity<Additive> for QuaternionBase<N, S>
+    where N: Real,
+          S: OwnedStorage<N, U4, U1>,
+          S::Alloc: OwnedAllocator<N, U4, U1, S> {
+    #[inline]
+    fn identity() -> Self {
+        Self::zero()
+    }
+}
+
+impl<N, S> AbstractMagma<Multiplicative> for QuaternionBase<N, S>
+    where N: Real,
+          S: OwnedStorage<N, U4, U1>,
+          S::Alloc: OwnedAllocator<N, U4, U1, S> {
+    #[inline]
+    fn operate(&self, rhs: &Self) -> Self {
+        self * rhs
+    }
+}
+
+impl<N, S> AbstractMagma<Additive> for QuaternionBase<N, S>
+    where N: Real,
+          S: OwnedStorage<N, U4, U1>,
+          S::Alloc: OwnedAllocator<N, U4, U1, S> {
+    #[inline]
+    fn operate(&self, rhs: &Self) -> Self {
+        self + rhs
+    }
+}
+
+impl<N, S> Inverse<Additive> for QuaternionBase<N, S>
+    where N: Real,
+          S: OwnedStorage<N, U4, U1>,
+          S::Alloc: OwnedAllocator<N, U4, U1, S> {
+    #[inline]
+    fn inverse(&self) -> Self {
+        -self
+    }
+}
+
+macro_rules! impl_structures(
+    ($Quaternion: ident; $($marker: ident<$operator: ident>),* $(,)*) => {$(
+        impl<N, S> $marker<$operator> for $Quaternion<N, S>
+            where N: Real,
+                  S: OwnedStorage<N, U4, U1>,
+                  S::Alloc: OwnedAllocator<N, U4, U1, S> { }
+    )*}
+);
+
+impl_structures!(
+    QuaternionBase;
+    AbstractSemigroup<Multiplicative>,
+    AbstractMonoid<Multiplicative>,
+
+    AbstractSemigroup<Additive>,
+    AbstractQuasigroup<Additive>,
+    AbstractMonoid<Additive>,
+    AbstractLoop<Additive>,
+    AbstractGroup<Additive>,
+    AbstractGroupAbelian<Additive>
+);
+
+
+/*
+ *
+ * Vector space.
+ *
+ */
+impl<N, S> AbstractModule for QuaternionBase<N, S>
+    where N: Real,
+          S: OwnedStorage<N, U4, U1>,
+          S::Alloc: OwnedAllocator<N, U4, U1, S> {
+    type AbstractRing = N;
+
+    #[inline]
+    fn multiply_by(&self, n: N) -> Self {
+        self * n
+    }
+}
+
+impl<N, S> Module for QuaternionBase<N, S>
+    where N: Real,
+          S: OwnedStorage<N, U4, U1>,
+          S::Alloc: OwnedAllocator<N, U4, U1, S> {
+    type Ring = N;
+}
+
+impl<N, S> VectorSpace for QuaternionBase<N, S>
+    where N: Real,
+          S: OwnedStorage<N, U4, U1>,
+          S::Alloc: OwnedAllocator<N, U4, U1, S> {
+    type Field = N;
+}
+
+impl<N, S> FiniteDimVectorSpace for QuaternionBase<N, S>
+    where N: Real,
+          S: OwnedStorage<N, U4, U1>,
+          S::Alloc: OwnedAllocator<N, U4, U1, S> {
+    #[inline]
+    fn dimension() -> usize {
+        4
+    }
+
+    #[inline]
+    fn canonical_basis_element(i: usize) -> Self {
+        Self::from_vector(ColumnVector::canonical_basis_element(i))
+    }
+
+    #[inline]
+    fn dot(&self, other: &Self) -> N {
+        self.coords.dot(&other.coords)
+    }
+
+    #[inline]
+    unsafe fn component_unchecked(&self, i: usize) -> &N {
+        self.coords.component_unchecked(i)
+    }
+
+    #[inline]
+    unsafe fn component_unchecked_mut(&mut self, i: usize) -> &mut N {
+        self.coords.component_unchecked_mut(i)
+    }
+}
+
+impl<N, S> NormedSpace for QuaternionBase<N, S>
+    where N: Real,
+          S: OwnedStorage<N, U4, U1>,
+          S::Alloc: OwnedAllocator<N, U4, U1, S> {
+    #[inline]
+    fn norm_squared(&self) -> N {
+        self.coords.norm_squared()
+    }
+
+    #[inline]
+    fn norm(&self) -> N {
+        self.as_vector().norm()
+    }
+
+    #[inline]
+    fn normalize(&self) -> Self {
+        let v = self.coords.normalize();
+        Self::from_vector(v)
+    }
+
+    #[inline]
+    fn normalize_mut(&mut self) -> N {
+        self.coords.normalize_mut()
+    }
+
+    #[inline]
+    fn try_normalize(&self, min_norm: N) -> Option<Self> {
+        if let Some(v) = self.coords.try_normalize(min_norm) {
+            Some(Self::from_vector(v))
+        }
+        else {
+            None
+        }
+    }
+
+    #[inline]
+    fn try_normalize_mut(&mut self, min_norm: N) -> Option<N> {
+        self.coords.try_normalize_mut(min_norm)
+    }
+}
+
+/*
+ *
+ * Implementations for UnitQuaternionBase.
+ *
+ */
+impl<N, S> Identity<Multiplicative> for UnitQuaternionBase<N, S>
+    where N: Real,
+          S: OwnedStorage<N, U4, U1>,
+          S::Alloc: OwnedAllocator<N, U4, U1, S> {
+    #[inline]
+    fn identity() -> Self {
+        Self::identity()
+    }
+}
+
+impl<N, S> AbstractMagma<Multiplicative> for UnitQuaternionBase<N, S>
+    where N: Real,
+          S: OwnedStorage<N, U4, U1>,
+          S::Alloc: OwnedAllocator<N, U4, U1, S> {
+    #[inline]
+    fn operate(&self, rhs: &Self) -> Self {
+        self * rhs
+    }
+}
+
+impl<N, S> Inverse<Multiplicative> for UnitQuaternionBase<N, S>
+    where N: Real,
+          S: OwnedStorage<N, U4, U1>,
+          S::Alloc: OwnedAllocator<N, U4, U1, S> {
+    #[inline]
+    fn inverse(&self) -> Self {
+        self.inverse()
+    }
+
+    #[inline]
+    fn inverse_mut(&mut self) {
+        self.inverse_mut()
+    }
+}
+
+impl_structures!(
+    UnitQuaternionBase;
+    AbstractSemigroup<Multiplicative>,
+    AbstractQuasigroup<Multiplicative>,
+    AbstractMonoid<Multiplicative>,
+    AbstractLoop<Multiplicative>,
+    AbstractGroup<Multiplicative>
+);
+
+impl<N, SA, SB> Transformation<PointBase<N, U3, SB>> for UnitQuaternionBase<N, SA>
+    where N: Real,
+          SA: OwnedStorage<N, U4, U1>,
+          SB: OwnedStorage<N, U3, U1, Alloc = SA::Alloc>,
+          SA::Alloc: OwnedAllocator<N, U4, U1, SA>,
+          SB::Alloc: OwnedAllocator<N, U3, U1, SB> {
+    #[inline]
+    fn transform_point(&self, pt: &PointBase<N, U3, SB>) -> PointBase<N, U3, SB> {
+        self * pt
+    }
+
+    #[inline]
+    fn transform_vector(&self, v: &ColumnVector<N, U3, SB>) -> ColumnVector<N, U3, SB> {
+        self * v
+    }
+}
+
+impl<N, SA, SB> ProjectiveTransformation<PointBase<N, U3, SB>> for UnitQuaternionBase<N, SA>
+    where N: Real,
+          SA: OwnedStorage<N, U4, U1>,
+          SB: OwnedStorage<N, U3, U1, Alloc = SA::Alloc>,
+          SA::Alloc: OwnedAllocator<N, U4, U1, SA>,
+          SB::Alloc: OwnedAllocator<N, U3, U1, SB> {
+    #[inline]
+    fn inverse_transform_point(&self, pt: &PointBase<N, U3, SB>) -> PointBase<N, U3, SB> {
+        // FIXME: would it be useful performancewise not to call inverse explicitly (i-e. implement
+        // the inverse transformation explicitly here) ?
+        self.inverse() * pt
+    }
+
+    #[inline]
+    fn inverse_transform_vector(&self, v: &ColumnVector<N, U3, SB>) -> ColumnVector<N, U3, SB> {
+        self.inverse() * v
+    }
+}
+
+impl<N, SA, SB> AffineTransformation<PointBase<N, U3, SB>> for UnitQuaternionBase<N, SA>
+    where N: Real,
+          SA: OwnedStorage<N, U4, U1>,
+          SB: OwnedStorage<N, U3, U1, Alloc = SA::Alloc>,
+          SA::Alloc: OwnedAllocator<N, U4, U1, SA>,
+          SB::Alloc: OwnedAllocator<N, U3, U1, SB> {
+    type Rotation          = Self;
+    type NonUniformScaling = Id;
+    type Translation       = Id;
+
+    #[inline]
+    fn decompose(&self) -> (Id, Self, Id, Self) {
+        (Id::new(), self.clone(), Id::new(), Self::identity())
+    }
+
+    #[inline]
+    fn append_translation(&self, _: &Self::Translation) -> Self {
+        self.clone()
+    }
+
+    #[inline]
+    fn prepend_translation(&self, _: &Self::Translation) -> Self {
+        self.clone()
+    }
+
+    #[inline]
+    fn append_rotation(&self, r: &Self::Rotation) -> Self {
+        r * self
+    }
+
+    #[inline]
+    fn prepend_rotation(&self, r: &Self::Rotation) -> Self {
+        self * r
+    }
+
+    #[inline]
+    fn append_scaling(&self, _: &Self::NonUniformScaling) -> Self {
+        self.clone()
+    }
+
+    #[inline]
+    fn prepend_scaling(&self, _: &Self::NonUniformScaling) -> Self {
+        self.clone()
+    }
+}
+
+impl<N, SA, SB> Similarity<PointBase<N, U3, SB>> for UnitQuaternionBase<N, SA>
+    where N: Real,
+          SA: OwnedStorage<N, U4, U1>,
+          SB: OwnedStorage<N, U3, U1, Alloc = SA::Alloc>,
+          SA::Alloc: OwnedAllocator<N, U4, U1, SA>,
+          SB::Alloc: OwnedAllocator<N, U3, U1, SB> {
+    type Scaling  = Id;
+
+    #[inline]
+    fn translation(&self) -> Id {
+        Id::new()
+    }
+
+    #[inline]
+    fn rotation(&self) -> Self {
+        self.clone()
+    }
+
+    #[inline]
+    fn scaling(&self) -> Id {
+        Id::new()
+    }
+}
+
+macro_rules! marker_impl(
+    ($($Trait: ident),*) => {$(
+        impl<N, SA, SB> $Trait<PointBase<N, U3, SB>> for UnitQuaternionBase<N, SA>
+        where N: Real,
+              SA: OwnedStorage<N, U4, U1>,
+              SB: OwnedStorage<N, U3, U1, Alloc = SA::Alloc>,
+              SA::Alloc: OwnedAllocator<N, U4, U1, SA>,
+              SB::Alloc: OwnedAllocator<N, U3, U1, SB> { }
+    )*}
+);
+
+marker_impl!(Isometry, DirectIsometry, OrthogonalTransformation);
+
+
+
+impl<N, SA, SB> Rotation<PointBase<N, U3, SB>> for UnitQuaternionBase<N, SA>
+    where N:  Real,
+          SA: OwnedStorage<N, U4, U1>,
+          SB: OwnedStorage<N, U3, U1, Alloc = SA::Alloc>,
+          SA::Alloc: OwnedAllocator<N, U4, U1, SA> + Allocator<N, U3, U1>,
+          SB::Alloc: OwnedAllocator<N, U3, U1, SB> {
+    #[inline]
+    fn powf(&self, n: N) -> Option<Self> {
+        Some(self.powf(n))
+    }
+
+    #[inline]
+    fn rotation_between(a: &ColumnVector<N, U3, SB>, b: &ColumnVector<N, U3, SB>) -> Option<Self> {
+        Self::rotation_between(a, b)
+    }
+
+    #[inline]
+    fn scaled_rotation_between(a: &ColumnVector<N, U3, SB>, b: &ColumnVector<N, U3, SB>, s: N) -> Option<Self> {
+        Self::scaled_rotation_between(a, b, s)
+    }
+}

src/geometry/quaternion_alias.rs

@@ -0,0 +1,10 @@
+use core::MatrixArray;
+use core::dimension::{U1, U4};
+
+use geometry::{QuaternionBase, UnitQuaternionBase};
+
+/// A statically-allocated quaternion.
+pub type Quaternion<N> = QuaternionBase<N, MatrixArray<N, U4, U1>>;
+
+/// A statically-allocated unit quaternion.
+pub type UnitQuaternion<N> = UnitQuaternionBase<N, MatrixArray<N, U4, U1>>;

src/geometry/quaternion_construction.rs

@@ -0,0 +1,369 @@
+#[cfg(feature = "arbitrary")]
+use quickcheck::{Arbitrary, Gen};
+
+use rand::{Rand, Rng};
+use num::{Zero, One};
+
+use alga::general::Real;
+
+use core::{Unit, ColumnVector, Vector3};
+use core::storage::{Storage, OwnedStorage};
+use core::allocator::{Allocator, OwnedAllocator};
+use core::dimension::{U1, U3, U4};
+
+use geometry::{QuaternionBase, UnitQuaternionBase, RotationBase, OwnedRotation};
+
+impl<N, S> QuaternionBase<N, S>
+    where N: Real,
+          S: Storage<N, U4, U1> {
+    /// Creates a quaternion from a 4D vector. The quaternion scalar part corresponds to the `w`
+    /// vector component.
+    #[inline]
+    pub fn from_vector(vector: ColumnVector<N, U4, S>) -> Self {
+        QuaternionBase {
+            coords: vector
+        }
+    }
+}
+
+impl<N, S> QuaternionBase<N, S>
+    where N: Real,
+          S: OwnedStorage<N, U4, U1>,
+          S::Alloc: OwnedAllocator<N, U4, U1, S> {
+    /// Creates a new quaternion from its individual components. Note that the arguments order does
+    /// **not** follow the storage order.
+    ///
+    /// The storage order is `[ x, y, z, w ]`.
+    #[inline]
+    pub fn new(w: N, x: N, y: N, z: N) -> Self {
+        let v = ColumnVector::<N, U4, S>::new(x, y, z, w);
+        Self::from_vector(v)
+    }
+
+    /// Creates a new quaternion from its scalar and vector parts. Note that the arguments order does
+    /// **not** follow the storage order.
+    ///
+    /// The storage order is [ vector, scalar ].
+    #[inline]
+    // FIXME: take a reference to `vector`?
+    pub fn from_parts<SB>(scalar: N, vector: ColumnVector<N, U3, SB>) -> Self
+        where SB: Storage<N, U3, U1> {
+
+        Self::new(scalar, vector[0], vector[1], vector[2])
+    }
+
+    /// Creates a new quaternion from its polar decomposition.
+    ///
+    /// Note that `axis` is assumed to be a unit vector.
+    // FIXME: take a reference to `axis`?
+    pub fn from_polar_decomposition<SB>(scale: N, theta: N, axis: Unit<ColumnVector<N, U3, SB>>) -> Self
+        where SB: Storage<N, U3, U1> {
+        let rot = UnitQuaternionBase::<N, S>::from_axis_angle(&axis, theta * ::convert(2.0f64));
+
+        rot.unwrap() * scale
+    }
+
+    /// The quaternion multiplicative identity.
+    #[inline]
+    pub fn identity() -> Self {
+        Self::new(N::one(), N::zero(), N::zero(), N::zero())
+    }
+}
+
+impl<N, S> One for QuaternionBase<N, S>
+    where N: Real,
+          S: OwnedStorage<N, U4, U1>,
+          S::Alloc: OwnedAllocator<N, U4, U1, S> {
+    #[inline]
+    fn one() -> Self {
+        Self::identity()
+    }
+}
+
+impl<N, S> Zero for QuaternionBase<N, S>
+    where N: Real,
+          S: OwnedStorage<N, U4, U1>,
+          S::Alloc: OwnedAllocator<N, U4, U1, S> {
+    #[inline]
+    fn zero() -> Self {
+        Self::new(N::zero(), N::zero(), N::zero(), N::zero())
+    }
+
+    #[inline]
+    fn is_zero(&self) -> bool {
+        self.coords.is_zero()
+    }
+}
+
+impl<N, S> Rand for QuaternionBase<N, S>
+    where N: Real + Rand,
+          S: OwnedStorage<N, U4, U1>,
+          S::Alloc: OwnedAllocator<N, U4, U1, S> {
+    #[inline]
+    fn rand<R: Rng>(rng: &mut R) -> Self {
+        QuaternionBase::new(rng.gen(), rng.gen(), rng.gen(), rng.gen())
+    }
+}
+
+#[cfg(feature="arbitrary")]
+impl<N, S> Arbitrary for QuaternionBase<N, S>
+    where N: Real + Arbitrary,
+          S: OwnedStorage<N, U4, U1> + Send,
+          S::Alloc: OwnedAllocator<N, U4, U1, S> {
+    #[inline]
+    fn arbitrary<G: Gen>(g: &mut G) -> Self {
+        QuaternionBase::new(N::arbitrary(g), N::arbitrary(g),
+                            N::arbitrary(g), N::arbitrary(g))
+    }
+}
+
+impl<N, S> UnitQuaternionBase<N, S>
+    where N: Real,
+          S: OwnedStorage<N, U4, U1>,
+          S::Alloc: OwnedAllocator<N, U4, U1, S> {
+    /// The quaternion multiplicative identity.
+    #[inline]
+    pub fn identity() -> Self {
+        Self::new_unchecked(QuaternionBase::identity())
+    }
+
+    /// Creates a new quaternion from a unit vector (the rotation axis) and an angle
+    /// (the rotation angle).
+    #[inline]
+    pub fn from_axis_angle<SB>(axis: &Unit<ColumnVector<N, U3, SB>>, angle: N) -> Self
+        where SB: Storage<N, U3, U1> {
+        let (sang, cang) = (angle / ::convert(2.0f64)).sin_cos();
+
+        let q = QuaternionBase::from_parts(cang, axis.as_ref() * sang);
+        Self::new_unchecked(q)
+    }
+
+    /// Creates a new unit quaternion from a quaternion.
+    ///
+    /// The input quaternion will be normalized.
+    #[inline]
+    pub fn from_quaternion(q: QuaternionBase<N, S>) -> Self {
+        Self::new_normalize(q)
+    }
+
+    /// Creates a new unit quaternion from Euler angles.
+    ///
+    /// The primitive rotations are applied in order: 1 roll − 2 pitch − 3 yaw.
+    #[inline]
+    pub fn from_euler_angles(roll: N, pitch: N, yaw: N) -> Self {
+        let (sr, cr) = (roll  * ::convert(0.5f64)).sin_cos();
+        let (sp, cp) = (pitch * ::convert(0.5f64)).sin_cos();
+        let (sy, cy) = (yaw   * ::convert(0.5f64)).sin_cos();
+
+        let q = QuaternionBase::new(
+                    cr * cp * cy + sr * sp * sy,
+                    sr * cp * cy - cr * sp * sy,
+                    cr * sp * cy + sr * cp * sy,
+                    cr * cp * sy - sr * sp * cy);
+
+        Self::new_unchecked(q)
+    }
+
+    /// Builds an unit quaternion from a rotation matrix.
+    #[inline]
+    pub fn from_rotation_matrix<SB>(rotmat: &RotationBase<N, U3, SB>) -> Self
+        where SB: Storage<N, U3, U3>,
+              SB::Alloc: Allocator<N, U3, U1> {
+
+        // Robust matrix to quaternion transformation.
+        // See http://www.euclideanspace.com/maths/geometry/rotations/conversions/matrixToQuaternion
+        let tr = rotmat[(0, 0)] + rotmat[(1, 1)] + rotmat[(2, 2)];
+        let res;
+
+        let _0_25: N = ::convert(0.25);
+
+        if tr > N::zero() {
+          let denom = (tr + N::one()).sqrt() * ::convert(2.0);
+          res = QuaternionBase::new(_0_25 * denom,
+                                    (rotmat[(2, 1)] - rotmat[(1, 2)]) / denom,
+                                    (rotmat[(0, 2)] - rotmat[(2, 0)]) / denom,
+                                    (rotmat[(1, 0)] - rotmat[(0, 1)]) / denom);
+        }
+        else if rotmat[(0, 0)] > rotmat[(1, 1)] && rotmat[(0, 0)] > rotmat[(2, 2)] {
+          let denom = (N::one() + rotmat[(0, 0)] - rotmat[(1, 1)] - rotmat[(2, 2)]).sqrt() * ::convert(2.0);
+          res = QuaternionBase::new((rotmat[(2, 1)] - rotmat[(1, 2)]) / denom,
+                                    _0_25 * denom,
+                                    (rotmat[(0, 1)] + rotmat[(1, 0)]) / denom,
+                                    (rotmat[(0, 2)] + rotmat[(2, 0)]) / denom);
+        }
+        else if rotmat[(1, 1)] > rotmat[(2, 2)] {
+          let denom = (N::one() + rotmat[(1, 1)] - rotmat[(0, 0)] - rotmat[(2, 2)]).sqrt() * ::convert(2.0);
+          res = QuaternionBase::new((rotmat[(0, 2)] - rotmat[(2, 0)]) / denom,
+                                    (rotmat[(0, 1)] + rotmat[(1, 0)]) / denom,
+                                    _0_25 * denom,
+                                    (rotmat[(1, 2)] + rotmat[(2, 1)]) / denom);
+        }
+        else {
+          let denom = (N::one() + rotmat[(2, 2)] - rotmat[(0, 0)] - rotmat[(1, 1)]).sqrt() * ::convert(2.0);
+          res = QuaternionBase::new((rotmat[(1, 0)] - rotmat[(0, 1)]) / denom,
+                                    (rotmat[(0, 2)] + rotmat[(2, 0)]) / denom,
+                                    (rotmat[(1, 2)] + rotmat[(2, 1)]) / denom,
+                                    _0_25 * denom);
+        }
+
+        Self::new_unchecked(res)
+    }
+
+    /// The unit quaternion needed to make `a` and `b` be collinear and point toward the same
+    /// direction.
+    #[inline]
+    pub fn rotation_between<SB, SC>(a: &ColumnVector<N, U3, SB>, b: &ColumnVector<N, U3, SC>) -> Option<Self>
+        where SB: Storage<N, U3, U1>,
+              SC: Storage<N, U3, U1> {
+        Self::scaled_rotation_between(a, b, N::one())
+    }
+
+    /// The smallest rotation needed to make `a` and `b` collinear and point toward the same
+    /// direction, raised to the power `s`.
+    #[inline]
+    pub fn scaled_rotation_between<SB, SC>(a: &ColumnVector<N, U3, SB>, b: &ColumnVector<N, U3, SC>, s: N) -> Option<Self>
+        where SB: Storage<N, U3, U1>,
+              SC: Storage<N, U3, U1> {
+        // FIXME: code duplication with RotationBase.
+        if let (Some(na), Some(nb)) = (a.try_normalize(N::zero()), b.try_normalize(N::zero())) {
+            let c = na.cross(&nb);
+
+            if let Some(axis) = Unit::try_new(c, N::default_epsilon()) {
+                return Some(Self::from_axis_angle(&axis, na.dot(&nb).acos() * s))
+            }
+
+            // Zero or PI.
+            if na.dot(&nb) < N::zero() {
+                // PI
+                //
+                // The rotation axis is undefined but the angle not zero. This is not a
+                // simple rotation.
+                return None;
+            }
+        }
+
+        Some(Self::identity())
+    }
+
+
+    /// Creates an unit quaternion that corresponds to the local frame of an observer standing at the
+    /// origin and looking toward `dir`.
+    ///
+    /// It maps the view direction `dir` to the positive `z` axis.
+    ///
+    /// # Arguments
+    ///   * dir - The look direction, that is, direction the matrix `z` axis will be aligned with.
+    ///   * up - The vertical direction. The only requirement of this parameter is to not be
+    ///   collinear
+    ///   to `dir`. Non-collinearity is not checked.
+    #[inline]
+    pub fn new_observer_frame<SB, SC>(dir: &ColumnVector<N, U3, SB>, up: &ColumnVector<N, U3, SC>) -> Self
+    where SB: Storage<N, U3, U1>,
+          SC: Storage<N, U3, U1>,
+          S::Alloc: Allocator<N, U3, U3> +
+                    Allocator<N, U3, U1> {
+        Self::from_rotation_matrix(&OwnedRotation::<N, U3, S::Alloc>::new_observer_frame(dir, up))
+    }
+
+
+    /// Builds a right-handed look-at view matrix without translation.
+    ///
+    /// This conforms to the common notion of right handed look-at matrix from the computer
+    /// graphics community.
+    ///
+    /// # Arguments
+    ///   * eye - The eye position.
+    ///   * target - The target position.
+    ///   * up - A vector approximately aligned with required the vertical axis. The only
+    ///   requirement of this parameter is to not be collinear to `target - eye`.
+    #[inline]
+    pub fn look_at_rh<SB, SC>(dir: &ColumnVector<N, U3, SB>, up: &ColumnVector<N, U3, SC>) -> Self
+    where SB: Storage<N, U3, U1>,
+          SC: Storage<N, U3, U1>,
+          S::Alloc: Allocator<N, U3, U3> +
+                    Allocator<N, U3, U1> {
+        Self::new_observer_frame(&-dir, up).inverse()
+    }
+
+    /// Builds a left-handed look-at view matrix without translation.
+    ///
+    /// This conforms to the common notion of left handed look-at matrix from the computer
+    /// graphics community.
+    ///
+    /// # Arguments
+    ///   * eye - The eye position.
+    ///   * target - The target position.
+    ///   * up - A vector approximately aligned with required the vertical axis. The only
+    ///   requirement of this parameter is to not be collinear to `target - eye`.
+    #[inline]
+    pub fn look_at_lh<SB, SC>(dir: &ColumnVector<N, U3, SB>, up: &ColumnVector<N, U3, SC>) -> Self
+    where SB: Storage<N, U3, U1>,
+          SC: Storage<N, U3, U1>,
+          S::Alloc: Allocator<N, U3, U3> +
+                    Allocator<N, U3, U1> {
+            Self::new_observer_frame(dir, up).inverse()
+    }
+}
+
+impl<N, S> UnitQuaternionBase<N, S>
+    where N: Real,
+          S: OwnedStorage<N, U4, U1>,
+          S::Alloc: OwnedAllocator<N, U4, U1, S> +
+                    Allocator<N, U3, U1> {
+    /// Creates a new unit quaternion rotation from a rotation axis scaled by the rotation angle.
+    ///
+    /// If `axisangle` is zero, this returns the indentity rotation.
+    #[inline]
+    pub fn new<SB>(axisangle: ColumnVector<N, U3, SB>) -> Self
+        where SB: Storage<N, U3, U1> {
+        let two: N = ::convert(2.0f64);
+        let q = QuaternionBase::<N, S>::from_parts(N::zero(), axisangle / two).exp();
+        Self::new_unchecked(q)
+    }
+
+    /// Creates a new unit quaternion rotation from a rotation axis scaled by the rotation angle.
+    ///
+    /// If `axisangle` is zero, this returns the indentity rotation.
+    /// Same as `Self::new(axisangle)`.
+    #[inline]
+    pub fn from_scaled_axis<SB>(axisangle: ColumnVector<N, U3, SB>) -> Self
+        where SB: Storage<N, U3, U1> {
+        Self::new(axisangle)
+    }
+}
+
+impl<N, S> One for UnitQuaternionBase<N, S>
+    where N: Real,
+          S: OwnedStorage<N, U4, U1>,
+          S::Alloc: OwnedAllocator<N, U4, U1, S> {
+    #[inline]
+    fn one() -> Self {
+        Self::identity()
+    }
+}
+
+impl<N, S> Rand for UnitQuaternionBase<N, S>
+    where N: Real + Rand,
+          S: OwnedStorage<N, U4, U1>,
+          S::Alloc: OwnedAllocator<N, U4, U1, S> +
+                    Allocator<N, U3, U1> {
+    #[inline]
+    fn rand<R: Rng>(rng: &mut R) -> Self {
+        let axisangle = Vector3::rand(rng);
+        UnitQuaternionBase::from_scaled_axis(axisangle)
+    }
+}
+
+#[cfg(feature="arbitrary")]
+impl<N, S> Arbitrary for UnitQuaternionBase<N, S>
+    where N: Real + Arbitrary,
+          S: OwnedStorage<N, U4, U1> + Send,
+          S::Alloc: OwnedAllocator<N, U4, U1, S> +
+                    Allocator<N, U3, U1> {
+    #[inline]
+    fn arbitrary<G: Gen>(g: &mut G) -> Self {
+        let axisangle = Vector3::arbitrary(g);
+        UnitQuaternionBase::from_scaled_axis(axisangle)
+
+    }
+}

src/geometry/quaternion_conversion.rs

@@ -0,0 +1,212 @@
+use num::Zero;
+
+use alga::general::{SubsetOf, SupersetOf, Real};
+use alga::linear::Rotation as AlgaRotation;
+
+use core::{ColumnVector, SquareMatrix};
+use core::dimension::{U1, U3, U4};
+use core::storage::OwnedStorage;
+use core::allocator::{Allocator, OwnedAllocator};
+use geometry::{PointBase, QuaternionBase, UnitQuaternionBase, OwnedUnitQuaternionBase, RotationBase,
+               OwnedRotation, IsometryBase, SimilarityBase, TransformBase, SuperTCategoryOf, TAffine, TranslationBase};
+
+/*
+ * This file provides the following conversions:
+ * =============================================
+ *
+ * Quaternion     -> Quaternion
+ * UnitQuaternion -> UnitQuaternion
+ * UnitQuaternion -> RotationBase<U3>
+ * UnitQuaternion -> IsometryBase<U3>
+ * UnitQuaternion -> SimilarityBase<U3>
+ * UnitQuaternion -> TransformBase<U3>
+ * UnitQuaternion -> Matrix<U4> (homogeneous)
+ *
+ * NOTE:
+ * UnitQuaternion -> Quaternion is already provided by: Unit<T> -> T
+ */
+
+impl<N1, N2, SA, SB> SubsetOf<QuaternionBase<N2, SB>> for QuaternionBase<N1, SA>
+    where N1: Real,
+          N2: Real + SupersetOf<N1>,
+          SA: OwnedStorage<N1, U4, U1>,
+          SB: OwnedStorage<N2, U4, U1>,
+          SA::Alloc: OwnedAllocator<N1, U4, U1, SA>,
+          SB::Alloc: OwnedAllocator<N2, U4, U1, SB> {
+    #[inline]
+    fn to_superset(&self) -> QuaternionBase<N2, SB> {
+        QuaternionBase::from_vector(self.coords.to_superset())
+    }
+
+    #[inline]
+    fn is_in_subset(q: &QuaternionBase<N2, SB>) -> bool {
+        ::is_convertible::<_, ColumnVector<N1, U4, SA>>(&q.coords)
+    }
+
+    #[inline]
+    unsafe fn from_superset_unchecked(q: &QuaternionBase<N2, SB>) -> Self {
+        Self::from_vector(q.coords.to_subset_unchecked())
+    }
+}
+
+impl<N1, N2, SA, SB> SubsetOf<UnitQuaternionBase<N2, SB>> for UnitQuaternionBase<N1, SA>
+    where N1: Real,
+          N2: Real + SupersetOf<N1>,
+          SA: OwnedStorage<N1, U4, U1>,
+          SB: OwnedStorage<N2, U4, U1>,
+          SA::Alloc: OwnedAllocator<N1, U4, U1, SA>,
+          SB::Alloc: OwnedAllocator<N2, U4, U1, SB> {
+    #[inline]
+    fn to_superset(&self) -> UnitQuaternionBase<N2, SB> {
+        UnitQuaternionBase::new_unchecked(self.as_ref().to_superset())
+    }
+
+    #[inline]
+    fn is_in_subset(uq: &UnitQuaternionBase<N2, SB>) -> bool {
+        ::is_convertible::<_, QuaternionBase<N1, SA>>(uq.as_ref())
+    }
+
+    #[inline]
+    unsafe fn from_superset_unchecked(uq: &UnitQuaternionBase<N2, SB>) -> Self {
+        Self::new_unchecked(::convert_ref_unchecked(uq.as_ref()))
+    }
+}
+
+impl<N1, N2, SA, SB> SubsetOf<RotationBase<N2, U3, SB>> for UnitQuaternionBase<N1, SA>
+    where N1: Real,
+          N2: Real + SupersetOf<N1>,
+          SA: OwnedStorage<N1, U4, U1>,
+          SB: OwnedStorage<N2, U3, U3>,
+          SA::Alloc: OwnedAllocator<N1, U4, U1, SA> +
+                     Allocator<N1, U3, U3>,
+          SB::Alloc: OwnedAllocator<N2, U3, U3, SB> +
+                     Allocator<N2, U4, U1> +
+                     Allocator<N2, U3, U1> {
+    #[inline]
+    fn to_superset(&self) -> RotationBase<N2, U3, SB> {
+        let q: OwnedUnitQuaternionBase<N2, SB::Alloc> = self.to_superset();
+        q.to_rotation_matrix()
+    }
+
+    #[inline]
+    fn is_in_subset(rot: &RotationBase<N2, U3, SB>) -> bool {
+        ::is_convertible::<_, OwnedRotation<N1, U3, SA::Alloc>>(rot)
+    }
+
+    #[inline]
+    unsafe fn from_superset_unchecked(rot: &RotationBase<N2, U3, SB>) -> Self {
+        let q = OwnedUnitQuaternionBase::<N2, SB::Alloc>::from_rotation_matrix(rot);
+        ::convert_unchecked(q)
+    }
+}
+
+
+impl<N1, N2, SA, SB, R> SubsetOf<IsometryBase<N2, U3, SB, R>> for UnitQuaternionBase<N1, SA>
+    where N1: Real,
+          N2: Real + SupersetOf<N1>,
+          SA: OwnedStorage<N1, U4, U1>,
+          SB: OwnedStorage<N2, U3, U1>,
+          R:  AlgaRotation<PointBase<N2, U3, SB>> + SupersetOf<UnitQuaternionBase<N1, SA>>,
+          SA::Alloc: OwnedAllocator<N1, U4, U1, SA>,
+          SB::Alloc: OwnedAllocator<N2, U3, U1, SB> {
+    #[inline]
+    fn to_superset(&self) -> IsometryBase<N2, U3, SB, R> {
+        IsometryBase::from_parts(TranslationBase::identity(), ::convert_ref(self))
+    }
+
+    #[inline]
+    fn is_in_subset(iso: &IsometryBase<N2, U3, SB, R>) -> bool {
+        iso.translation.vector.is_zero()
+    }
+
+    #[inline]
+    unsafe fn from_superset_unchecked(iso: &IsometryBase<N2, U3, SB, R>) -> Self {
+        ::convert_ref_unchecked(&iso.rotation)
+    }
+}
+
+
+impl<N1, N2, SA, SB, R> SubsetOf<SimilarityBase<N2, U3, SB, R>> for UnitQuaternionBase<N1, SA>
+    where N1: Real,
+          N2: Real + SupersetOf<N1>,
+          SA: OwnedStorage<N1, U4, U1>,
+          SB: OwnedStorage<N2, U3, U1>,
+          R:  AlgaRotation<PointBase<N2, U3, SB>> + SupersetOf<UnitQuaternionBase<N1, SA>>,
+          SA::Alloc: OwnedAllocator<N1, U4, U1, SA>,
+          SB::Alloc: OwnedAllocator<N2, U3, U1, SB> {
+    #[inline]
+    fn to_superset(&self) -> SimilarityBase<N2, U3, SB, R> {
+        SimilarityBase::from_isometry(::convert_ref(self), N2::one())
+    }
+
+    #[inline]
+    fn is_in_subset(sim: &SimilarityBase<N2, U3, SB, R>) -> bool {
+        sim.isometry.translation.vector.is_zero() &&
+        sim.scaling() == N2::one()
+    }
+
+    #[inline]
+    unsafe fn from_superset_unchecked(sim: &SimilarityBase<N2, U3, SB, R>) -> Self {
+        ::convert_ref_unchecked(&sim.isometry)
+    }
+}
+
+
+impl<N1, N2, SA, SB, C> SubsetOf<TransformBase<N2, U3, SB, C>> for UnitQuaternionBase<N1, SA>
+    where N1: Real,
+          N2: Real + SupersetOf<N1>,
+          SA: OwnedStorage<N1, U4, U1>,
+          SB: OwnedStorage<N2, U4, U4>,
+          C:  SuperTCategoryOf<TAffine>,
+          SA::Alloc: OwnedAllocator<N1, U4, U1, SA> +
+                     Allocator<N1, U3, U3> +
+                     Allocator<N1, U3, U1> +
+                     Allocator<N1, U4, U4>,
+          SB::Alloc: OwnedAllocator<N2, U4, U4, SB> +
+                     Allocator<N2, U3, U3> +
+                     Allocator<N2, U1, U3> {
+    #[inline]
+    fn to_superset(&self) -> TransformBase<N2, U3, SB, C> {
+        TransformBase::from_matrix_unchecked(self.to_homogeneous().to_superset())
+    }
+
+    #[inline]
+    fn is_in_subset(t: &TransformBase<N2, U3, SB, C>) -> bool {
+        <Self as SubsetOf<_>>::is_in_subset(t.matrix())
+    }
+
+    #[inline]
+    unsafe fn from_superset_unchecked(t: &TransformBase<N2, U3, SB, C>) -> Self {
+        Self::from_superset_unchecked(t.matrix())
+    }
+}
+
+
+impl<N1, N2, SA, SB> SubsetOf<SquareMatrix<N2, U4, SB>> for UnitQuaternionBase<N1, SA>
+    where N1: Real,
+          N2: Real + SupersetOf<N1>,
+          SA: OwnedStorage<N1, U4, U1>,
+          SB: OwnedStorage<N2, U4, U4>,
+          SA::Alloc: OwnedAllocator<N1, U4, U1, SA> +
+                     Allocator<N1, U3, U3> +
+                     Allocator<N1, U3, U1> +
+                     Allocator<N1, U4, U4>,
+          SB::Alloc: OwnedAllocator<N2, U4, U4, SB> +
+                     Allocator<N2, U3, U3> +
+                     Allocator<N2, U1, U3> {
+    #[inline]
+    fn to_superset(&self) -> SquareMatrix<N2, U4, SB> {
+        self.to_homogeneous().to_superset()
+    }
+
+    #[inline]
+    fn is_in_subset(m: &SquareMatrix<N2, U4, SB>) -> bool {
+        ::is_convertible::<_, OwnedRotation<N1, U3, SA::Alloc>>(m)
+    }
+
+    #[inline]
+    unsafe fn from_superset_unchecked(m: &SquareMatrix<N2, U4, SB>) -> Self {
+        let rot: OwnedRotation<N1, U3, SA::Alloc> = ::convert_ref_unchecked(m);
+        Self::from_rotation_matrix(&rot)
+    }
+}

src/geometry/quaternion_coordinates.rs

@@ -0,0 +1,34 @@
+use std::mem;
+use std::ops::{Deref, DerefMut};
+
+use alga::general::Real;
+
+use core::coordinates::IJKW;
+use core::storage::OwnedStorage;
+use core::allocator::OwnedAllocator;
+use core::dimension::{U1, U4};
+
+use geometry::QuaternionBase;
+
+
+impl<N, S> Deref for QuaternionBase<N, S>
+    where N: Real,
+          S: OwnedStorage<N, U4, U1>,
+          S::Alloc: OwnedAllocator<N, U4, U1, S> {
+    type Target = IJKW<N>;
+
+    #[inline]
+    fn deref(&self) -> &Self::Target {
+        unsafe { mem::transmute(self) }
+    }
+}
+
+impl<N, S> DerefMut for QuaternionBase<N, S>
+    where N: Real,
+          S: OwnedStorage<N, U4, U1>,
+          S::Alloc: OwnedAllocator<N, U4, U1, S> {
+    #[inline]
+    fn deref_mut(&mut self) -> &mut Self::Target {
+        unsafe { mem::transmute(self) }
+    }
+}

src/geometry/quaternion_ops.rs

@@ -0,0 +1,729 @@
+/*
+ * This file provides:
+ * ===================
+ *
+ *
+ * (Quaternion)
+ *
+ * Index<usize>
+ * IndexMut<usize>
+ * Quaternion × Quaternion
+ * Quaternion + Quaternion
+ * Quaternion - Quaternion
+ * -Quaternion
+ * Quaternion × Scalar
+ * Quaternion ÷ Scalar
+ * Scalar × Quaternion
+ *
+ * (Unit Quaternion)
+ * UnitQuaternion × UnitQuaternion
+ * UnitQuaternion × RotationBase   -> UnitQuaternion
+ * RotationBase   × UnitQuaternion -> UnitQuaternion
+ *
+ * UnitQuaternion ÷ UnitQuaternion
+ * UnitQuaternion ÷ RotationBase   -> UnitQuaternion
+ * RotationBase   ÷ UnitQuaternion -> UnitQuaternion
+ *
+ *
+ * UnitQuaternion × PointBase
+ * UnitQuaternion × ColumnVector
+ * UnitQuaternion × Unit<ColumnVector>
+ *
+ * NOTE: -UnitQuaternion is already provided by `Unit<T>`.
+ *
+ *
+ * (Assignment Operators)
+ *
+ * Quaternion ×= Scalar
+ * Quaternion ×= Quaternion
+ * Quaternion += Quaternion
+ * Quaternion -= Quaternion
+ *
+ * UnitQuaternion ×= UnitQuaternion
+ * UnitQuaternion ×= RotationBase
+ *
+ * UnitQuaternion ÷= UnitQuaternion
+ * UnitQuaternion ÷= RotationBase
+ *
+ * FIXME: RotationBase ×= UnitQuaternion
+ * FIXME: RotationBase ÷= UnitQuaternion
+ *
+ */
+
+use std::ops::{Index, IndexMut, Neg, Add, AddAssign, Mul, MulAssign, Div, DivAssign, Sub, SubAssign};
+
+use alga::general::Real;
+
+use core::{ColumnVector, OwnedColumnVector, Unit};
+use core::storage::{Storage, StorageMut};
+use core::allocator::Allocator;
+use core::dimension::{U1, U3, U4};
+
+use geometry::{QuaternionBase, OwnedQuaternionBase, UnitQuaternionBase, OwnedUnitQuaternionBase,
+               PointBase, OwnedPoint, RotationBase};
+
+impl<N, S> Index<usize> for QuaternionBase<N, S>
+    where N: Real,
+          S: Storage<N, U4, U1> {
+    type Output = N;
+
+    #[inline]
+    fn index(&self, i: usize) -> &N {
+        &self.coords[i]
+    }
+}
+
+impl<N, S> IndexMut<usize> for QuaternionBase<N, S>
+    where N: Real,
+          S: StorageMut<N, U4, U1> {
+
+    #[inline]
+    fn index_mut(&mut self, i: usize) -> &mut N {
+        &mut self.coords[i]
+    }
+}
+
+macro_rules! quaternion_op_impl(
+    ($Op: ident, $op: ident;
+     ($LhsRDim: ident, $LhsCDim: ident), ($RhsRDim: ident, $RhsCDim: ident);
+     $lhs: ident: $Lhs: ty, $rhs: ident: $Rhs: ty, Output = $Result: ty $(=> $VDimA: ty, $VDimB: ty)*;
+     $action: expr; $($lives: tt),*) => {
+        impl<$($lives ,)* N, SA, SB> $Op<$Rhs> for $Lhs
+            where N: Real,
+                  SA: Storage<N, $LhsRDim, $LhsCDim>,
+                  SB: Storage<N, $RhsRDim, $RhsCDim>,
+                  $(SA::Alloc: Allocator<N, $VDimA, U1>,
+                    SB::Alloc: Allocator<N, $VDimB, U1>)* {
+            type Output = $Result;
+
+            #[inline]
+            fn $op($lhs, $rhs: $Rhs) -> Self::Output {
+                $action
+            }
+        }
+    }
+);
+
+
+// Quaternion + Quaternion
+quaternion_op_impl!(
+    Add, add;
+    (U4, U1), (U4, U1);
+    self: &'a QuaternionBase<N, SA>, rhs: &'b QuaternionBase<N, SB>, Output = OwnedQuaternionBase<N, SA::Alloc>;
+    QuaternionBase::from_vector(&self.coords + &rhs.coords);
+    'a, 'b);
+
+quaternion_op_impl!(
+    Add, add;
+    (U4, U1), (U4, U1);
+    self: &'a QuaternionBase<N, SA>, rhs: QuaternionBase<N, SB>, Output = OwnedQuaternionBase<N, SB::Alloc>;
+    QuaternionBase::from_vector(&self.coords + rhs.coords);
+    'a);
+
+quaternion_op_impl!(
+    Add, add;
+    (U4, U1), (U4, U1);
+    self: QuaternionBase<N, SA>, rhs: &'b QuaternionBase<N, SB>, Output = OwnedQuaternionBase<N, SA::Alloc>;
+    QuaternionBase::from_vector(self.coords + &rhs.coords);
+    'b);
+
+quaternion_op_impl!(
+    Add, add;
+    (U4, U1), (U4, U1);
+    self: QuaternionBase<N, SA>, rhs: QuaternionBase<N, SB>, Output = OwnedQuaternionBase<N, SA::Alloc>;
+    QuaternionBase::from_vector(self.coords + rhs.coords);
+    );
+
+
+// Quaternion - Quaternion
+quaternion_op_impl!(
+    Sub, sub;
+    (U4, U1), (U4, U1);
+    self: &'a QuaternionBase<N, SA>, rhs: &'b QuaternionBase<N, SB>, Output = OwnedQuaternionBase<N, SA::Alloc>;
+    QuaternionBase::from_vector(&self.coords - &rhs.coords);
+    'a, 'b);
+
+quaternion_op_impl!(
+    Sub, sub;
+    (U4, U1), (U4, U1);
+    self: &'a QuaternionBase<N, SA>, rhs: QuaternionBase<N, SB>, Output = OwnedQuaternionBase<N, SB::Alloc>;
+    QuaternionBase::from_vector(&self.coords - rhs.coords);
+    'a);
+
+quaternion_op_impl!(
+    Sub, sub;
+    (U4, U1), (U4, U1);
+    self: QuaternionBase<N, SA>, rhs: &'b QuaternionBase<N, SB>, Output = OwnedQuaternionBase<N, SA::Alloc>;
+    QuaternionBase::from_vector(self.coords - &rhs.coords);
+    'b);
+
+quaternion_op_impl!(
+    Sub, sub;
+    (U4, U1), (U4, U1);
+    self: QuaternionBase<N, SA>, rhs: QuaternionBase<N, SB>, Output = OwnedQuaternionBase<N, SA::Alloc>;
+    QuaternionBase::from_vector(self.coords - rhs.coords);
+    );
+
+
+// Quaternion × Quaternion
+quaternion_op_impl!(
+    Mul, mul;
+    (U4, U1), (U4, U1);
+    self: &'a QuaternionBase<N, SA>, rhs: &'b QuaternionBase<N, SB>, Output = OwnedQuaternionBase<N, SA::Alloc>;
+    QuaternionBase::new(
+        self[3] * rhs[3] - self[0] * rhs[0] - self[1] * rhs[1] - self[2] * rhs[2],
+        self[3] * rhs[0] + self[0] * rhs[3] + self[1] * rhs[2] - self[2] * rhs[1],
+        self[3] * rhs[1] - self[0] * rhs[2] + self[1] * rhs[3] + self[2] * rhs[0],
+        self[3] * rhs[2] + self[0] * rhs[1] - self[1] * rhs[0] + self[2] * rhs[3]);
+    'a, 'b);
+
+quaternion_op_impl!(
+    Mul, mul;
+    (U4, U1), (U4, U1);
+    self: &'a QuaternionBase<N, SA>, rhs: QuaternionBase<N, SB>, Output = OwnedQuaternionBase<N, SA::Alloc>;
+    self * &rhs;
+    'a);
+
+quaternion_op_impl!(
+    Mul, mul;
+    (U4, U1), (U4, U1);
+    self: QuaternionBase<N, SA>, rhs: &'b QuaternionBase<N, SB>, Output = OwnedQuaternionBase<N, SA::Alloc>;
+    &self * rhs;
+    'b);
+
+quaternion_op_impl!(
+    Mul, mul;
+    (U4, U1), (U4, U1);
+    self: QuaternionBase<N, SA>, rhs: QuaternionBase<N, SB>, Output = OwnedQuaternionBase<N, SA::Alloc>;
+    &self * &rhs;
+    );
+
+// UnitQuaternion × UnitQuaternion
+quaternion_op_impl!(
+    Mul, mul;
+    (U4, U1), (U4, U1);
+    self: &'a UnitQuaternionBase<N, SA>, rhs: &'b UnitQuaternionBase<N, SB>, Output = OwnedUnitQuaternionBase<N, SA::Alloc>;
+    UnitQuaternionBase::new_unchecked(self.quaternion() * rhs.quaternion());
+    'a, 'b);
+
+quaternion_op_impl!(
+    Mul, mul;
+    (U4, U1), (U4, U1);
+    self: &'a UnitQuaternionBase<N, SA>, rhs: UnitQuaternionBase<N, SB>, Output = OwnedUnitQuaternionBase<N, SA::Alloc>;
+    self * &rhs;
+    'a);
+
+quaternion_op_impl!(
+    Mul, mul;
+    (U4, U1), (U4, U1);
+    self: UnitQuaternionBase<N, SA>, rhs: &'b UnitQuaternionBase<N, SB>, Output = OwnedUnitQuaternionBase<N, SA::Alloc>;
+    &self * rhs;
+    'b);
+
+quaternion_op_impl!(
+    Mul, mul;
+    (U4, U1), (U4, U1);
+    self: UnitQuaternionBase<N, SA>, rhs: UnitQuaternionBase<N, SB>, Output = OwnedUnitQuaternionBase<N, SA::Alloc>;
+    &self * &rhs;
+    );
+
+// UnitQuaternion ÷ UnitQuaternion
+quaternion_op_impl!(
+    Div, div;
+    (U4, U1), (U4, U1);
+    self: &'a UnitQuaternionBase<N, SA>, rhs: &'b UnitQuaternionBase<N, SB>, Output = OwnedUnitQuaternionBase<N, SA::Alloc>;
+    self * rhs.inverse();
+    'a, 'b);
+
+quaternion_op_impl!(
+    Div, div;
+    (U4, U1), (U4, U1);
+    self: &'a UnitQuaternionBase<N, SA>, rhs: UnitQuaternionBase<N, SB>, Output = OwnedUnitQuaternionBase<N, SA::Alloc>;
+    self / &rhs;
+    'a);
+
+quaternion_op_impl!(
+    Div, div;
+    (U4, U1), (U4, U1);
+    self: UnitQuaternionBase<N, SA>, rhs: &'b UnitQuaternionBase<N, SB>, Output = OwnedUnitQuaternionBase<N, SA::Alloc>;
+    &self / rhs;
+    'b);
+
+quaternion_op_impl!(
+    Div, div;
+    (U4, U1), (U4, U1);
+    self: UnitQuaternionBase<N, SA>, rhs: UnitQuaternionBase<N, SB>, Output = OwnedUnitQuaternionBase<N, SA::Alloc>;
+    &self / &rhs;
+    );
+
+// UnitQuaternion × RotationBase
+quaternion_op_impl!(
+    Mul, mul;
+    (U4, U1), (U3, U3);
+    self: &'a UnitQuaternionBase<N, SA>, rhs: &'b RotationBase<N, U3, SB>,
+    Output = OwnedUnitQuaternionBase<N, SA::Alloc> => U3, U3;
+    // FIXME: can we avoid the conversion from a rotation matrix?
+    self * OwnedUnitQuaternionBase::<N, SA::Alloc>::from_rotation_matrix(rhs);
+    'a, 'b);
+
+quaternion_op_impl!(
+    Mul, mul;
+    (U4, U1), (U3, U3);
+    self: &'a UnitQuaternionBase<N, SA>, rhs: RotationBase<N, U3, SB>,
+    Output = OwnedUnitQuaternionBase<N, SA::Alloc> => U3, U3;
+    self * OwnedUnitQuaternionBase::<N, SA::Alloc>::from_rotation_matrix(&rhs);
+    'a);
+
+quaternion_op_impl!(
+    Mul, mul;
+    (U4, U1), (U3, U3);
+    self: UnitQuaternionBase<N, SA>, rhs: &'b RotationBase<N, U3, SB>,
+    Output = OwnedUnitQuaternionBase<N, SA::Alloc> => U3, U3;
+    self * OwnedUnitQuaternionBase::<N, SA::Alloc>::from_rotation_matrix(rhs);
+    'b);
+
+quaternion_op_impl!(
+    Mul, mul;
+    (U4, U1), (U3, U3);
+    self: UnitQuaternionBase<N, SA>, rhs: RotationBase<N, U3, SB>,
+    Output = OwnedUnitQuaternionBase<N, SA::Alloc> => U3, U3;
+    self * OwnedUnitQuaternionBase::<N, SA::Alloc>::from_rotation_matrix(&rhs);
+    );
+
+// UnitQuaternion ÷ RotationBase
+quaternion_op_impl!(
+    Div, div;
+    (U4, U1), (U3, U3);
+    self: &'a UnitQuaternionBase<N, SA>, rhs: &'b RotationBase<N, U3, SB>,
+    Output = OwnedUnitQuaternionBase<N, SA::Alloc> => U3, U3;
+    // FIXME: can we avoid the conversion to a rotation matrix?
+    self / OwnedUnitQuaternionBase::<N, SA::Alloc>::from_rotation_matrix(rhs);
+    'a, 'b);
+
+quaternion_op_impl!(
+    Div, div;
+    (U4, U1), (U3, U3);
+    self: &'a UnitQuaternionBase<N, SA>, rhs: RotationBase<N, U3, SB>,
+    Output = OwnedUnitQuaternionBase<N, SA::Alloc> => U3, U3;
+    self / OwnedUnitQuaternionBase::<N, SA::Alloc>::from_rotation_matrix(&rhs);
+    'a);
+
+quaternion_op_impl!(
+    Div, div;
+    (U4, U1), (U3, U3);
+    self: UnitQuaternionBase<N, SA>, rhs: &'b RotationBase<N, U3, SB>,
+    Output = OwnedUnitQuaternionBase<N, SA::Alloc> => U3, U3;
+    self / OwnedUnitQuaternionBase::<N, SA::Alloc>::from_rotation_matrix(rhs);
+    'b);
+
+quaternion_op_impl!(
+    Div, div;
+    (U4, U1), (U3, U3);
+    self: UnitQuaternionBase<N, SA>, rhs: RotationBase<N, U3, SB>,
+    Output = OwnedUnitQuaternionBase<N, SA::Alloc> => U3, U3;
+    self / OwnedUnitQuaternionBase::<N, SA::Alloc>::from_rotation_matrix(&rhs);
+    );
+
+// RotationBase × UnitQuaternion
+quaternion_op_impl!(
+    Mul, mul;
+    (U3, U3), (U4, U1);
+    self: &'a RotationBase<N, U3, SA>, rhs: &'b UnitQuaternionBase<N, SB>,
+    Output = OwnedUnitQuaternionBase<N, SB::Alloc> => U3, U3;
+    // FIXME: can we avoid the conversion from a rotation matrix?
+    OwnedUnitQuaternionBase::<N, SB::Alloc>::from_rotation_matrix(self) * rhs;
+    'a, 'b);
+
+quaternion_op_impl!(
+    Mul, mul;
+    (U3, U3), (U4, U1);
+    self: &'a RotationBase<N, U3, SA>, rhs: UnitQuaternionBase<N, SB>,
+    Output = OwnedUnitQuaternionBase<N, SB::Alloc> => U3, U3;
+    OwnedUnitQuaternionBase::<N, SB::Alloc>::from_rotation_matrix(self) * rhs;
+    'a);
+
+quaternion_op_impl!(
+    Mul, mul;
+    (U3, U3), (U4, U1);
+    self: RotationBase<N, U3, SA>, rhs: &'b UnitQuaternionBase<N, SB>,
+    Output = OwnedUnitQuaternionBase<N, SB::Alloc> => U3, U3;
+    OwnedUnitQuaternionBase::<N, SB::Alloc>::from_rotation_matrix(&self) * rhs;
+    'b);
+
+quaternion_op_impl!(
+    Mul, mul;
+    (U3, U3), (U4, U1);
+    self: RotationBase<N, U3, SA>, rhs: UnitQuaternionBase<N, SB>,
+    Output = OwnedUnitQuaternionBase<N, SB::Alloc> => U3, U3;
+    OwnedUnitQuaternionBase::<N, SB::Alloc>::from_rotation_matrix(&self) * rhs;
+    );
+
+// RotationBase ÷ UnitQuaternion
+quaternion_op_impl!(
+    Div, div;
+    (U3, U3), (U4, U1);
+    self: &'a RotationBase<N, U3, SA>, rhs: &'b UnitQuaternionBase<N, SB>,
+    Output = OwnedUnitQuaternionBase<N, SB::Alloc> => U3, U3;
+    // FIXME: can we avoid the conversion from a rotation matrix?
+    OwnedUnitQuaternionBase::<N, SB::Alloc>::from_rotation_matrix(self) / rhs;
+    'a, 'b);
+
+quaternion_op_impl!(
+    Div, div;
+    (U3, U3), (U4, U1);
+    self: &'a RotationBase<N, U3, SA>, rhs: UnitQuaternionBase<N, SB>,
+    Output = OwnedUnitQuaternionBase<N, SB::Alloc> => U3, U3;
+    OwnedUnitQuaternionBase::<N, SB::Alloc>::from_rotation_matrix(self) / rhs;
+    'a);
+
+quaternion_op_impl!(
+    Div, div;
+    (U3, U3), (U4, U1);
+    self: RotationBase<N, U3, SA>, rhs: &'b UnitQuaternionBase<N, SB>,
+    Output = OwnedUnitQuaternionBase<N, SB::Alloc> => U3, U3;
+    OwnedUnitQuaternionBase::<N, SB::Alloc>::from_rotation_matrix(&self) / rhs;
+    'b);
+
+quaternion_op_impl!(
+    Div, div;
+    (U3, U3), (U4, U1);
+    self: RotationBase<N, U3, SA>, rhs: UnitQuaternionBase<N, SB>,
+    Output = OwnedUnitQuaternionBase<N, SB::Alloc> => U3, U3;
+    OwnedUnitQuaternionBase::<N, SB::Alloc>::from_rotation_matrix(&self) / rhs;
+    );
+
+// UnitQuaternion × Vector
+quaternion_op_impl!(
+    Mul, mul;
+    (U4, U1), (U3, U1);
+    self: &'a UnitQuaternionBase<N, SA>, rhs: &'b ColumnVector<N, U3, SB>,
+    Output = OwnedColumnVector<N, U3, SA::Alloc> => U3, U4;
+    {
+        let two: N = ::convert(2.0f64);
+        let t = self.as_ref().vector().cross(rhs) * two;
+        let cross = self.as_ref().vector().cross(&t);
+
+        t * self.as_ref().scalar() + cross + rhs
+    };
+    'a, 'b);
+
+quaternion_op_impl!(
+    Mul, mul;
+    (U4, U1), (U3, U1);
+    self: &'a UnitQuaternionBase<N, SA>, rhs: ColumnVector<N, U3, SB>,
+    Output = OwnedColumnVector<N, U3, SA::Alloc> => U3, U4;
+    self * &rhs;
+    'a);
+
+quaternion_op_impl!(
+    Mul, mul;
+    (U4, U1), (U3, U1);
+    self: UnitQuaternionBase<N, SA>, rhs: &'b ColumnVector<N, U3, SB>,
+    Output = OwnedColumnVector<N, U3, SA::Alloc> => U3, U4;
+    &self * rhs;
+    'b);
+
+quaternion_op_impl!(
+    Mul, mul;
+    (U4, U1), (U3, U1);
+    self: UnitQuaternionBase<N, SA>, rhs: ColumnVector<N, U3, SB>,
+    Output = OwnedColumnVector<N, U3, SA::Alloc> => U3, U4;
+    &self * &rhs;
+    );
+
+// UnitQuaternion × PointBase
+quaternion_op_impl!(
+    Mul, mul;
+    (U4, U1), (U3, U1);
+    self: &'a UnitQuaternionBase<N, SA>, rhs: &'b PointBase<N, U3, SB>,
+    Output = OwnedPoint<N, U3, SA::Alloc> => U3, U4;
+    PointBase::from_coordinates(self * &rhs.coords);
+    'a, 'b);
+
+quaternion_op_impl!(
+    Mul, mul;
+    (U4, U1), (U3, U1);
+    self: &'a UnitQuaternionBase<N, SA>, rhs: PointBase<N, U3, SB>,
+    Output = OwnedPoint<N, U3, SA::Alloc> => U3, U4;
+    PointBase::from_coordinates(self * rhs.coords);
+    'a);
+
+quaternion_op_impl!(
+    Mul, mul;
+    (U4, U1), (U3, U1);
+    self: UnitQuaternionBase<N, SA>, rhs: &'b PointBase<N, U3, SB>,
+    Output = OwnedPoint<N, U3, SA::Alloc> => U3, U4;
+    PointBase::from_coordinates(self * &rhs.coords);
+    'b);
+
+quaternion_op_impl!(
+    Mul, mul;
+    (U4, U1), (U3, U1);
+    self: UnitQuaternionBase<N, SA>, rhs: PointBase<N, U3, SB>,
+    Output = OwnedPoint<N, U3, SA::Alloc> => U3, U4;
+    PointBase::from_coordinates(self * rhs.coords);
+    );
+
+// UnitQuaternion × Unit<Vector>
+quaternion_op_impl!(
+    Mul, mul;
+    (U4, U1), (U3, U1);
+    self: &'a UnitQuaternionBase<N, SA>, rhs: &'b Unit<ColumnVector<N, U3, SB>>,
+    Output = Unit<OwnedColumnVector<N, U3, SA::Alloc>> => U3, U4;
+    Unit::new_unchecked(self * rhs.as_ref());
+    'a, 'b);
+
+quaternion_op_impl!(
+    Mul, mul;
+    (U4, U1), (U3, U1);
+    self: &'a UnitQuaternionBase<N, SA>, rhs: Unit<ColumnVector<N, U3, SB>>,
+    Output = Unit<OwnedColumnVector<N, U3, SA::Alloc>> => U3, U4;
+    Unit::new_unchecked(self * rhs.unwrap());
+    'a);
+
+quaternion_op_impl!(
+    Mul, mul;
+    (U4, U1), (U3, U1);
+    self: UnitQuaternionBase<N, SA>, rhs: &'b Unit<ColumnVector<N, U3, SB>>,
+    Output = Unit<OwnedColumnVector<N, U3, SA::Alloc>> => U3, U4;
+    Unit::new_unchecked(self * rhs.as_ref());
+    'b);
+
+quaternion_op_impl!(
+    Mul, mul;
+    (U4, U1), (U3, U1);
+    self: UnitQuaternionBase<N, SA>, rhs: Unit<ColumnVector<N, U3, SB>>,
+    Output = Unit<OwnedColumnVector<N, U3, SA::Alloc>> => U3, U4;
+    Unit::new_unchecked(self * rhs.unwrap());
+    );
+
+
+
+macro_rules! scalar_op_impl(
+    ($($Op: ident, $op: ident, $OpAssign: ident, $op_assign: ident);* $(;)*) => {$(
+        impl<N, S> $Op<N> for QuaternionBase<N, S>
+            where N: Real,
+                  S: Storage<N, U4, U1> {
+            type Output = OwnedQuaternionBase<N, S::Alloc>;
+
+            #[inline]
+            fn $op(self, n: N) -> Self::Output {
+                QuaternionBase::from_vector(self.coords.$op(n))
+            }
+        }
+
+        impl<'a, N, S> $Op<N> for &'a QuaternionBase<N, S>
+            where N: Real,
+                  S: Storage<N, U4, U1> {
+            type Output = OwnedQuaternionBase<N, S::Alloc>;
+
+            #[inline]
+            fn $op(self, n: N) -> Self::Output {
+                QuaternionBase::from_vector((&self.coords).$op(n))
+            }
+        }
+
+        impl<N, S> $OpAssign<N> for QuaternionBase<N, S>
+            where N: Real,
+                  S: StorageMut<N, U4, U1> {
+
+            #[inline]
+            fn $op_assign(&mut self, n: N) {
+                self.coords.$op_assign(n)
+            }
+        }
+    )*}
+);
+
+scalar_op_impl!(
+    Mul, mul, MulAssign, mul_assign;
+    Div, div, DivAssign, div_assign;
+);
+
+macro_rules! left_scalar_mul_impl(
+    ($($T: ty),* $(,)*) => {$(
+        impl<S> Mul<QuaternionBase<$T, S>> for $T
+            where S: Storage<$T, U4, U1> {
+            type Output = OwnedQuaternionBase<$T, S::Alloc>;
+
+            #[inline]
+            fn mul(self, right: QuaternionBase<$T, S>) -> Self::Output {
+                QuaternionBase::from_vector(self * right.coords)
+            }
+        }
+
+        impl<'b, S> Mul<&'b QuaternionBase<$T, S>> for $T
+            where S: Storage<$T, U4, U1> {
+            type Output = OwnedQuaternionBase<$T, S::Alloc>;
+
+            #[inline]
+            fn mul(self, right: &'b QuaternionBase<$T, S>) -> Self::Output {
+                QuaternionBase::from_vector(self * &right.coords)
+            }
+        }
+    )*}
+);
+
+left_scalar_mul_impl!(f32, f64);
+
+impl<N, S> Neg for QuaternionBase<N, S>
+    where N: Real,
+          S: Storage<N, U4, U1> {
+    type Output = OwnedQuaternionBase<N, S::Alloc>;
+
+    #[inline]
+    fn neg(self) -> Self::Output {
+        QuaternionBase::from_vector(-self.coords)
+    }
+}
+
+impl<'a, N, S> Neg for &'a QuaternionBase<N, S>
+    where N: Real,
+          S: Storage<N, U4, U1> {
+    type Output = OwnedQuaternionBase<N, S::Alloc>;
+
+    #[inline]
+    fn neg(self) -> Self::Output {
+        QuaternionBase::from_vector(-&self.coords)
+    }
+}
+
+macro_rules! quaternion_op_impl(
+    ($OpAssign: ident, $op_assign: ident;
+     ($LhsRDim: ident, $LhsCDim: ident), ($RhsRDim: ident, $RhsCDim: ident);
+     $lhs: ident: $Lhs: ty, $rhs: ident: $Rhs: ty $(=> $VDimA: ty, $VDimB: ty)*;
+     $action: expr; $($lives: tt),*) => {
+        impl<$($lives ,)* N, SA, SB> $OpAssign<$Rhs> for $Lhs
+            where N: Real,
+                  SA: StorageMut<N, $LhsRDim, $LhsCDim>,
+                  SB: Storage<N, $RhsRDim, $RhsCDim>,
+                  $(SA::Alloc: Allocator<N, $VDimA, U1> + Allocator<N, U4, U1>,
+                    //                                    ^^^^^^^^^^^^^^^^^^^^
+                    //                                    XXX: For some reasons, the compiler needs
+                    //                                    this bound to compile UnitQuat *= RotationBase.
+                    //                                    Though in theory this bound is already
+                    //                                    inherited from `SA: StorageMut`…
+                    SB::Alloc: Allocator<N, $VDimB, U1>)* {
+
+            #[inline]
+            fn $op_assign(&mut $lhs, $rhs: $Rhs) {
+                $action
+            }
+        }
+    }
+);
+
+// Quaternion += Quaternion
+quaternion_op_impl!(
+    AddAssign, add_assign;
+    (U4, U1), (U4, U1);
+    self: QuaternionBase<N, SA>, rhs: &'b QuaternionBase<N, SB>;
+    self.coords += &rhs.coords;
+    'b);
+
+quaternion_op_impl!(
+    AddAssign, add_assign;
+    (U4, U1), (U4, U1);
+    self: QuaternionBase<N, SA>, rhs: QuaternionBase<N, SB>;
+    self.coords += rhs.coords; );
+
+
+// Quaternion -= Quaternion
+quaternion_op_impl!(
+    SubAssign, sub_assign;
+    (U4, U1), (U4, U1);
+    self: QuaternionBase<N, SA>, rhs: &'b QuaternionBase<N, SB>;
+    self.coords -= &rhs.coords;
+    'b);
+
+quaternion_op_impl!(
+    SubAssign, sub_assign;
+    (U4, U1), (U4, U1);
+    self: QuaternionBase<N, SA>, rhs: QuaternionBase<N, SB>;
+    self.coords -= rhs.coords; );
+
+// Quaternion ×= Quaternion
+quaternion_op_impl!(
+    MulAssign, mul_assign;
+    (U4, U1), (U4, U1);
+    self: QuaternionBase<N, SA>, rhs: &'b QuaternionBase<N, SB>;
+    {
+        let res = &*self * rhs;
+        // FIXME: will this be optimized away?
+        self.coords.copy_from(&res.coords);
+    };
+    'b);
+
+quaternion_op_impl!(
+    MulAssign, mul_assign;
+    (U4, U1), (U4, U1);
+    self: QuaternionBase<N, SA>, rhs: QuaternionBase<N, SB>;
+    *self *= &rhs; );
+
+// UnitQuaternion ×= UnitQuaternion
+quaternion_op_impl!(
+    MulAssign, mul_assign;
+    (U4, U1), (U4, U1);
+    self: UnitQuaternionBase<N, SA>, rhs: &'b UnitQuaternionBase<N, SB>;
+    {
+        let res = &*self * rhs;
+        self.as_mut_unchecked().coords.copy_from(&res.as_ref().coords);
+    };
+    'b);
+
+quaternion_op_impl!(
+    MulAssign, mul_assign;
+    (U4, U1), (U4, U1);
+    self: UnitQuaternionBase<N, SA>, rhs: UnitQuaternionBase<N, SB>;
+    *self *= &rhs; );
+
+// UnitQuaternion ÷= UnitQuaternion
+quaternion_op_impl!(
+    DivAssign, div_assign;
+    (U4, U1), (U4, U1);
+    self: UnitQuaternionBase<N, SA>, rhs: &'b UnitQuaternionBase<N, SB>;
+    {
+        let res = &*self / rhs;
+        self.as_mut_unchecked().coords.copy_from(&res.as_ref().coords);
+    };
+    'b);
+
+quaternion_op_impl!(
+    DivAssign, div_assign;
+    (U4, U1), (U4, U1);
+    self: UnitQuaternionBase<N, SA>, rhs: UnitQuaternionBase<N, SB>;
+    *self /= &rhs; );
+
+// UnitQuaternion ×= RotationBase
+quaternion_op_impl!(
+    MulAssign, mul_assign;
+    (U4, U1), (U3, U3);
+    self: UnitQuaternionBase<N, SA>, rhs: &'b RotationBase<N, U3, SB> => U3, U3;
+    {
+        let res = &*self * rhs;
+        self.as_mut_unchecked().coords.copy_from(&res.as_ref().coords);
+    };
+    'b);
+
+quaternion_op_impl!(
+    MulAssign, mul_assign;
+    (U4, U1), (U3, U3);
+    self: UnitQuaternionBase<N, SA>, rhs: RotationBase<N, U3, SB> => U3, U3;
+    *self *= &rhs; );
+
+// UnitQuaternion ÷= RotationBase
+quaternion_op_impl!(
+    DivAssign, div_assign;
+    (U4, U1), (U3, U3);
+    self: UnitQuaternionBase<N, SA>, rhs: &'b RotationBase<N, U3, SB> => U3, U3;
+    {
+        let res = &*self / rhs;
+        self.as_mut_unchecked().coords.copy_from(&res.as_ref().coords);
+    };
+    'b);
+
+quaternion_op_impl!(
+    DivAssign, div_assign;
+    (U4, U1), (U3, U3);
+    self: UnitQuaternionBase<N, SA>, rhs: RotationBase<N, U3, SB> => U3, U3;
+    *self /= &rhs; );

src/geometry/rotation.rs

@@ -0,0 +1,174 @@
+use num::{Zero, One};
+use std::fmt;
+use approx::ApproxEq;
+
+use alga::general::Real;
+
+use core::{SquareMatrix, Scalar, OwnedSquareMatrix};
+use core::dimension::{DimName, DimNameSum, DimNameAdd, U1};
+use core::storage::{Storage, StorageMut};
+use core::allocator::Allocator;
+
+
+/// A rotation matrix with an owned storage.
+pub type OwnedRotation<N, D, A> = RotationBase<N, D, <A as Allocator<N, D, D>>::Buffer>;
+
+/// A rotation matrix.
+#[repr(C)]
+#[derive(Hash, Debug, Clone, Copy)]
+#[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
+pub struct RotationBase<N: Scalar, D: DimName, S> {
+    matrix: SquareMatrix<N, D, S>
+}
+
+impl<N: Scalar, D: DimName, S: Storage<N, D, D>> RotationBase<N, D, S>
+    where N: Scalar,
+          S: Storage<N, D, D> {
+    /// A reference to the underlying matrix representation of this rotation.
+    #[inline]
+    pub fn matrix(&self) -> &SquareMatrix<N, D, S> {
+        &self.matrix
+    }
+
+    /// A mutable reference to the underlying matrix representation of this rotation.
+    ///
+    /// This is unsafe because this allows the user to replace the matrix by another one that is
+    /// non-square, non-inversible, or non-orthonormal. If one of those properties is broken,
+    /// subsequent method calls may be UB.
+    #[inline]
+    pub unsafe fn matrix_mut(&mut self) -> &mut SquareMatrix<N, D, S> {
+        &mut self.matrix
+    }
+
+    /// Unwraps the underlying matrix.
+    #[inline]
+    pub fn unwrap(self) -> SquareMatrix<N, D, S> {
+        self.matrix
+    }
+
+    /// Converts this rotation into its equivalent homogeneous transformation matrix.
+    #[inline]
+    pub fn to_homogeneous(&self) -> OwnedSquareMatrix<N, DimNameSum<D, U1>, S::Alloc>
+        where N: Zero + One,
+              D: DimNameAdd<U1>,
+              S::Alloc: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1>> {
+        let mut res = OwnedSquareMatrix::<N, _, S::Alloc>::identity();
+        res.fixed_slice_mut::<D, D>(0, 0).copy_from(&self.matrix);
+
+        res
+    }
+}
+
+impl<N: Scalar, D: DimName, S: Storage<N, D, D>> RotationBase<N, D, S> {
+    /// Creates a new rotation from the given square matrix.
+    ///
+    /// The matrix squareness is checked but not its orthonormality.
+    #[inline]
+    pub fn from_matrix_unchecked(matrix: SquareMatrix<N, D, S>) -> RotationBase<N, D, S> {
+        assert!(matrix.is_square(), "Unable to create a rotation from a non-square matrix.");
+
+        RotationBase {
+            matrix: matrix
+        }
+    }
+
+    /// Transposes `self`.
+    #[inline]
+    pub fn transpose(&self) -> OwnedRotation<N, D, S::Alloc> {
+        RotationBase::from_matrix_unchecked(self.matrix.transpose())
+    }
+
+    /// Inverts `self`.
+    #[inline]
+    pub fn inverse(&self) -> OwnedRotation<N, D, S::Alloc> {
+        self.transpose()
+    }
+}
+
+
+impl<N: Scalar, D: DimName, S: StorageMut<N, D, D>> RotationBase<N, D, S> {
+    /// Transposes `self` in-place.
+    #[inline]
+    pub fn transpose_mut(&mut self) {
+        self.matrix.transpose_mut()
+    }
+
+    /// Inverts `self` in-place.
+    #[inline]
+    pub fn inverse_mut(&mut self) {
+        self.transpose_mut()
+    }
+}
+
+impl<N: Scalar + Eq, D: DimName, S: Storage<N, D, D>> Eq for RotationBase<N, D, S> { }
+
+impl<N: Scalar + PartialEq, D: DimName, S: Storage<N, D, D>> PartialEq for RotationBase<N, D, S> {
+    #[inline]
+    fn eq(&self, right: &RotationBase<N, D, S>) -> bool {
+        self.matrix == right.matrix
+    }
+}
+
+impl<N, D: DimName, S> ApproxEq for RotationBase<N, D, S>
+    where N: Scalar + ApproxEq,
+          S: Storage<N, D, D>,
+          N::Epsilon: Copy {
+    type Epsilon = N::Epsilon;
+
+    #[inline]
+    fn default_epsilon() -> Self::Epsilon {
+        N::default_epsilon()
+    }
+
+    #[inline]
+    fn default_max_relative() -> Self::Epsilon {
+        N::default_max_relative()
+    }
+
+    #[inline]
+    fn default_max_ulps() -> u32 {
+        N::default_max_ulps()
+    }
+
+    #[inline]
+    fn relative_eq(&self, other: &Self, epsilon: Self::Epsilon, max_relative: Self::Epsilon) -> bool {
+        self.matrix.relative_eq(&other.matrix, epsilon, max_relative)
+    }
+
+    #[inline]
+    fn ulps_eq(&self, other: &Self, epsilon: Self::Epsilon, max_ulps: u32) -> bool {
+        self.matrix.ulps_eq(&other.matrix, epsilon, max_ulps)
+    }
+}
+
+/*
+ *
+ * Display
+ *
+ */
+impl<N, D: DimName, S> fmt::Display for RotationBase<N, D, S>
+    where N: Real + fmt::Display,
+          S: Storage<N, D, D>,
+          S::Alloc: Allocator<usize, D, D> {
+    fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
+        let precision = f.precision().unwrap_or(3);
+
+        try!(writeln!(f, "RotationBase matrix {{"));
+        try!(write!(f, "{:.*}", precision, self.matrix));
+        writeln!(f, "}}")
+    }
+}
+
+//          //         /*
+//          //          *
+//          //          * Absolute
+//          //          *
+//          //          */
+//          //         impl<N: Absolute> Absolute for $t<N> {
+//          //             type AbsoluteValue = $submatrix<N::AbsoluteValue>;
+//          //
+//          //             #[inline]
+//          //             fn abs(m: &$t<N>) -> $submatrix<N::AbsoluteValue> {
+//          //                 Absolute::abs(&m.submatrix)
+//          //             }
+//          //         }

src/geometry/rotation_alga.rs

@@ -0,0 +1,301 @@
+use alga::general::{AbstractMagma, AbstractGroup, AbstractLoop, AbstractMonoid, AbstractQuasigroup,
+                    AbstractSemigroup, Real, Inverse, Multiplicative, Identity, Id};
+use alga::linear::{Transformation, Similarity, AffineTransformation, Isometry, DirectIsometry,
+                   OrthogonalTransformation, ProjectiveTransformation, Rotation};
+
+use core::ColumnVector;
+use core::dimension::{DimName, U1};
+use core::storage::OwnedStorage;
+use core::allocator::OwnedAllocator;
+
+use geometry::{RotationBase, PointBase};
+
+
+
+/*
+ *
+ * Algebraic structures.
+ *
+ */
+impl<N, D: DimName, S> Identity<Multiplicative> for RotationBase<N, D, S>
+    where N: Real,
+          S: OwnedStorage<N, D, D>,
+          S::Alloc: OwnedAllocator<N, D, D, S> {
+    #[inline]
+    fn identity() -> Self {
+        Self::identity()
+    }
+}
+
+impl<N, D: DimName, S> Inverse<Multiplicative> for RotationBase<N, D, S>
+    where N: Real,
+          S: OwnedStorage<N, D, D>,
+          S::Alloc: OwnedAllocator<N, D, D, S> {
+    #[inline]
+    fn inverse(&self) -> Self {
+        self.transpose()
+    }
+
+    #[inline]
+    fn inverse_mut(&mut self) {
+        self.transpose_mut()
+    }
+}
+
+impl<N, D: DimName, S> AbstractMagma<Multiplicative> for RotationBase<N, D, S>
+    where N: Real,
+          S: OwnedStorage<N, D, D>,
+          S::Alloc: OwnedAllocator<N, D, D, S> {
+    #[inline]
+    fn operate(&self, rhs: &Self) -> Self {
+        self * rhs
+    }
+}
+
+macro_rules! impl_multiplicative_structures(
+    ($($marker: ident<$operator: ident>),* $(,)*) => {$(
+        impl<N, D: DimName, S> $marker<$operator> for RotationBase<N, D, S>
+            where N: Real,
+                  S: OwnedStorage<N, D, D>,
+                  S::Alloc: OwnedAllocator<N, D, D, S> { }
+    )*}
+);
+
+impl_multiplicative_structures!(
+    AbstractSemigroup<Multiplicative>,
+    AbstractMonoid<Multiplicative>,
+    AbstractQuasigroup<Multiplicative>,
+    AbstractLoop<Multiplicative>,
+    AbstractGroup<Multiplicative>
+);
+
+/*
+ *
+ * Transformation groups.
+ *
+ */
+impl<N, D: DimName, SA, SB> Transformation<PointBase<N, D, SB>> for RotationBase<N, D, SA>
+    where N:  Real,
+          SA: OwnedStorage<N, D, D>,
+          SB: OwnedStorage<N, D, U1, Alloc = SA::Alloc>,
+          SA::Alloc: OwnedAllocator<N, D, D, SA>,
+          SB::Alloc: OwnedAllocator<N, D, U1, SB> {
+    #[inline]
+    fn transform_point(&self, pt: &PointBase<N, D, SB>) -> PointBase<N, D, SB> {
+        self * pt
+    }
+
+    #[inline]
+    fn transform_vector(&self, v: &ColumnVector<N, D, SB>) -> ColumnVector<N, D, SB> {
+        self * v
+    }
+}
+
+impl<N, D: DimName, SA, SB> ProjectiveTransformation<PointBase<N, D, SB>> for RotationBase<N, D, SA>
+    where N:  Real,
+          SA: OwnedStorage<N, D, D>,
+          SB: OwnedStorage<N, D, U1, Alloc = SA::Alloc>,
+          SA::Alloc: OwnedAllocator<N, D, D, SA>,
+          SB::Alloc: OwnedAllocator<N, D, U1, SB> {
+    #[inline]
+    fn inverse_transform_point(&self, pt: &PointBase<N, D, SB>) -> PointBase<N, D, SB> {
+        PointBase::from_coordinates(self.inverse_transform_vector(&pt.coords))
+    }
+
+    #[inline]
+    fn inverse_transform_vector(&self, v: &ColumnVector<N, D, SB>) -> ColumnVector<N, D, SB> {
+        self.matrix().tr_mul(v)
+    }
+}
+
+impl<N, D: DimName, SA, SB> AffineTransformation<PointBase<N, D, SB>> for RotationBase<N, D, SA>
+    where N:  Real,
+          SA: OwnedStorage<N, D, D>,
+          SB: OwnedStorage<N, D, U1, Alloc = SA::Alloc>,
+          SA::Alloc: OwnedAllocator<N, D, D, SA>,
+          SB::Alloc: OwnedAllocator<N, D, U1, SB> {
+    type Rotation          = Self;
+    type NonUniformScaling = Id;
+    type Translation       = Id;
+
+    #[inline]
+    fn decompose(&self) -> (Id, Self, Id, Self) {
+        (Id::new(), self.clone(), Id::new(), Self::identity())
+    }
+
+    #[inline]
+    fn append_translation(&self, _: &Self::Translation) -> Self {
+        self.clone()
+    }
+
+    #[inline]
+    fn prepend_translation(&self, _: &Self::Translation) -> Self {
+        self.clone()
+    }
+
+    #[inline]
+    fn append_rotation(&self, r: &Self::Rotation) -> Self {
+        r * self
+    }
+
+    #[inline]
+    fn prepend_rotation(&self, r: &Self::Rotation) -> Self {
+        self * r
+    }
+
+    #[inline]
+    fn append_scaling(&self, _: &Self::NonUniformScaling) -> Self {
+        self.clone()
+    }
+
+    #[inline]
+    fn prepend_scaling(&self, _: &Self::NonUniformScaling) -> Self {
+        self.clone()
+    }
+}
+
+
+impl<N, D: DimName, SA, SB> Similarity<PointBase<N, D, SB>> for RotationBase<N, D, SA>
+    where N:  Real,
+          SA: OwnedStorage<N, D, D>,
+          SB: OwnedStorage<N, D, U1, Alloc = SA::Alloc>,
+          SA::Alloc: OwnedAllocator<N, D, D, SA>,
+          SB::Alloc: OwnedAllocator<N, D, U1, SB> {
+    type Scaling  = Id;
+
+    #[inline]
+    fn translation(&self) -> Id {
+        Id::new()
+    }
+
+    #[inline]
+    fn rotation(&self) -> Self {
+        self.clone()
+    }
+
+    #[inline]
+    fn scaling(&self) -> Id {
+        Id::new()
+    }
+}
+
+macro_rules! marker_impl(
+    ($($Trait: ident),*) => {$(
+        impl<N, D: DimName, SA, SB> $Trait<PointBase<N, D, SB>> for RotationBase<N, D, SA>
+        where N:  Real,
+              SA: OwnedStorage<N, D, D>,
+              SB: OwnedStorage<N, D, U1, Alloc = SA::Alloc>,
+              SA::Alloc: OwnedAllocator<N, D, D, SA>,
+              SB::Alloc: OwnedAllocator<N, D, U1, SB> { }
+    )*}
+);
+
+marker_impl!(Isometry, DirectIsometry, OrthogonalTransformation);
+
+
+/// Subgroups of the n-dimensional rotation group `SO(n)`.
+impl<N, D: DimName, SA, SB> Rotation<PointBase<N, D, SB>> for RotationBase<N, D, SA>
+    where N:  Real,
+          SA: OwnedStorage<N, D, D>,
+          SB: OwnedStorage<N, D, U1, Alloc = SA::Alloc>,
+          SA::Alloc: OwnedAllocator<N, D, D, SA>,
+          SB::Alloc: OwnedAllocator<N, D, U1, SB> {
+    #[inline]
+    fn powf(&self, _: N) -> Option<Self> {
+        // XXX: Add the general case.
+        // XXX: Use specialization for 2D and 3D.
+        unimplemented!()
+    }
+
+    #[inline]
+    fn rotation_between(_: &ColumnVector<N, D, SB>, _: &ColumnVector<N, D, SB>) -> Option<Self> {
+        // XXX: Add the general case.
+        // XXX: Use specialization for 2D and 3D.
+        unimplemented!()
+    }
+
+    #[inline]
+    fn scaled_rotation_between(_: &ColumnVector<N, D, SB>, _: &ColumnVector<N, D, SB>, _: N) -> Option<Self> {
+        // XXX: Add the general case.
+        // XXX: Use specialization for 2D and 3D.
+        unimplemented!()
+    }
+}
+
+/*
+impl<N: Real> Matrix for RotationBase<N> {
+    type Field     = N;
+    type Row       = Matrix<N>;
+    type Column    = Matrix<N>;
+    type Transpose = Self;
+
+    #[inline]
+    fn nrows(&self) -> usize {
+        self.submatrix.nrows()
+    }
+
+    #[inline]
+    fn ncolumns(&self) -> usize {
+        self.submatrix.ncolumns()
+    }
+
+    #[inline]
+    fn row(&self, i: usize) -> Self::Row {
+        self.submatrix.row(i)
+    }
+
+    #[inline]
+    fn column(&self, i: usize) -> Self::Column {
+        self.submatrix.column(i)
+    }
+
+    #[inline]
+    fn get(&self, i: usize, j: usize) -> Self::Field {
+        self.submatrix[(i, j)]
+    }
+
+    #[inline]
+    unsafe fn get_unchecked(&self, i: usize, j: usize) -> Self::Field {
+        self.submatrix.at_fast(i, j)
+    }
+
+    #[inline]
+    fn transpose(&self) -> Self::Transpose {
+        RotationBase::from_matrix_unchecked(self.submatrix.transpose())
+    }
+}
+
+impl<N: Real> SquareMatrix for RotationBase<N> {
+    type Vector = Matrix<N>;
+
+    #[inline]
+    fn diagonal(&self) -> Self::Coordinates {
+        self.submatrix.diagonal()
+    }
+
+    #[inline]
+    fn determinant(&self) -> Self::Field {
+        ::one()
+    }
+
+    #[inline]
+    fn try_inverse(&self) -> Option<Self> {
+        Some(::transpose(self))
+    }
+
+    #[inline]
+    fn try_inverse_mut(&mut self) -> bool {
+        self.transpose_mut();
+        true
+    }
+
+    #[inline]
+    fn transpose_mut(&mut self) {
+        self.submatrix.transpose_mut()
+    }
+}
+
+impl<N: Real> InversibleSquareMatrix for RotationBase<N> { }
+*/
+
+

src/geometry/rotation_alias.rs

@@ -0,0 +1,13 @@
+use core::MatrixArray;
+use core::dimension::{U2, U3};
+
+use geometry::RotationBase;
+
+/// A D-dimensional rotation matrix.
+pub type Rotation<N, D> = RotationBase<N, D, MatrixArray<N, D, D>>;
+
+/// A 2-dimensional rotation matrix.
+pub type Rotation2<N> = Rotation<N, U2>;
+
+/// A 3-dimensional rotation matrix.
+pub type Rotation3<N> = Rotation<N, U3>;

src/geometry/rotation_construction.rs

@@ -0,0 +1,31 @@
+use num::{Zero, One};
+
+use alga::general::{ClosedAdd, ClosedMul};
+
+use core::{SquareMatrix, Scalar};
+use core::dimension::DimName;
+use core::storage::OwnedStorage;
+use core::allocator::OwnedAllocator;
+
+use geometry::RotationBase;
+
+impl<N, D: DimName, S> RotationBase<N, D, S>
+    where N: Scalar + Zero + One,
+          S: OwnedStorage<N, D, D>,
+          S::Alloc: OwnedAllocator<N, D, D, S> {
+    /// Creates a new square identity rotation of the given `dimension`.
+    #[inline]
+    pub fn identity() -> RotationBase<N, D, S> {
+        Self::from_matrix_unchecked(SquareMatrix::<N, D, S>::identity())
+    }
+}
+
+impl<N, D: DimName, S> One for RotationBase<N, D, S>
+    where N: Scalar + Zero + One + ClosedAdd + ClosedMul,
+          S: OwnedStorage<N, D, D>,
+          S::Alloc: OwnedAllocator<N, D, D, S> {
+    #[inline]
+    fn one() -> Self {
+        Self::identity()
+    }
+}

src/geometry/rotation_conversion.rs

@@ -0,0 +1,195 @@
+use num::Zero;
+
+use alga::general::{Real, SubsetOf, SupersetOf};
+use alga::linear::Rotation as AlgaRotation;
+
+use core::{Matrix, SquareMatrix};
+use core::dimension::{DimName, DimNameSum, DimNameAdd, U1, U3, U4};
+use core::storage::OwnedStorage;
+use core::allocator::{OwnedAllocator, Allocator};
+
+use geometry::{PointBase, TranslationBase, RotationBase, UnitQuaternionBase, OwnedUnitQuaternionBase, IsometryBase,
+               SimilarityBase, TransformBase, SuperTCategoryOf, TAffine};
+
+/*
+ * This file provides the following conversions:
+ * =============================================
+ *
+ * RotationBase  -> RotationBase
+ * Rotation3 -> UnitQuaternion
+ * RotationBase  -> IsometryBase
+ * RotationBase  -> SimilarityBase
+ * RotationBase  -> TransformBase
+ * RotationBase  -> Matrix (homogeneous)
+ */
+
+
+impl<N1, N2, D: DimName, SA, SB> SubsetOf<RotationBase<N2, D, SB>> for RotationBase<N1, D, SA>
+    where N1: Real,
+          N2: Real + SupersetOf<N1>,
+          SA: OwnedStorage<N1, D, D>,
+          SB: OwnedStorage<N2, D, D>,
+          SA::Alloc: OwnedAllocator<N1, D, D, SA>,
+          SB::Alloc: OwnedAllocator<N2, D, D, SB> {
+    #[inline]
+    fn to_superset(&self) -> RotationBase<N2, D, SB> {
+        RotationBase::from_matrix_unchecked(self.matrix().to_superset())
+    }
+
+    #[inline]
+    fn is_in_subset(rot: &RotationBase<N2, D, SB>) -> bool {
+        ::is_convertible::<_, Matrix<N1, D, D, SA>>(rot.matrix())
+    }
+
+    #[inline]
+    unsafe fn from_superset_unchecked(rot: &RotationBase<N2, D, SB>) -> Self {
+        RotationBase::from_matrix_unchecked(rot.matrix().to_subset_unchecked())
+    }
+}
+
+
+impl<N1, N2, SA, SB> SubsetOf<UnitQuaternionBase<N2, SB>> for RotationBase<N1, U3, SA>
+    where N1: Real,
+          N2: Real + SupersetOf<N1>,
+          SA: OwnedStorage<N1, U3, U3>,
+          SB: OwnedStorage<N2, U4, U1>,
+          SA::Alloc: OwnedAllocator<N1, U3, U3, SA> +
+                     Allocator<N1, U4, U1> +
+                     Allocator<N1, U3, U1>,
+          SB::Alloc: OwnedAllocator<N2, U4, U1, SB> +
+                     Allocator<N2, U3, U3> {
+    #[inline]
+    fn to_superset(&self) -> UnitQuaternionBase<N2, SB> {
+        let q = OwnedUnitQuaternionBase::<N1, SA::Alloc>::from_rotation_matrix(self);
+        q.to_superset()
+    }
+
+    #[inline]
+    fn is_in_subset(q: &UnitQuaternionBase<N2, SB>) -> bool {
+        ::is_convertible::<_, OwnedUnitQuaternionBase<N1, SA::Alloc>>(q)
+    }
+
+    #[inline]
+    unsafe fn from_superset_unchecked(q: &UnitQuaternionBase<N2, SB>) -> Self {
+        let q: OwnedUnitQuaternionBase<N1, SA::Alloc> = ::convert_ref_unchecked(q);
+        q.to_rotation_matrix()
+    }
+}
+
+
+impl<N1, N2, D: DimName, SA, SB, R> SubsetOf<IsometryBase<N2, D, SB, R>> for RotationBase<N1, D, SA>
+    where N1: Real,
+          N2: Real + SupersetOf<N1>,
+          SA: OwnedStorage<N1, D, D>,
+          SB: OwnedStorage<N2, D, U1>,
+          R:  AlgaRotation<PointBase<N2, D, SB>> + SupersetOf<RotationBase<N1, D, SA>>,
+          SA::Alloc: OwnedAllocator<N1, D, D, SA>,
+          SB::Alloc: OwnedAllocator<N2, D, U1, SB> {
+    #[inline]
+    fn to_superset(&self) -> IsometryBase<N2, D, SB, R> {
+        IsometryBase::from_parts(TranslationBase::identity(), ::convert_ref(self))
+    }
+
+    #[inline]
+    fn is_in_subset(iso: &IsometryBase<N2, D, SB, R>) -> bool {
+        iso.translation.vector.is_zero()
+    }
+
+    #[inline]
+    unsafe fn from_superset_unchecked(iso: &IsometryBase<N2, D, SB, R>) -> Self {
+        ::convert_ref_unchecked(&iso.rotation)
+    }
+}
+
+
+impl<N1, N2, D: DimName, SA, SB, R> SubsetOf<SimilarityBase<N2, D, SB, R>> for RotationBase<N1, D, SA>
+    where N1: Real,
+          N2: Real + SupersetOf<N1>,
+          SA: OwnedStorage<N1, D, D>,
+          SB: OwnedStorage<N2, D, U1>,
+          R:  AlgaRotation<PointBase<N2, D, SB>> + SupersetOf<RotationBase<N1, D, SA>>,
+          SA::Alloc: OwnedAllocator<N1, D, D, SA>,
+          SB::Alloc: OwnedAllocator<N2, D, U1, SB> {
+    #[inline]
+    fn to_superset(&self) -> SimilarityBase<N2, D, SB, R> {
+        SimilarityBase::from_parts(TranslationBase::identity(), ::convert_ref(self), N2::one())
+    }
+
+    #[inline]
+    fn is_in_subset(sim: &SimilarityBase<N2, D, SB, R>) -> bool {
+        sim.isometry.translation.vector.is_zero() &&
+        sim.scaling() == N2::one()
+    }
+
+    #[inline]
+    unsafe fn from_superset_unchecked(sim: &SimilarityBase<N2, D, SB, R>) -> Self {
+        ::convert_ref_unchecked(&sim.isometry.rotation)
+    }
+}
+
+
+impl<N1, N2, D, SA, SB, C> SubsetOf<TransformBase<N2, D, SB, C>> for RotationBase<N1, D, SA>
+    where N1: Real,
+          N2: Real + SupersetOf<N1>,
+          SA: OwnedStorage<N1, D, D>,
+          SB: OwnedStorage<N2, DimNameSum<D, U1>, DimNameSum<D, U1>>,
+          C:  SuperTCategoryOf<TAffine>,
+          D:  DimNameAdd<U1>,
+          SA::Alloc: OwnedAllocator<N1, D, D, SA> +
+                     Allocator<N1, DimNameSum<D, U1>, DimNameSum<D, U1>>,
+          SB::Alloc: OwnedAllocator<N2, DimNameSum<D, U1>, DimNameSum<D, U1>, SB> +
+                     Allocator<N2, D, D> +
+                     Allocator<N2, U1, D> {
+    #[inline]
+    fn to_superset(&self) -> TransformBase<N2, D, SB, C> {
+        TransformBase::from_matrix_unchecked(self.to_homogeneous().to_superset())
+    }
+
+    #[inline]
+    fn is_in_subset(t: &TransformBase<N2, D, SB, C>) -> bool {
+        <Self as SubsetOf<_>>::is_in_subset(t.matrix())
+    }
+
+    #[inline]
+    unsafe fn from_superset_unchecked(t: &TransformBase<N2, D, SB, C>) -> Self {
+        Self::from_superset_unchecked(t.matrix())
+    }
+}
+
+
+impl<N1, N2, D, SA, SB> SubsetOf<SquareMatrix<N2, DimNameSum<D, U1>, SB>> for RotationBase<N1, D, SA>
+    where N1: Real,
+          N2: Real + SupersetOf<N1>,
+          SA: OwnedStorage<N1, D, D>,
+          SB: OwnedStorage<N2, DimNameSum<D, U1>, DimNameSum<D, U1>>,
+          D:  DimNameAdd<U1>,
+          SA::Alloc: OwnedAllocator<N1, D, D, SA> +
+                     Allocator<N1, DimNameSum<D, U1>, DimNameSum<D, U1>>,
+          SB::Alloc: OwnedAllocator<N2, DimNameSum<D, U1>, DimNameSum<D, U1>, SB> +
+                     Allocator<N2, D, D> +
+                     Allocator<N2, U1, D> {
+    #[inline]
+    fn to_superset(&self) -> SquareMatrix<N2, DimNameSum<D, U1>, SB> {
+        self.to_homogeneous().to_superset()
+    }
+
+    #[inline]
+    fn is_in_subset(m: &SquareMatrix<N2, DimNameSum<D, U1>, SB>) -> bool {
+        let rot    = m.fixed_slice::<D, D>(0, 0);
+        let bottom = m.fixed_slice::<U1, D>(D::dim(), 0);
+
+        // Scalar types agree.
+        m.iter().all(|e| SupersetOf::<N1>::is_in_subset(e)) &&
+        // The block part is a rotation.
+        rot.is_special_orthogonal(N2::default_epsilon() * ::convert(100.0)) &&
+        // The bottom row is (0, 0, ..., 1)
+        bottom.iter().all(|e| e.is_zero()) &&
+        m[(D::dim(), D::dim())] == N2::one()
+    }
+
+    #[inline]
+    unsafe fn from_superset_unchecked(m: &SquareMatrix<N2, DimNameSum<D, U1>, SB>) -> Self {
+        let r = m.fixed_slice::<D, D>(0, 0);
+        Self::from_matrix_unchecked(::convert_unchecked(r.into_owned()))
+    }
+}

src/geometry/rotation_ops.rs

@@ -0,0 +1,162 @@
+/*
+ *
+ * This provides the following operator overladings:
+ *
+ * Index<(usize, usize)>
+ *
+ * RotationBase × RotationBase
+ * RotationBase ÷ RotationBase
+ * RotationBase × Matrix
+ * Matrix       × RotationBase
+ * Matrix       ÷ RotationBase
+ * RotationBase × PointBase
+ *
+ *
+ * RotationBase ×= RotationBase
+ * Matrix       ×= RotationBase
+ */
+
+
+use std::ops::{Mul, MulAssign, Div, DivAssign, Index};
+use num::Zero;
+
+use alga::general::{ClosedMul, ClosedAdd};
+
+use core::{Scalar, Matrix, MatrixMul};
+use core::dimension::{Dim, DimName, U1};
+use core::constraint::{ShapeConstraint, AreMultipliable};
+use core::storage::{OwnedStorage, Storage};
+use core::allocator::{OwnedAllocator, Allocator};
+
+use geometry::{PointBase, PointMul, RotationBase, OwnedRotation};
+
+impl<N: Scalar, D: DimName, S: Storage<N, D, D>> Index<(usize, usize)> for RotationBase<N, D, S> {
+    type Output = N;
+
+    #[inline]
+    fn index(&self, row_col: (usize, usize)) -> &N {
+        self.matrix().index(row_col)
+    }
+}
+
+// RotationBase × RotationBase
+md_impl_all!(
+    Mul, mul;
+    (D, D), (D, D) for D: DimName;
+    self: RotationBase<N, D, SA>, right: RotationBase<N, D, SB>, Output = OwnedRotation<N, D, SA::Alloc>;
+    [val val] => RotationBase::from_matrix_unchecked(self.unwrap() * right.unwrap());
+    [ref val] => RotationBase::from_matrix_unchecked(self.matrix() * right.unwrap());
+    [val ref] => RotationBase::from_matrix_unchecked(self.unwrap() * right.matrix());
+    [ref ref] => RotationBase::from_matrix_unchecked(self.matrix() * right.matrix());
+);
+
+// RotationBase ÷ RotationBase
+// FIXME: instead of calling inverse explicitely, could we just add a `mul_tr` or `mul_inv` method?
+md_impl_all!(
+    Div, div;
+    (D, D), (D, D) for D: DimName;
+    self: RotationBase<N, D, SA>, right: RotationBase<N, D, SB>, Output = OwnedRotation<N, D, SA::Alloc>;
+    [val val] => self * right.inverse();
+    [ref val] => self * right.inverse();
+    [val ref] => self * right.inverse();
+    [ref ref] => self * right.inverse();
+);
+
+// RotationBase × Matrix
+md_impl_all!(
+    Mul, mul;
+    (D1, D1), (R2, C2) for D1: DimName, R2: Dim, C2: Dim
+    where SA::Alloc: Allocator<N, D1, C2>
+    where ShapeConstraint: AreMultipliable<D1, D1, R2, C2>;
+    self: RotationBase<N, D1, SA>, right: Matrix<N, R2, C2, SB>, Output = MatrixMul<N, D1, D1, C2 , SA>;
+    [val val] => self.unwrap() * right;
+    [ref val] => self.matrix() * right;
+    [val ref] => self.unwrap() * right;
+    [ref ref] => self.matrix() * right;
+);
+
+// Matrix × RotationBase
+md_impl_all!(
+    Mul, mul;
+    (R1, C1), (D2, D2) for R1: Dim, C1: Dim, D2: DimName
+    where SA::Alloc: Allocator<N, R1, D2>
+    where ShapeConstraint: AreMultipliable<R1, C1, D2, D2>;
+    self: Matrix<N, R1, C1, SA>, right: RotationBase<N, D2, SB>, Output = MatrixMul<N, R1, C1, D2, SA>;
+    [val val] => self * right.unwrap();
+    [ref val] => self * right.unwrap();
+    [val ref] => self * right.matrix();
+    [ref ref] => self * right.matrix();
+);
+
+// Matrix ÷ RotationBase
+md_impl_all!(
+    Div, div;
+    (R1, C1), (D2, D2) for R1: Dim, C1: Dim, D2: DimName
+    where SA::Alloc: Allocator<N, R1, D2>
+    where ShapeConstraint: AreMultipliable<R1, C1, D2, D2>;
+    self: Matrix<N, R1, C1, SA>, right: RotationBase<N, D2, SB>, Output = MatrixMul<N, R1, C1, D2, SA>;
+    [val val] => self * right.inverse();
+    [ref val] => self * right.inverse();
+    [val ref] => self * right.inverse();
+    [ref ref] => self * right.inverse();
+);
+
+
+// RotationBase × PointBase
+// FIXME: we don't handle properly non-zero origins here. Do we want this to be the intended
+// behavior?
+md_impl_all!(
+    Mul, mul;
+    (D, D), (D, U1) for D: DimName
+    where SA::Alloc: Allocator<N, D, U1>
+    where ShapeConstraint: AreMultipliable<D, D, D, U1>;
+    self: RotationBase<N, D, SA>, right: PointBase<N, D, SB>, Output = PointMul<N, D, D, SA>;
+    [val val] => self.unwrap() * right;
+    [ref val] => self.matrix() * right;
+    [val ref] => self.unwrap() * right;
+    [ref ref] => self.matrix() * right;
+);
+
+
+// RotationBase ×= RotationBase
+// FIXME: try not to call `inverse()` explicitly.
+
+md_assign_impl_all!(
+    MulAssign, mul_assign;
+    (D, D), (D, D) for D: DimName;
+    self: RotationBase<N, D, SA>, right: RotationBase<N, D, SB>;
+    [val] => unsafe { self.matrix_mut().mul_assign(right.unwrap()) };
+    [ref] => unsafe { self.matrix_mut().mul_assign(right.matrix()) };
+);
+
+
+md_assign_impl_all!(
+    DivAssign, div_assign;
+    (D, D), (D, D) for D: DimName;
+    self: RotationBase<N, D, SA>, right: RotationBase<N, D, SB>;
+    [val] => unsafe { self.matrix_mut().mul_assign(right.inverse().unwrap()) };
+    [ref] => unsafe { self.matrix_mut().mul_assign(right.inverse().matrix()) };
+);
+
+// Matrix *= RotationBase
+// FIXME: try not to call `inverse()` explicitly.
+// FIXME: this shares the same limitations as for the current impl. of MulAssign for matrices.
+// (In particular the number of matrix column must be equal to the number of rotation columns,
+// i.e., equal to the rotation dimension.
+
+md_assign_impl_all!(
+    MulAssign, mul_assign;
+    (R1, C1), (C1, C1) for R1: DimName, C1: DimName;
+    self: Matrix<N, R1, C1, SA>, right: RotationBase<N, C1, SB>;
+    [val] => self.mul_assign(right.unwrap());
+    [ref] => self.mul_assign(right.matrix());
+);
+
+
+md_assign_impl_all!(
+    DivAssign, div_assign;
+    (R1, C1), (C1, C1) for R1: DimName, C1: DimName;
+    self: Matrix<N, R1, C1, SA>, right: RotationBase<N, C1, SB>;
+    [val] => self.mul_assign(right.inverse().unwrap());
+    [ref] => self.mul_assign(right.inverse().matrix());
+);

src/geometry/rotation_specialization.rs

@@ -0,0 +1,381 @@
+#[cfg(feature = "arbitrary")]
+use quickcheck::{Arbitrary, Gen};
+
+use std::ops::Neg;
+use num::Zero;
+use rand::{Rand, Rng};
+use alga::general::Real;
+
+use core::{Unit, ColumnVector, SquareMatrix, OwnedSquareMatrix, OwnedColumnVector, Vector3};
+use core::dimension::{U1, U2, U3};
+use core::storage::{Storage, OwnedStorage};
+use core::allocator::{Allocator, OwnedAllocator};
+
+use geometry::{RotationBase, OwnedRotation, UnitComplex};
+
+
+/*
+ *
+ * 2D RotationBase matrix.
+ *
+ */
+impl<N, S> RotationBase<N, U2, S>
+where N: Real,
+      S: OwnedStorage<N, U2, U2>,
+      S::Alloc: OwnedAllocator<N, U2, U2, S> {
+    /// Builds a 2 dimensional rotation matrix from an angle in radian.
+    pub fn new(angle: N) -> Self {
+        let (sia, coa) = angle.sin_cos();
+        Self::from_matrix_unchecked(SquareMatrix::<N, U2, S>::new(coa, -sia, sia, coa))
+    }
+
+    /// Builds a 2 dimensional rotation matrix from an angle in radian wrapped in a 1-dimensional vector.
+    ///
+    /// Equivalent to `Self::new(axisangle[0])`.
+    #[inline]
+    pub fn from_scaled_axis<SB: Storage<N, U1, U1>>(axisangle: ColumnVector<N, U1, SB>) -> Self {
+        Self::new(axisangle[0])
+    }
+
+    /// The rotation matrix required to align `a` and `b` but with its angl.
+    ///
+    /// This is the rotation `R` such that `(R * a).angle(b) == 0 && (R * a).dot(b).is_positive()`.
+    #[inline]
+    pub fn rotation_between<SB, SC>(a: &ColumnVector<N, U2, SB>, b: &ColumnVector<N, U2, SC>) -> Self
+        where SB: Storage<N, U2, U1>,
+              SC: Storage<N, U2, U1> {
+        ::convert(UnitComplex::rotation_between(a, b).to_rotation_matrix())
+    }
+
+    /// The smallest rotation needed to make `a` and `b` collinear and point toward the same
+    /// direction, raised to the power `s`.
+    #[inline]
+    pub fn scaled_rotation_between<SB, SC>(a: &ColumnVector<N, U2, SB>, b: &ColumnVector<N, U2, SC>, s: N) -> Self
+        where SB: Storage<N, U2, U1>,
+              SC: Storage<N, U2, U1> {
+        ::convert(UnitComplex::scaled_rotation_between(a, b, s).to_rotation_matrix())
+    }
+}
+
+impl<N, S> RotationBase<N, U2, S>
+where N: Real,
+      S: Storage<N, U2, U2> {
+    /// The rotation angle.
+    #[inline]
+    pub fn angle(&self) -> N {
+        self.matrix()[(1, 0)].atan2(self.matrix()[(0, 0)])
+    }
+
+    /// The rotation angle needed to make `self` and `other` coincide.
+    #[inline]
+    pub fn angle_to<SB: Storage<N, U2, U2>>(&self, other: &RotationBase<N, U2, SB>) -> N {
+        self.rotation_to(other).angle()
+    }
+
+    /// The rotation matrix needed to make `self` and `other` coincide.
+    ///
+    /// The result is such that: `self.rotation_to(other) * self == other`.
+    #[inline]
+    pub fn rotation_to<SB>(&self, other: &RotationBase<N, U2, SB>) -> OwnedRotation<N, U2, SB::Alloc>
+        where SB: Storage<N, U2, U2> {
+        other * self.inverse()
+    }
+
+    /// Raise the quaternion to a given floating power, i.e., returns the rotation with the angle
+    /// of `self` multiplied by `n`.
+    #[inline]
+    pub fn powf(&self, n: N) -> OwnedRotation<N, U2, S::Alloc> {
+        OwnedRotation::<_, _, S::Alloc>::new(self.angle() * n)
+    }
+
+    /// The rotation angle returned as a 1-dimensional vector.
+    #[inline]
+    pub fn scaled_axis(&self) -> OwnedColumnVector<N, U1, S::Alloc>
+        where S::Alloc: Allocator<N, U1, U1> {
+        ColumnVector::<_, U1, _>::new(self.angle())
+    }
+}
+
+impl<N, S> Rand for RotationBase<N, U2, S>
+where N: Real + Rand,
+      S: OwnedStorage<N, U2, U2>,
+      S::Alloc: OwnedAllocator<N, U2, U2, S> {
+    #[inline]
+    fn rand<R: Rng>(rng: &mut R) -> Self {
+        Self::new(rng.gen())
+    }
+}
+
+#[cfg(feature="arbitrary")]
+impl<N, S> Arbitrary for RotationBase<N, U2, S>
+where N: Real + Arbitrary,
+      S: OwnedStorage<N, U2, U2> + Send,
+      S::Alloc: OwnedAllocator<N, U2, U2, S> {
+    #[inline]
+    fn arbitrary<G: Gen>(g: &mut G) -> Self {
+        Self::new(N::arbitrary(g))
+    }
+}
+
+
+/*
+ *
+ * 3D RotationBase matrix.
+ *
+ */
+impl<N, S> RotationBase<N, U3, S>
+where N: Real,
+      S: OwnedStorage<N, U3, U3>,
+      S::Alloc: OwnedAllocator<N, U3, U3, S> {
+
+    /// Builds a 3 dimensional rotation matrix from an axis and an angle.
+    ///
+    /// # Arguments
+    ///   * `axisangle` - A vector representing the rotation. Its magnitude is the amount of rotation
+    ///   in radian. Its direction is the axis of rotation.
+    pub fn new<SB: Storage<N, U3, U1>>(axisangle: ColumnVector<N, U3, SB>) -> Self {
+        let (axis, angle) = Unit::new_and_get(axisangle.into_owned());
+        Self::from_axis_angle(&axis, angle)
+    }
+
+    /// Builds a 3D rotation matrix from an axis scaled by the rotation angle.
+    pub fn from_scaled_axis<SB: Storage<N, U3, U1>>(axisangle: ColumnVector<N, U3, SB>) -> Self {
+        Self::new(axisangle)
+    }
+
+    /// Builds a 3D rotation matrix from an axis and a rotation angle.
+    pub fn from_axis_angle<SB>(axis: &Unit<ColumnVector<N, U3, SB>>, angle: N) -> Self
+        where SB: Storage<N, U3, U1> {
+        if angle.is_zero() {
+            Self::identity()
+        }
+        else {
+            let ux         = axis.as_ref()[0];
+            let uy         = axis.as_ref()[1];
+            let uz         = axis.as_ref()[2];
+            let sqx        = ux * ux;
+            let sqy        = uy * uy;
+            let sqz        = uz * uz;
+            let (sin, cos) = angle.sin_cos();
+            let one_m_cos  = N::one() - cos;
+
+            Self::from_matrix_unchecked(
+                SquareMatrix::<N, U3, S>::new(
+                    (sqx + (N::one() - sqx) * cos),
+                    (ux * uy * one_m_cos - uz * sin),
+                    (ux * uz * one_m_cos + uy * sin),
+
+                    (ux * uy * one_m_cos + uz * sin),
+                    (sqy + (N::one() - sqy) * cos),
+                    (uy * uz * one_m_cos - ux * sin),
+
+                    (ux * uz * one_m_cos - uy * sin),
+                    (uy * uz * one_m_cos + ux * sin),
+                    (sqz + (N::one() - sqz) * cos)))
+        }
+    }
+
+    /// Creates a new rotation from Euler angles.
+    ///
+    /// The primitive rotations are applied in order: 1 roll − 2 pitch − 3 yaw.
+    pub fn from_euler_angles(roll: N, pitch: N, yaw: N) -> Self {
+        let (sr, cr) = roll.sin_cos();
+        let (sp, cp) = pitch.sin_cos();
+        let (sy, cy) = yaw.sin_cos();
+
+        Self::from_matrix_unchecked(
+            SquareMatrix::<N, U3, S>::new(
+                cy * cp, cy * sp * sr - sy * cr, cy * sp * cr + sy * sr,
+                sy * cp, sy * sp * sr + cy * cr, sy * sp * cr - cy * sr,
+                -sp,     cp * sr,                cp * cr)
+            )
+    }
+
+    /// Creates a rotation that corresponds to the local frame of an observer standing at the
+    /// origin and looking toward `dir`.
+    ///
+    /// It maps the view direction `dir` to the positive `z` axis.
+    ///
+    /// # Arguments
+    ///   * dir - The look direction, that is, direction the matrix `z` axis will be aligned with.
+    ///   * up - The vertical direction. The only requirement of this parameter is to not be
+    ///   collinear
+    ///   to `dir`. Non-collinearity is not checked.
+    #[inline]
+    pub fn new_observer_frame<SB, SC>(dir: &ColumnVector<N, U3, SB>, up: &ColumnVector<N, U3, SC>) -> Self
+    where SB: Storage<N, U3, U1>,
+          SC: Storage<N, U3, U1> {
+        let zaxis = dir.normalize();
+        let xaxis = up.cross(&zaxis).normalize();
+        let yaxis = zaxis.cross(&xaxis).normalize();
+
+        Self::from_matrix_unchecked(SquareMatrix::<N, U3, S>::new(
+                xaxis.x, yaxis.x, zaxis.x,
+                xaxis.y, yaxis.y, zaxis.y,
+                xaxis.z, yaxis.z, zaxis.z))
+    }
+
+
+    /// Builds a right-handed look-at view matrix without translation.
+    ///
+    /// This conforms to the common notion of right handed look-at matrix from the computer
+    /// graphics community.
+    ///
+    /// # Arguments
+    ///   * eye - The eye position.
+    ///   * target - The target position.
+    ///   * up - A vector approximately aligned with required the vertical axis. The only
+    ///   requirement of this parameter is to not be collinear to `target - eye`.
+    #[inline]
+    pub fn look_at_rh<SB, SC>(dir: &ColumnVector<N, U3, SB>, up: &ColumnVector<N, U3, SC>) -> Self
+    where SB: Storage<N, U3, U1>,
+          SC: Storage<N, U3, U1> {
+        Self::new_observer_frame(&dir.neg(), up).inverse()
+    }
+
+    /// Builds a left-handed look-at view matrix without translation.
+    ///
+    /// This conforms to the common notion of left handed look-at matrix from the computer
+    /// graphics community.
+    ///
+    /// # Arguments
+    ///   * eye - The eye position.
+    ///   * target - The target position.
+    ///   * up - A vector approximately aligned with required the vertical axis. The only
+    ///   requirement of this parameter is to not be collinear to `target - eye`.
+    #[inline]
+    pub fn look_at_lh<SB, SC>(dir: &ColumnVector<N, U3, SB>, up: &ColumnVector<N, U3, SC>) -> Self
+    where SB: Storage<N, U3, U1>,
+          SC: Storage<N, U3, U1> {
+            Self::new_observer_frame(dir, up).inverse()
+    }
+
+    /// The rotation matrix required to align `a` and `b` but with its angl.
+    ///
+    /// This is the rotation `R` such that `(R * a).angle(b) == 0 && (R * a).dot(b).is_positive()`.
+    #[inline]
+    pub fn rotation_between<SB, SC>(a: &ColumnVector<N, U3, SB>, b: &ColumnVector<N, U3, SC>) -> Option<Self>
+        where SB: Storage<N, U3, U1>,
+              SC: Storage<N, U3, U1> {
+        Self::scaled_rotation_between(a, b, N::one())
+    }
+
+    /// The smallest rotation needed to make `a` and `b` collinear and point toward the same
+    /// direction, raised to the power `s`.
+    #[inline]
+    pub fn scaled_rotation_between<SB, SC>(a: &ColumnVector<N, U3, SB>, b: &ColumnVector<N, U3, SC>, n: N)
+        -> Option<Self>
+        where SB: Storage<N, U3, U1>,
+              SC: Storage<N, U3, U1> {
+        // FIXME: code duplication with RotationBase.
+        if let (Some(na), Some(nb)) = (a.try_normalize(N::zero()), b.try_normalize(N::zero())) {
+            let c = na.cross(&nb);
+
+            if let Some(axis) = Unit::try_new(c, N::default_epsilon()) {
+                return Some(Self::from_axis_angle(&axis, na.dot(&nb).acos() * n))
+            }
+
+            // Zero or PI.
+            if na.dot(&nb) < N::zero() {
+                // PI
+                //
+                // The rotation axis is undefined but the angle not zero. This is not a
+                // simple rotation.
+                return None;
+            }
+        }
+
+        Some(Self::identity())
+    }
+}
+
+impl<N, S> RotationBase<N, U3, S>
+where N: Real,
+      S: Storage<N, U3, U3> {
+    /// The rotation angle.
+    #[inline]
+    pub fn angle(&self) -> N {
+        ((self.matrix()[(0, 0)] + self.matrix()[(1, 1)] + self.matrix()[(2, 2)] - N::one()) / ::convert(2.0)).acos()
+    }
+}
+
+impl<N, S> RotationBase<N, U3, S>
+where N: Real,
+      S: Storage<N, U3, U3>,
+      S::Alloc: Allocator<N, U3, U1> {
+    /// The rotation axis. Returns `None` if the rotation angle is zero or PI.
+    #[inline]
+    pub fn axis(&self) -> Option<Unit<OwnedColumnVector<N, U3, S::Alloc>>> {
+        let axis = OwnedColumnVector::<N, U3, S::Alloc>::new(
+            self.matrix()[(2, 1)] - self.matrix()[(1, 2)],
+            self.matrix()[(0, 2)] - self.matrix()[(2, 0)],
+            self.matrix()[(1, 0)] - self.matrix()[(0, 1)]);
+
+        Unit::try_new(axis, N::default_epsilon())
+    }
+
+    /// The rotation axis multiplied by the rotation angle.
+    #[inline]
+    pub fn scaled_axis(&self) -> OwnedColumnVector<N, U3, S::Alloc> {
+        if let Some(axis) = self.axis() {
+            axis.unwrap() * self.angle()
+        }
+        else {
+            ColumnVector::zero()
+        }
+    }
+
+    /// The rotation angle needed to make `self` and `other` coincide.
+    #[inline]
+    pub fn angle_to<SB: Storage<N, U3, U3>>(&self, other: &RotationBase<N, U3, SB>) -> N {
+        self.rotation_to(other).angle()
+    }
+
+    /// The rotation matrix needed to make `self` and `other` coincide.
+    ///
+    /// The result is such that: `self.rotation_to(other) * self == other`.
+    #[inline]
+    pub fn rotation_to<SB>(&self, other: &RotationBase<N, U3, SB>) -> OwnedRotation<N, U3, SB::Alloc>
+        where SB: Storage<N, U3, U3> {
+        other * self.inverse()
+    }
+
+    /// Raise the quaternion to a given floating power, i.e., returns the rotation with the same
+    /// axis as `self` and an angle equal to `self.angle()` multiplied by `n`.
+    #[inline]
+    pub fn powf(&self, n: N) -> OwnedRotation<N, U3, S::Alloc> {
+        if let Some(axis) = self.axis() {
+            OwnedRotation::<_, _, S::Alloc>::from_axis_angle(&axis, self.angle() * n)
+        }
+        else if self.matrix()[(0, 0)] < N::zero() {
+            let minus_id = OwnedSquareMatrix::<N, U3, S::Alloc>::from_diagonal_element(-N::one());
+            OwnedRotation::<_, _, S::Alloc>::from_matrix_unchecked(minus_id)
+        }
+        else {
+            OwnedRotation::<_, _, S::Alloc>::identity()
+        }
+    }
+}
+
+impl<N, S> Rand for RotationBase<N, U3, S>
+where N: Real + Rand,
+      S: OwnedStorage<N, U3, U3>,
+      S::Alloc: OwnedAllocator<N, U3, U3, S> +
+                Allocator<N, U3, U1> {
+    #[inline]
+    fn rand<R: Rng>(rng: &mut R) -> Self {
+        Self::new(Vector3::rand(rng))
+    }
+}
+
+#[cfg(feature="arbitrary")]
+impl<N, S> Arbitrary for RotationBase<N, U3, S>
+where N: Real + Arbitrary,
+      S: OwnedStorage<N, U3, U3> + Send,
+      S::Alloc: OwnedAllocator<N, U3, U3, S> +
+                Allocator<N, U3, U1> {
+    #[inline]
+    fn arbitrary<G: Gen>(g: &mut G) -> Self {
+        Self::new(Vector3::arbitrary(g))
+    }
+}

src/geometry/similarity.rs

@@ -0,0 +1,276 @@
+use std::fmt;
+use approx::ApproxEq;
+
+use alga::general::{ClosedMul, Real, SubsetOf};
+use alga::linear::Rotation;
+
+use core::{Scalar, OwnedSquareMatrix};
+use core::dimension::{DimName, DimNameSum, DimNameAdd, U1};
+use core::storage::{Storage, OwnedStorage};
+use core::allocator::{Allocator, OwnedAllocator};
+use geometry::{PointBase, TranslationBase, IsometryBase};
+
+
+/// A similarity that uses a data storage deduced from the allocator `A`.
+pub type OwnedSimilarityBase<N, D, A, R> =
+    SimilarityBase<N, D, <A as Allocator<N, D, U1>>::Buffer, R>;
+
+/// A similarity, i.e., an uniform scaling, followed by a rotation, followed by a translation.
+#[repr(C)]
+#[derive(Hash, Debug, Clone, Copy)]
+#[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
+pub struct SimilarityBase<N: Scalar, D: DimName, S, R> {
+    /// The part of this similarity that does not include the scaling factor.
+    pub isometry: IsometryBase<N, D, S, R>,
+    scaling:      N
+}
+
+impl<N, D: DimName, S, R> SimilarityBase<N, D, S, R>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          R: Rotation<PointBase<N, D, S>>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    /// Creates a new similarity from its rotational and translational parts.
+    #[inline]
+    pub fn from_parts(translation: TranslationBase<N, D, S>, rotation: R, scaling: N) -> SimilarityBase<N, D, S, R> {
+        SimilarityBase::from_isometry(IsometryBase::from_parts(translation, rotation), scaling)
+    }
+
+    /// Creates a new similarity from its rotational and translational parts.
+    #[inline]
+    pub fn from_isometry(isometry: IsometryBase<N, D, S, R>, scaling: N) -> SimilarityBase<N, D, S, R> {
+        assert!(!relative_eq!(scaling, N::zero()), "The scaling factor must not be zero.");
+
+        SimilarityBase {
+            isometry: isometry,
+            scaling:  scaling
+        }
+    }
+
+    /// Creates a new similarity that applies only a scaling factor.
+    #[inline]
+    pub fn from_scaling(scaling: N) -> SimilarityBase<N, D, S, R> {
+        Self::from_isometry(IsometryBase::identity(), scaling)
+    }
+
+    /// Inverts `self`.
+    #[inline]
+    pub fn inverse(&self) -> SimilarityBase<N, D, S, R> {
+        let mut res = self.clone();
+        res.inverse_mut();
+        res
+    }
+
+    /// Inverts `self` in-place.
+    #[inline]
+    pub fn inverse_mut(&mut self) {
+        self.scaling = N::one() / self.scaling;
+        self.isometry.inverse_mut();
+        self.isometry.translation.vector *= self.scaling;
+    }
+
+    /// The scaling factor of this similarity transformation.
+    #[inline]
+    pub fn set_scaling(&mut self, scaling: N) {
+        assert!(!relative_eq!(scaling, N::zero()), "The similarity scaling factor must not be zero.");
+
+        self.scaling = scaling;
+    }
+
+    /// The similarity transformation that applies a scaling factor `scaling` before `self`.
+    #[inline]
+    pub fn prepend_scaling(&self, scaling: N) -> Self {
+        assert!(!relative_eq!(scaling, N::zero()), "The similarity scaling factor must not be zero.");
+
+        Self::from_isometry(self.isometry.clone(), self.scaling * scaling)
+    }
+
+    /// The similarity transformation that applies a scaling factor `scaling` after `self`.
+    #[inline]
+    pub fn append_scaling(&self, scaling: N) -> Self {
+        assert!(!relative_eq!(scaling, N::zero()), "The similarity scaling factor must not be zero.");
+
+        Self::from_parts(
+            TranslationBase::from_vector(&self.isometry.translation.vector * scaling),
+            self.isometry.rotation.clone(),
+            self.scaling * scaling)
+    }
+
+    /// Sets `self` to the similarity transformation that applies a scaling factor `scaling` before `self`.
+    #[inline]
+    pub fn prepend_scaling_mut(&mut self, scaling: N) {
+        assert!(!relative_eq!(scaling, N::zero()), "The similarity scaling factor must not be zero.");
+
+        self.scaling *= scaling
+    }
+
+    /// Sets `self` to the similarity transformation that applies a scaling factor `scaling` after `self`.
+    #[inline]
+    pub fn append_scaling_mut(&mut self, scaling: N) {
+        assert!(!relative_eq!(scaling, N::zero()), "The similarity scaling factor must not be zero.");
+
+        self.isometry.translation.vector *= scaling;
+        self.scaling *= scaling;
+    }
+
+    /// Appends to `self` the given translation in-place.
+    #[inline]
+    pub fn append_translation_mut(&mut self, t: &TranslationBase<N, D, S>) {
+        self.isometry.append_translation_mut(t)
+    }
+
+    /// Appends to `self` the given rotation in-place.
+    #[inline]
+    pub fn append_rotation_mut(&mut self, r: &R) {
+        self.isometry.append_rotation_mut(r)
+    }
+
+    /// Appends in-place to `self` a rotation centered at the point `p`, i.e., the rotation that
+    /// lets `p` invariant.
+    #[inline]
+    pub fn append_rotation_wrt_point_mut(&mut self, r: &R, p: &PointBase<N, D, S>) {
+        self.isometry.append_rotation_wrt_point_mut(r, p)
+    }
+
+    /// Appends in-place to `self` a rotation centered at the point with coordinates
+    /// `self.translation`.
+    #[inline]
+    pub fn append_rotation_wrt_center_mut(&mut self, r: &R) {
+        self.isometry.append_rotation_wrt_center_mut(r)
+    }
+}
+
+
+// NOTE: we don't require `R: Rotation<...>` here becaus this is not useful for the implementation
+// and makes it harde to use it, e.g., for Transform × Isometry implementation.
+// This is OK since all constructors of the isometry enforce the Rotation bound already (and
+// explicit struct construction is prevented by the private scaling factor).
+impl<N, D: DimName, S, R> SimilarityBase<N, D, S, R>
+    where N: Scalar + ClosedMul,
+          S: Storage<N, D, U1> {
+    /// Converts this similarity into its equivalent homogeneous transformation matrix.
+    #[inline]
+    pub fn to_homogeneous(&self) -> OwnedSquareMatrix<N, DimNameSum<D, U1>, S::Alloc>
+        where D: DimNameAdd<U1>,
+              R: SubsetOf<OwnedSquareMatrix<N, DimNameSum<D, U1>, S::Alloc>>,
+              S::Alloc: Allocator<N, D, D> +
+                        Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1>> {
+        let mut res = self.isometry.to_homogeneous();
+
+        for e in res.fixed_slice_mut::<D, D>(0, 0).iter_mut() {
+            *e *= self.scaling
+        }
+
+        res
+    }
+
+    /// The scaling factor of this similarity transformation.
+    #[inline]
+    pub fn scaling(&self) -> N {
+        self.scaling
+    }
+}
+
+
+impl<N, D: DimName, S, R> Eq for SimilarityBase<N, D, S, R>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          R: Rotation<PointBase<N, D, S>> + Eq,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+}
+
+impl<N, D: DimName, S, R> PartialEq for SimilarityBase<N, D, S, R>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          R: Rotation<PointBase<N, D, S>> + PartialEq,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    #[inline]
+    fn eq(&self, right: &SimilarityBase<N, D, S, R>) -> bool {
+        self.isometry == right.isometry && self.scaling == right.scaling
+    }
+}
+
+impl<N, D: DimName, S, R> ApproxEq for SimilarityBase<N, D, S, R>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          R: Rotation<PointBase<N, D, S>> + ApproxEq<Epsilon = N::Epsilon>,
+          S::Alloc: OwnedAllocator<N, D, U1, S>,
+          N::Epsilon: Copy {
+    type Epsilon = N::Epsilon;
+
+    #[inline]
+    fn default_epsilon() -> Self::Epsilon {
+        N::default_epsilon()
+    }
+
+    #[inline]
+    fn default_max_relative() -> Self::Epsilon {
+        N::default_max_relative()
+    }
+
+    #[inline]
+    fn default_max_ulps() -> u32 {
+        N::default_max_ulps()
+    }
+
+    #[inline]
+    fn relative_eq(&self, other: &Self, epsilon: Self::Epsilon, max_relative: Self::Epsilon) -> bool {
+        self.isometry.relative_eq(&other.isometry, epsilon, max_relative) &&
+        self.scaling.relative_eq(&other.scaling, epsilon, max_relative)
+    }
+
+    #[inline]
+    fn ulps_eq(&self, other: &Self, epsilon: Self::Epsilon, max_ulps: u32) -> bool {
+        self.isometry.ulps_eq(&other.isometry, epsilon, max_ulps) &&
+        self.scaling.ulps_eq(&other.scaling, epsilon, max_ulps)
+    }
+}
+
+/*
+ *
+ * Display
+ *
+ */
+impl<N, D: DimName, S, R> fmt::Display for SimilarityBase<N, D, S, R>
+    where N: Real + fmt::Display,
+          S: OwnedStorage<N, D, U1>,
+          R: Rotation<PointBase<N, D, S>> + fmt::Display,
+          S::Alloc: OwnedAllocator<N, D, U1, S> + Allocator<usize, D, U1> {
+    fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
+        let precision = f.precision().unwrap_or(3);
+
+        try!(writeln!(f, "SimilarityBase {{"));
+        try!(write!(f, "{:.*}", precision, self.isometry));
+        try!(write!(f, "Scaling: {:.*}", precision, self.scaling));
+        writeln!(f, "}}")
+    }
+}
+
+/*
+//         /*
+//          *
+//          * ToHomogeneous
+//          *
+//          */
+//         impl<N: Real> ToHomogeneous<$homogeneous<N>> for $t<N> {
+//             #[inline]
+//             fn to_homogeneous(&self) -> $homogeneous<N> {
+//                 self.vector.to_homogeneous()
+//             }
+//         }
+
+
+//         /*
+//          *
+//          * Absolute
+//          *
+//          */
+//         impl<N: Absolute> Absolute for $t<N> {
+//             type AbsoluteValue = $submatrix<N::AbsoluteValue>;
+//
+//             #[inline]
+//             fn abs(m: &$t<N>) -> $submatrix<N::AbsoluteValue> {
+//                 Absolute::abs(&m.submatrix)
+//             }
+//         }
+*/

src/geometry/similarity_alga.rs

@@ -0,0 +1,184 @@
+use alga::general::{AbstractMagma, AbstractGroup, AbstractLoop, AbstractMonoid, AbstractQuasigroup,
+                    AbstractSemigroup, Real, Inverse, Multiplicative, Identity};
+use alga::linear::{Transformation, AffineTransformation, Rotation, Similarity, ProjectiveTransformation};
+
+use core::ColumnVector;
+use core::dimension::{DimName, U1};
+use core::storage::OwnedStorage;
+use core::allocator::OwnedAllocator;
+
+use geometry::{SimilarityBase, TranslationBase, PointBase};
+
+
+/*
+ *
+ * Algebraic structures.
+ *
+ */
+impl<N, D: DimName, S, R> Identity<Multiplicative> for SimilarityBase<N, D, S, R>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          R: Rotation<PointBase<N, D, S>>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    #[inline]
+    fn identity() -> Self {
+        Self::identity()
+    }
+}
+
+impl<N, D: DimName, S, R> Inverse<Multiplicative> for SimilarityBase<N, D, S, R>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          R: Rotation<PointBase<N, D, S>>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    #[inline]
+    fn inverse(&self) -> Self {
+        self.inverse()
+    }
+
+    #[inline]
+    fn inverse_mut(&mut self) {
+        self.inverse_mut()
+    }
+}
+
+impl<N, D: DimName, S, R> AbstractMagma<Multiplicative> for SimilarityBase<N, D, S, R>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          R: Rotation<PointBase<N, D, S>>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    #[inline]
+    fn operate(&self, rhs: &Self) -> Self {
+        self * rhs
+    }
+}
+
+macro_rules! impl_multiplicative_structures(
+    ($($marker: ident<$operator: ident>),* $(,)*) => {$(
+        impl<N, D: DimName, S, R> $marker<$operator> for SimilarityBase<N, D, S, R>
+            where N: Real,
+                  S: OwnedStorage<N, D, U1>,
+                  R: Rotation<PointBase<N, D, S>>,
+                  S::Alloc: OwnedAllocator<N, D, U1, S> { }
+    )*}
+);
+
+impl_multiplicative_structures!(
+    AbstractSemigroup<Multiplicative>,
+    AbstractMonoid<Multiplicative>,
+    AbstractQuasigroup<Multiplicative>,
+    AbstractLoop<Multiplicative>,
+    AbstractGroup<Multiplicative>
+);
+
+/*
+ *
+ * Transformation groups.
+ *
+ */
+impl<N, D: DimName, S, R> Transformation<PointBase<N, D, S>> for SimilarityBase<N, D, S, R>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          R: Rotation<PointBase<N, D, S>>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    #[inline]
+    fn transform_point(&self, pt: &PointBase<N, D, S>) -> PointBase<N, D, S> {
+        self * pt
+    }
+
+    #[inline]
+    fn transform_vector(&self, v: &ColumnVector<N, D, S>) -> ColumnVector<N, D, S> {
+        self * v
+    }
+}
+
+impl<N, D: DimName, S, R> ProjectiveTransformation<PointBase<N, D, S>> for SimilarityBase<N, D, S, R>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          R: Rotation<PointBase<N, D, S>>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    #[inline]
+    fn inverse_transform_point(&self, pt: &PointBase<N, D, S>) -> PointBase<N, D, S> {
+        self.isometry.inverse_transform_point(pt) / self.scaling()
+    }
+
+    #[inline]
+    fn inverse_transform_vector(&self, v: &ColumnVector<N, D, S>) -> ColumnVector<N, D, S> {
+        self.isometry.inverse_transform_vector(v) / self.scaling()
+    }
+}
+
+impl<N, D: DimName, S, R> AffineTransformation<PointBase<N, D, S>> for SimilarityBase<N, D, S, R>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          R: Rotation<PointBase<N, D, S>>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    type NonUniformScaling = N;
+    type Rotation          = R;
+    type Translation       = TranslationBase<N, D, S>;
+
+    #[inline]
+    fn decompose(&self) -> (TranslationBase<N, D, S>, R, N, R) {
+        (self.isometry.translation.clone(), self.isometry.rotation.clone(), self.scaling(), R::identity())
+    }
+
+    #[inline]
+    fn append_translation(&self, t: &Self::Translation) -> Self {
+        t * self
+    }
+
+    #[inline]
+    fn prepend_translation(&self, t: &Self::Translation) -> Self {
+        self * t
+    }
+
+    #[inline]
+    fn append_rotation(&self, r: &Self::Rotation) -> Self {
+        SimilarityBase::from_isometry(self.isometry.append_rotation(r), self.scaling())
+    }
+
+    #[inline]
+    fn prepend_rotation(&self, r: &Self::Rotation) -> Self {
+        self * r
+    }
+
+    #[inline]
+    fn append_scaling(&self, s: &Self::NonUniformScaling) -> Self {
+        self.append_scaling(*s)
+    }
+
+    #[inline]
+    fn prepend_scaling(&self, s: &Self::NonUniformScaling) -> Self {
+        self.prepend_scaling(*s)
+    }
+
+    #[inline]
+    fn append_rotation_wrt_point(&self, r: &Self::Rotation, p: &PointBase<N, D, S>) -> Option<Self> {
+        let mut res = self.clone();
+        res.append_rotation_wrt_point_mut(r, p);
+        Some(res)
+    }
+}
+
+impl<N, D: DimName, S, R> Similarity<PointBase<N, D, S>> for SimilarityBase<N, D, S, R>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          R: Rotation<PointBase<N, D, S>>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    type Scaling = N;
+
+    #[inline]
+    fn translation(&self) -> TranslationBase<N, D, S> {
+        self.isometry.translation()
+    }
+
+    #[inline]
+    fn rotation(&self) -> R {
+        self.isometry.rotation()
+    }
+
+    #[inline]
+    fn scaling(&self) -> N {
+        self.scaling()
+    }
+}

src/geometry/similarity_alias.rs

@@ -0,0 +1,19 @@
+use core::MatrixArray;
+use core::dimension::{U1, U2, U3};
+
+use geometry::{Rotation, SimilarityBase, UnitQuaternion, UnitComplex};
+
+/// A D-dimensional similarity.
+pub type Similarity<N, D> = SimilarityBase<N, D, MatrixArray<N, D, U1>, Rotation<N, D>>;
+
+/// A 2-dimensional similarity.
+pub type Similarity2<N> = SimilarityBase<N, U2, MatrixArray<N, U2, U1>, UnitComplex<N>>;
+
+/// A 3-dimensional similarity.
+pub type Similarity3<N> = SimilarityBase<N, U3, MatrixArray<N, U3, U1>, UnitQuaternion<N>>;
+
+/// A 2-dimensional similarity using a rotation matrix for its rotation part.
+pub type SimilarityMatrix2<N> = Similarity<N, U2>;
+
+/// A 3-dimensional similarity using a rotation matrix for its rotation part.
+pub type SimilarityMatrix3<N> = Similarity<N, U3>;

src/geometry/similarity_construction.rs

@@ -0,0 +1,200 @@
+#[cfg(feature = "arbitrary")]
+use quickcheck::{Arbitrary, Gen};
+
+use num::One;
+use rand::{Rng, Rand};
+
+use alga::general::Real;
+use alga::linear::Rotation as AlgaRotation;
+
+use core::ColumnVector;
+use core::dimension::{DimName, U1, U2, U3, U4};
+use core::allocator::{OwnedAllocator, Allocator};
+use core::storage::OwnedStorage;
+
+use geometry::{PointBase, TranslationBase, RotationBase, SimilarityBase,
+               UnitComplex, UnitQuaternionBase, IsometryBase};
+
+
+impl<N, D: DimName, S, R> SimilarityBase<N, D, S, R>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          R: AlgaRotation<PointBase<N, D, S>>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    /// Creates a new identity similarity.
+    #[inline]
+    pub fn identity() -> Self {
+        Self::from_isometry(IsometryBase::identity(), N::one())
+    }
+}
+
+impl<N, D: DimName, S, R> One for SimilarityBase<N, D, S, R>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          R: AlgaRotation<PointBase<N, D, S>>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    /// Creates a new identity similarity.
+    #[inline]
+    fn one() -> Self {
+        Self::identity()
+    }
+}
+
+impl<N, D: DimName, S, R> Rand for SimilarityBase<N, D, S, R>
+    where N: Real + Rand,
+          S: OwnedStorage<N, D, U1>,
+          R: AlgaRotation<PointBase<N, D, S>> + Rand,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    #[inline]
+    fn rand<G: Rng>(rng: &mut G) -> Self {
+        let mut s = rng.gen();
+        while relative_eq!(s, N::zero()) {
+            s = rng.gen()
+        }
+
+        Self::from_isometry(rng.gen(), s)
+    }
+}
+
+impl<N, D: DimName, S, R> SimilarityBase<N, D, S, R>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          R: AlgaRotation<PointBase<N, D, S>>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    /// The similarity that applies tha scaling factor `scaling`, followed by the rotation `r` with
+    /// its axis passing through the point `p`.
+    #[inline]
+    pub fn rotation_wrt_point(r: R, p: PointBase<N, D, S>, scaling: N) -> Self {
+        let shift = r.transform_vector(&-&p.coords);
+        Self::from_parts(TranslationBase::from_vector(shift + p.coords), r, scaling)
+    }
+}
+
+#[cfg(feature = "arbitrary")]
+impl<N, D: DimName, S, R> Arbitrary for SimilarityBase<N, D, S, R>
+    where N: Real + Arbitrary + Send,
+          S: OwnedStorage<N, D, U1> + Send,
+          R: AlgaRotation<PointBase<N, D, S>> + Arbitrary + Send,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    #[inline]
+    fn arbitrary<G: Gen>(rng: &mut G) -> Self {
+        let mut s = Arbitrary::arbitrary(rng);
+        while relative_eq!(s, N::zero()) {
+            s = Arbitrary::arbitrary(rng)
+        }
+
+        Self::from_isometry(Arbitrary::arbitrary(rng), s)
+    }
+}
+
+/*
+ *
+ * Constructors for various static dimensions.
+ *
+ */
+
+// 2D rotation.
+impl<N, S, SR> SimilarityBase<N, U2, S, RotationBase<N, U2, SR>>
+    where N: Real,
+          S:  OwnedStorage<N, U2, U1, Alloc = SR::Alloc>,
+          SR: OwnedStorage<N, U2, U2>,
+          S::Alloc:  OwnedAllocator<N, U2, U1, S>,
+          SR::Alloc: OwnedAllocator<N, U2, U2, SR> {
+    /// Creates a new similarity from a translation and a rotation angle.
+    #[inline]
+    pub fn new(translation: ColumnVector<N, U2, S>, angle: N, scaling: N) -> Self {
+        Self::from_parts(TranslationBase::from_vector(translation), RotationBase::<N, U2, SR>::new(angle), scaling)
+    }
+}
+
+impl<N, S> SimilarityBase<N, U2, S, UnitComplex<N>>
+    where N: Real,
+          S: OwnedStorage<N, U2, U1>,
+          S::Alloc: OwnedAllocator<N, U2, U1, S> {
+    /// Creates a new similarity from a translation and a rotation angle.
+    #[inline]
+    pub fn new(translation: ColumnVector<N, U2, S>, angle: N, scaling: N) -> Self {
+        Self::from_parts(TranslationBase::from_vector(translation), UnitComplex::new(angle), scaling)
+    }
+}
+
+// 3D rotation.
+macro_rules! similarity_construction_impl(
+    ($Rot: ty, $RotId: ident, $RRDim: ty, $RCDim: ty) => {
+        impl<N, S, SR> SimilarityBase<N, U3, S, $Rot>
+            where N: Real,
+                  S:  OwnedStorage<N, U3, U1, Alloc = SR::Alloc>,
+                  SR: OwnedStorage<N, $RRDim, $RCDim>,
+                  S::Alloc:  OwnedAllocator<N, U3, U1, S>,
+                  SR::Alloc: OwnedAllocator<N, $RRDim, $RCDim, SR> +
+                             Allocator<N, U3, U3> {
+            /// Creates a new similarity from a translation, rotation axis-angle, and scaling
+            /// factor.
+            #[inline]
+            pub fn new(translation: ColumnVector<N, U3, S>, axisangle: ColumnVector<N, U3, S>, scaling: N) -> Self {
+                Self::from_isometry(IsometryBase::<_, _, _, $Rot>::new(translation, axisangle), scaling)
+            }
+
+            /// Creates an similarity that corresponds to the a scaling factor and a local frame of
+            /// an observer standing at the point `eye` and looking toward `target`.
+            ///
+            /// It maps the view direction `target - eye` to the positive `z` axis and the origin to the
+            /// `eye`.
+            ///
+            /// # Arguments
+            ///   * eye - The observer position.
+            ///   * target - The target position.
+            ///   * up - Vertical direction. The only requirement of this parameter is to not be collinear
+            ///   to `eye - at`. Non-collinearity is not checked.
+            #[inline]
+            pub fn new_observer_frame(eye:    &PointBase<N, U3, S>,
+                                      target: &PointBase<N, U3, S>,
+                                      up:     &ColumnVector<N, U3, S>,
+                                      scaling: N)
+                                      -> Self {
+                Self::from_isometry(IsometryBase::<_, _, _, $Rot>::new_observer_frame(eye, target, up), scaling)
+            }
+
+            /// Builds a right-handed look-at view matrix including scaling factor.
+            ///
+            /// This conforms to the common notion of right handed look-at matrix from the computer
+            /// graphics community.
+            ///
+            /// # Arguments
+            ///   * eye - The eye position.
+            ///   * target - The target position.
+            ///   * up - A vector approximately aligned with required the vertical axis. The only
+            ///   requirement of this parameter is to not be collinear to `target - eye`.
+            #[inline]
+            pub fn look_at_rh(eye:     &PointBase<N, U3, S>,
+                              target:  &PointBase<N, U3, S>,
+                              up:      &ColumnVector<N, U3, S>,
+                              scaling: N)
+                              -> Self {
+                Self::from_isometry(IsometryBase::<_, _, _, $Rot>::look_at_rh(eye, target, up), scaling)
+            }
+
+            /// Builds a left-handed look-at view matrix including a scaling factor.
+            ///
+            /// This conforms to the common notion of left handed look-at matrix from the computer
+            /// graphics community.
+            ///
+            /// # Arguments
+            ///   * eye - The eye position.
+            ///   * target - The target position.
+            ///   * up - A vector approximately aligned with required the vertical axis. The only
+            ///   requirement of this parameter is to not be collinear to `target - eye`.
+            #[inline]
+            pub fn look_at_lh(eye:     &PointBase<N, U3, S>,
+                              target:  &PointBase<N, U3, S>,
+                              up:      &ColumnVector<N, U3, S>,
+                              scaling: N)
+                              -> Self {
+                Self::from_isometry(IsometryBase::<_, _, _, $Rot>::look_at_lh(eye, target, up), scaling)
+            }
+        }
+    }
+);
+
+similarity_construction_impl!(RotationBase<N, U3, SR>, RotationBase, U3, U3);
+similarity_construction_impl!(UnitQuaternionBase<N, SR>, UnitQuaternionBase, U4, U1);

src/geometry/similarity_conversion.rs

@@ -0,0 +1,163 @@
+use alga::general::{Real, SubsetOf, SupersetOf};
+use alga::linear::Rotation;
+
+use core::{SquareMatrix, OwnedSquareMatrix};
+use core::dimension::{DimName, DimNameAdd, DimNameSum, U1};
+use core::storage::OwnedStorage;
+use core::allocator::{Allocator, OwnedAllocator};
+
+use geometry::{PointBase, TranslationBase, IsometryBase, SimilarityBase, TransformBase, SuperTCategoryOf, TAffine};
+
+/*
+ * This file provides the following conversions:
+ * =============================================
+ *
+ * SimilarityBase -> SimilarityBase
+ * SimilarityBase -> TransformBase
+ * SimilarityBase -> Matrix (homogeneous)
+ */
+
+
+impl<N1, N2, D: DimName, SA, SB, R1, R2> SubsetOf<SimilarityBase<N2, D, SB, R2>> for SimilarityBase<N1, D, SA, R1>
+    where N1: Real + SubsetOf<N2>,
+          N2: Real + SupersetOf<N1>,
+          R1: Rotation<PointBase<N1, D, SA>> + SubsetOf<R2>,
+          R2: Rotation<PointBase<N2, D, SB>>,
+          SA: OwnedStorage<N1, D, U1>,
+          SB: OwnedStorage<N2, D, U1>,
+          SA::Alloc: OwnedAllocator<N1, D, U1, SA>,
+          SB::Alloc: OwnedAllocator<N2, D, U1, SB> {
+    #[inline]
+    fn to_superset(&self) -> SimilarityBase<N2, D, SB, R2> {
+        SimilarityBase::from_isometry(
+            self.isometry.to_superset(),
+            self.scaling().to_superset()
+        )
+    }
+
+    #[inline]
+    fn is_in_subset(sim: &SimilarityBase<N2, D, SB, R2>) -> bool {
+        ::is_convertible::<_, IsometryBase<N1, D, SA, R1>>(&sim.isometry) &&
+        ::is_convertible::<_, N1>(&sim.scaling())
+    }
+
+    #[inline]
+    unsafe fn from_superset_unchecked(sim: &SimilarityBase<N2, D, SB, R2>) -> Self {
+        SimilarityBase::from_isometry(
+            sim.isometry.to_subset_unchecked(),
+            sim.scaling().to_subset_unchecked()
+        )
+    }
+}
+
+
+impl<N1, N2, D, SA, SB, R, C> SubsetOf<TransformBase<N2, D, SB, C>> for SimilarityBase<N1, D, SA, R>
+    where N1: Real,
+          N2: Real + SupersetOf<N1>,
+          SA: OwnedStorage<N1, D, U1>,
+          SB: OwnedStorage<N2, DimNameSum<D, U1>, DimNameSum<D, U1>>,
+          C:  SuperTCategoryOf<TAffine>,
+          R: Rotation<PointBase<N1, D, SA>> +
+             SubsetOf<OwnedSquareMatrix<N1, DimNameSum<D, U1>, SA::Alloc>> + // needed by: .to_homogeneous()
+             SubsetOf<SquareMatrix<N2, DimNameSum<D, U1>, SB>>,              // needed by: ::convert_unchecked(mm)
+          D:  DimNameAdd<U1>,
+          SA::Alloc: OwnedAllocator<N1, D, U1, SA> +
+                     Allocator<N1, D, D> +                                 // needed by R
+                     Allocator<N1, DimNameSum<D, U1>, DimNameSum<D, U1>> + // needed by: .to_homogeneous()
+                     Allocator<N2, DimNameSum<D, U1>, DimNameSum<D, U1>>,  // needed by R
+          SB::Alloc: OwnedAllocator<N2, DimNameSum<D, U1>, DimNameSum<D, U1>, SB> +
+                     Allocator<N2, D, D> +  // needed by: mm.fixed_slice_mut
+                     Allocator<N2, D, U1> + // needed by: m.fixed_slice
+                     Allocator<N2, U1, D> { // needed by: m.fixed_slice
+    #[inline]
+    fn to_superset(&self) -> TransformBase<N2, D, SB, C> {
+        TransformBase::from_matrix_unchecked(self.to_homogeneous().to_superset())
+    }
+
+    #[inline]
+    fn is_in_subset(t: &TransformBase<N2, D, SB, C>) -> bool {
+        <Self as SubsetOf<_>>::is_in_subset(t.matrix())
+    }
+
+    #[inline]
+    unsafe fn from_superset_unchecked(t: &TransformBase<N2, D, SB, C>) -> Self {
+        Self::from_superset_unchecked(t.matrix())
+    }
+}
+
+
+impl<N1, N2, D, SA, SB, R> SubsetOf<SquareMatrix<N2, DimNameSum<D, U1>, SB>> for SimilarityBase<N1, D, SA, R>
+    where N1: Real,
+          N2: Real + SupersetOf<N1>,
+          SA: OwnedStorage<N1, D, U1>,
+          SB: OwnedStorage<N2, DimNameSum<D, U1>, DimNameSum<D, U1>>,
+          R: Rotation<PointBase<N1, D, SA>> +
+             SubsetOf<OwnedSquareMatrix<N1, DimNameSum<D, U1>, SA::Alloc>> + // needed by: .to_homogeneous()
+             SubsetOf<SquareMatrix<N2, DimNameSum<D, U1>, SB>>,              // needed by: ::convert_unchecked(mm)
+          D:  DimNameAdd<U1>,
+          SA::Alloc: OwnedAllocator<N1, D, U1, SA> +
+                     Allocator<N1, D, D> +                                 // needed by R
+                     Allocator<N1, DimNameSum<D, U1>, DimNameSum<D, U1>> + // needed by: .to_homogeneous()
+                     Allocator<N2, DimNameSum<D, U1>, DimNameSum<D, U1>>,  // needed by R
+          SB::Alloc: OwnedAllocator<N2, DimNameSum<D, U1>, DimNameSum<D, U1>, SB> +
+                     Allocator<N2, D, D>  + // needed by: mm.fixed_slice_mut
+                     Allocator<N2, D, U1> + // needed by: m.fixed_slice
+                     Allocator<N2, U1, D> { // needed by: m.fixed_slice
+    #[inline]
+    fn to_superset(&self) -> SquareMatrix<N2, DimNameSum<D, U1>, SB> {
+        self.to_homogeneous().to_superset()
+    }
+
+    #[inline]
+    fn is_in_subset(m: &SquareMatrix<N2, DimNameSum<D, U1>, SB>) -> bool {
+        let mut rot = m.fixed_slice::<D, D>(0, 0).clone_owned();
+        if rot.fixed_columns_mut::<U1>(0).try_normalize_mut(N2::zero()).is_some() &&
+           rot.fixed_columns_mut::<U1>(1).try_normalize_mut(N2::zero()).is_some() &&
+           rot.fixed_columns_mut::<U1>(2).try_normalize_mut(N2::zero()).is_some() {
+
+            // FIXME: could we avoid explicit the computation of the determinant?
+            // (its sign is needed to see if the scaling factor is negative).
+            if rot.determinant() < N2::zero() {
+                rot.fixed_columns_mut::<U1>(0).neg_mut();
+                rot.fixed_columns_mut::<U1>(1).neg_mut();
+                rot.fixed_columns_mut::<U1>(2).neg_mut();
+            }
+
+            let bottom = m.fixed_slice::<U1, D>(D::dim(), 0);
+            // Scalar types agree.
+            m.iter().all(|e| SupersetOf::<N1>::is_in_subset(e)) &&
+            // The normalized block part is a rotation.
+            // rot.is_special_orthogonal(N2::default_epsilon().sqrt()) &&
+            // The bottom row is (0, 0, ..., 1)
+            bottom.iter().all(|e| e.is_zero()) &&
+            m[(D::dim(), D::dim())] == N2::one()
+        }
+        else {
+            false
+        }
+    }
+
+    #[inline]
+    unsafe fn from_superset_unchecked(m: &SquareMatrix<N2, DimNameSum<D, U1>, SB>) -> Self {
+        let mut mm = m.clone_owned();
+        let na = mm.fixed_slice_mut::<D, U1>(0, 0).normalize_mut();
+        let nb = mm.fixed_slice_mut::<D, U1>(0, 1).normalize_mut();
+        let nc = mm.fixed_slice_mut::<D, U1>(0, 2).normalize_mut();
+
+        let mut scale = (na + nb + nc) / ::convert(3.0); // We take the mean, for robustness.
+
+        // FIXME: could we avoid the explicit computation of the determinant?
+        // (its sign is needed to see if the scaling factor is negative).
+        if mm.fixed_slice::<D, D>(0, 0).determinant() < N2::zero() {
+            mm.fixed_slice_mut::<D, U1>(0, 0).neg_mut();
+            mm.fixed_slice_mut::<D, U1>(0, 1).neg_mut();
+            mm.fixed_slice_mut::<D, U1>(0, 2).neg_mut();
+            scale = -scale;
+        }
+
+        let t = m.fixed_slice::<D, U1>(0, D::dim()).into_owned();
+        let t = TranslationBase::from_vector(::convert_unchecked(t));
+
+        Self::from_parts(t, ::convert_unchecked(mm), ::convert_unchecked(scale))
+    }
+}

src/geometry/similarity_ops.rs

@@ -0,0 +1,512 @@
+use std::ops::{Mul, MulAssign, Div, DivAssign};
+
+use alga::general::Real;
+use alga::linear::Rotation;
+
+use core::ColumnVector;
+use core::dimension::{DimName, U1, U3, U4};
+use core::storage::OwnedStorage;
+use core::allocator::OwnedAllocator;
+
+use geometry::{PointBase, RotationBase, SimilarityBase, TranslationBase, UnitQuaternionBase, IsometryBase};
+
+// FIXME: there are several cloning of rotations that we could probably get rid of (but we didn't
+// yet because that would require to add a bound like `where for<'a, 'b> &'a R: Mul<&'b R, Output = R>`
+// which is quite ugly.
+
+/*
+ *
+ * In this file, we provide:
+ * =========================
+ *
+ *
+ * (Operators)
+ *
+ * SimilarityBase × SimilarityBase
+ * SimilarityBase × R
+ * SimilarityBase × IsometryBase
+ *
+ * IsometryBase × SimilarityBase
+ * IsometryBase ÷ SimilarityBase
+ *
+ *
+ * SimilarityBase ÷ SimilarityBase
+ * SimilarityBase ÷ R
+ * SimilarityBase ÷ IsometryBase
+ *
+ * SimilarityBase × PointBase
+ * SimilarityBase × ColumnVector
+ *
+ *
+ * SimilarityBase  × TranslationBase
+ * TranslationBase × SimilarityBase
+ *
+ * NOTE: The following are provided explicitly because we can't have R × SimilarityBase.
+ * RotationBase   × SimilarityBase<RotationBase>
+ * UnitQuaternion × SimilarityBase<UnitQuaternion>
+ *
+ * RotationBase   ÷ SimilarityBase<RotationBase>
+ * UnitQuaternion ÷ SimilarityBase<UnitQuaternion>
+ *
+ * (Assignment Operators)
+ *
+ * SimilarityBase ×= TranslationBase
+ *
+ * SimilarityBase ×= SimilarityBase
+ * SimilarityBase ×= IsometryBase
+ * SimilarityBase ×= R
+ *
+ * SimilarityBase ÷= SimilarityBase
+ * SimilarityBase ÷= IsometryBase
+ * SimilarityBase ÷= R
+ *
+ */
+
+
+// XXX: code duplication: those macros are the same as for the isometry.
+macro_rules! similarity_binop_impl(
+    ($Op: ident, $op: ident;
+     $lhs: ident: $Lhs: ty, $rhs: ident: $Rhs: ty, Output = $Output: ty;
+     $action: expr; $($lives: tt),*) => {
+        impl<$($lives ,)* N, D: DimName, S, R> $Op<$Rhs> for $Lhs
+            where N: Real,
+                  S: OwnedStorage<N, D, U1>,
+                  R: Rotation<PointBase<N, D, S>>,
+                  S::Alloc: OwnedAllocator<N, D, U1, S> {
+            type Output = $Output;
+
+            #[inline]
+            fn $op($lhs, $rhs: $Rhs) -> Self::Output {
+                $action
+            }
+        }
+    }
+);
+
+macro_rules! similarity_binop_impl_all(
+    ($Op: ident, $op: ident;
+     $lhs: ident: $Lhs: ty, $rhs: ident: $Rhs: ty, Output = $Output: ty;
+     [val val] => $action_val_val: expr;
+     [ref val] => $action_ref_val: expr;
+     [val ref] => $action_val_ref: expr;
+     [ref ref] => $action_ref_ref: expr;) => {
+        similarity_binop_impl!(
+            $Op, $op;
+            $lhs: $Lhs, $rhs: $Rhs, Output = $Output;
+            $action_val_val; );
+
+        similarity_binop_impl!(
+            $Op, $op;
+            $lhs: &'a $Lhs, $rhs: $Rhs, Output = $Output;
+            $action_ref_val; 'a);
+
+        similarity_binop_impl!(
+            $Op, $op;
+            $lhs: $Lhs, $rhs: &'b $Rhs, Output = $Output;
+            $action_val_ref; 'b);
+
+        similarity_binop_impl!(
+            $Op, $op;
+            $lhs: &'a $Lhs, $rhs: &'b $Rhs, Output = $Output;
+            $action_ref_ref; 'a, 'b);
+    }
+);
+
+macro_rules! similarity_binop_assign_impl_all(
+    ($OpAssign: ident, $op_assign: ident;
+     $lhs: ident: $Lhs: ty, $rhs: ident: $Rhs: ty;
+     [val] => $action_val: expr;
+     [ref] => $action_ref: expr;) => {
+        impl<N, D: DimName, S, R> $OpAssign<$Rhs> for $Lhs
+            where N: Real,
+                  S: OwnedStorage<N, D, U1>,
+                  R: Rotation<PointBase<N, D, S>>,
+                  S::Alloc: OwnedAllocator<N, D, U1, S> {
+            #[inline]
+            fn $op_assign(&mut $lhs, $rhs: $Rhs) {
+                $action_val
+            }
+        }
+
+        impl<'b, N, D: DimName, S, R> $OpAssign<&'b $Rhs> for $Lhs
+            where N: Real,
+                  S: OwnedStorage<N, D, U1>,
+                  R: Rotation<PointBase<N, D, S>>,
+                  S::Alloc: OwnedAllocator<N, D, U1, S> {
+            #[inline]
+            fn $op_assign(&mut $lhs, $rhs: &'b $Rhs) {
+                $action_ref
+            }
+        }
+    }
+);
+
+// SimilarityBase × SimilarityBase
+// SimilarityBase ÷ SimilarityBase
+similarity_binop_impl_all!(
+    Mul, mul;
+    self: SimilarityBase<N, D, S, R>, rhs: SimilarityBase<N, D, S, R>, Output = SimilarityBase<N, D, S, R>;
+    [val val] => &self * &rhs;
+    [ref val] =>  self * &rhs;
+    [val ref] => &self *  rhs;
+    [ref ref] => {
+        let mut res = self * &rhs.isometry;
+        res.prepend_scaling_mut(rhs.scaling());
+        res
+    };
+);
+
+
+similarity_binop_impl_all!(
+    Div, div;
+    self: SimilarityBase<N, D, S, R>, rhs: SimilarityBase<N, D, S, R>, Output = SimilarityBase<N, D, S, R>;
+    [val val] => self * rhs.inverse();
+    [ref val] => self * rhs.inverse();
+    [val ref] => self * rhs.inverse();
+    [ref ref] => self * rhs.inverse();
+);
+
+
+// SimilarityBase ×= TranslationBase
+similarity_binop_assign_impl_all!(
+    MulAssign, mul_assign;
+    self: SimilarityBase<N, D, S, R>, rhs: TranslationBase<N, D, S>;
+    [val] => *self *= &rhs;
+    [ref] => {
+        let shift = self.isometry.rotation.transform_vector(&rhs.vector) * self.scaling();
+        self.isometry.translation.vector += shift;
+    };
+);
+
+
+// SimilarityBase ×= SimilarityBase
+// SimilarityBase ÷= SimilarityBase
+similarity_binop_assign_impl_all!(
+    MulAssign, mul_assign;
+    self: SimilarityBase<N, D, S, R>, rhs: SimilarityBase<N, D, S, R>;
+    [val] => *self *= &rhs;
+    [ref] => {
+        *self *= &rhs.isometry;
+        self.prepend_scaling_mut(rhs.scaling());
+    };
+);
+
+
+similarity_binop_assign_impl_all!(
+    DivAssign, div_assign;
+    self: SimilarityBase<N, D, S, R>, rhs: SimilarityBase<N, D, S, R>;
+    [val] => *self /= &rhs;
+    // FIXME: don't invert explicitly.
+    [ref] => *self *= rhs.inverse();
+);
+
+
+// SimilarityBase ×= IsometryBase
+// SimilarityBase ÷= IsometryBase
+similarity_binop_assign_impl_all!(
+    MulAssign, mul_assign;
+    self: SimilarityBase<N, D, S, R>, rhs: IsometryBase<N, D, S, R>;
+    [val] => *self *= &rhs;
+    [ref] => {
+        let shift = self.isometry.rotation.transform_vector(&rhs.translation.vector) * self.scaling();
+        self.isometry.translation.vector += shift;
+        self.isometry.rotation *= rhs.rotation.clone();
+    };
+);
+
+
+similarity_binop_assign_impl_all!(
+    DivAssign, div_assign;
+    self: SimilarityBase<N, D, S, R>, rhs: IsometryBase<N, D, S, R>;
+    [val] => *self /= &rhs;
+    // FIXME: don't invert explicitly.
+    [ref] => *self *= rhs.inverse();
+);
+
+
+// SimilarityBase ×= R
+// SimilarityBase ÷= R
+similarity_binop_assign_impl_all!(
+    MulAssign, mul_assign;
+    self: SimilarityBase<N, D, S, R>, rhs: R;
+    [val] => self.isometry.rotation *= rhs;
+    [ref] => self.isometry.rotation *= rhs.clone();
+);
+
+
+similarity_binop_assign_impl_all!(
+    DivAssign, div_assign;
+    self: SimilarityBase<N, D, S, R>, rhs: R;
+    // FIXME: don't invert explicitly?
+    [val] => *self *= rhs.inverse();
+    [ref] => *self *= rhs.inverse();
+);
+
+
+// SimilarityBase × R
+// SimilarityBase ÷ R
+similarity_binop_impl_all!(
+    Mul, mul;
+    self: SimilarityBase<N, D, S, R>, rhs: R, Output = SimilarityBase<N, D, S, R>;
+    [val val] => {
+        let scaling = self.scaling();
+        SimilarityBase::from_isometry(self.isometry * rhs, scaling)
+    };
+    [ref val] => SimilarityBase::from_isometry(&self.isometry * rhs, self.scaling());
+    [val ref] => {
+        let scaling = self.scaling();
+        SimilarityBase::from_isometry(self.isometry * rhs, scaling)
+    };
+    [ref ref] => SimilarityBase::from_isometry(&self.isometry * rhs, self.scaling());
+);
+
+
+
+similarity_binop_impl_all!(
+    Div, div;
+    self: SimilarityBase<N, D, S, R>, rhs: R, Output = SimilarityBase<N, D, S, R>;
+    [val val] => {
+        let scaling = self.scaling();
+        SimilarityBase::from_isometry(self.isometry / rhs, scaling)
+    };
+    [ref val] => SimilarityBase::from_isometry(&self.isometry / rhs, self.scaling());
+    [val ref] => {
+        let scaling = self.scaling();
+        SimilarityBase::from_isometry(self.isometry / rhs, scaling)
+    };
+    [ref ref] => SimilarityBase::from_isometry(&self.isometry / rhs, self.scaling());
+);
+
+// SimilarityBase × IsometryBase
+// SimilarityBase ÷ IsometryBase
+similarity_binop_impl_all!(
+    Mul, mul;
+    self: SimilarityBase<N, D, S, R>, rhs: IsometryBase<N, D, S, R>, Output = SimilarityBase<N, D, S, R>;
+    [val val] => &self * &rhs;
+    [ref val] => self * &rhs;
+    [val ref] => &self * rhs;
+    [ref ref] => {
+        let shift = self.isometry.rotation.transform_vector(&rhs.translation.vector) * self.scaling();
+        SimilarityBase::from_parts(
+            TranslationBase::from_vector(&self.isometry.translation.vector + shift),
+            self.isometry.rotation.clone() * rhs.rotation.clone(),
+            self.scaling())
+    };
+);
+
+
+
+similarity_binop_impl_all!(
+    Div, div;
+    self: SimilarityBase<N, D, S, R>, rhs: IsometryBase<N, D, S, R>, Output = SimilarityBase<N, D, S, R>;
+    [val val] => self * rhs.inverse();
+    [ref val] => self * rhs.inverse();
+    [val ref] => self * rhs.inverse();
+    [ref ref] => self * rhs.inverse();
+);
+
+// IsometryBase × SimilarityBase
+// IsometryBase ÷ SimilarityBase
+similarity_binop_impl_all!(
+    Mul, mul;
+    self: IsometryBase<N, D, S, R>, rhs: SimilarityBase<N, D, S, R>, Output = SimilarityBase<N, D, S, R>;
+    [val val] => {
+        let scaling = rhs.scaling();
+        SimilarityBase::from_isometry(self * rhs.isometry, scaling)
+    };
+    [ref val] => {
+        let scaling = rhs.scaling();
+        SimilarityBase::from_isometry(self * rhs.isometry, scaling)
+    };
+    [val ref] => {
+        let scaling = rhs.scaling();
+        SimilarityBase::from_isometry(self * &rhs.isometry, scaling)
+    };
+    [ref ref] => {
+        let scaling = rhs.scaling();
+        SimilarityBase::from_isometry(self * &rhs.isometry, scaling)
+    };
+);
+
+
+similarity_binop_impl_all!(
+    Div, div;
+    self: IsometryBase<N, D, S, R>, rhs: SimilarityBase<N, D, S, R>, Output = SimilarityBase<N, D, S, R>;
+    [val val] => self * rhs.inverse();
+    [ref val] => self * rhs.inverse();
+    [val ref] => self * rhs.inverse();
+    [ref ref] => self * rhs.inverse();
+);
+
+
+// SimilarityBase × PointBase
+similarity_binop_impl_all!(
+    Mul, mul;
+    self: SimilarityBase<N, D, S, R>, right: PointBase<N, D, S>, Output = PointBase<N, D, S>;
+    [val val] => {
+        let scaling = self.scaling();
+        self.isometry.translation * (self.isometry.rotation.transform_point(&right) * scaling)
+    };
+    [ref val] => &self.isometry.translation * (self.isometry.rotation.transform_point(&right) * self.scaling());
+    [val ref] => {
+        let scaling = self.scaling();
+        self.isometry.translation * (self.isometry.rotation.transform_point(right) * scaling)
+    };
+    [ref ref] => &self.isometry.translation * (self.isometry.rotation.transform_point(right) * self.scaling());
+);
+
+
+// SimilarityBase × Vector
+similarity_binop_impl_all!(
+    Mul, mul;
+    self: SimilarityBase<N, D, S, R>, right: ColumnVector<N, D, S>, Output = ColumnVector<N, D, S>;
+    [val val] => self.isometry.rotation.transform_vector(&right) * self.scaling();
+    [ref val] => self.isometry.rotation.transform_vector(&right) * self.scaling();
+    [val ref] => self.isometry.rotation.transform_vector(right) * self.scaling();
+    [ref ref] => self.isometry.rotation.transform_vector(right) * self.scaling();
+);
+
+
+// SimilarityBase × TranslationBase
+similarity_binop_impl_all!(
+    Mul, mul;
+    self: SimilarityBase<N, D, S, R>, right: TranslationBase<N, D, S>, Output = SimilarityBase<N, D, S, R>;
+    [val val] => &self * &right;
+    [ref val] => self * &right;
+    [val ref] => &self * right;
+    [ref ref] => {
+        let shift = self.isometry.rotation.transform_vector(&right.vector) * self.scaling();
+        SimilarityBase::from_parts(
+            TranslationBase::from_vector(&self.isometry.translation.vector + shift),
+            self.isometry.rotation.clone(),
+            self.scaling())
+    };
+);
+
+
+// TranslationBase × SimilarityBase
+similarity_binop_impl_all!(
+    Mul, mul;
+    self: TranslationBase<N, D, S>, right: SimilarityBase<N, D, S, R>, Output = SimilarityBase<N, D, S, R>;
+    [val val] => {
+        let scaling = right.scaling();
+        SimilarityBase::from_isometry(self * right.isometry, scaling)
+    };
+    [ref val] => {
+        let scaling = right.scaling();
+        SimilarityBase::from_isometry(self * right.isometry, scaling)
+    };
+    [val ref] => SimilarityBase::from_isometry(self * &right.isometry, right.scaling());
+    [ref ref] => SimilarityBase::from_isometry(self * &right.isometry, right.scaling());
+);
+
+
+macro_rules! similarity_from_composition_impl(
+    ($Op: ident, $op: ident;
+     ($R1: ty, $C1: ty),($R2: ty, $C2: ty) $(for $Dims: ident: $DimsBound: ident),*;
+     $lhs: ident: $Lhs: ty, $rhs: ident: $Rhs: ty, Output = $Output: ty;
+     $action: expr; $($lives: tt),*) => {
+        impl<$($lives ,)* N $(, $Dims: $DimsBound)*, SA, SB> $Op<$Rhs> for $Lhs
+            where N: Real,
+                  SA: OwnedStorage<N, $R1, $C1>,
+                  SB: OwnedStorage<N, $R2, $C2, Alloc = SA::Alloc>,
+                  SA::Alloc: OwnedAllocator<N, $R1, $C1, SA>,
+                  SB::Alloc: OwnedAllocator<N, $R2, $C2, SB> {
+            type Output = $Output;
+
+            #[inline]
+            fn $op($lhs, $rhs: $Rhs) -> Self::Output {
+                $action
+            }
+        }
+    }
+);
+
+macro_rules! similarity_from_composition_impl_all(
+    ($Op: ident, $op: ident;
+     ($R1: ty, $C1: ty),($R2: ty, $C2: ty) $(for $Dims: ident: $DimsBound: ident),*;
+     $lhs: ident: $Lhs: ty, $rhs: ident: $Rhs: ty, Output = $Output: ty;
+     [val val] => $action_val_val: expr;
+     [ref val] => $action_ref_val: expr;
+     [val ref] => $action_val_ref: expr;
+     [ref ref] => $action_ref_ref: expr;) => {
+
+        similarity_from_composition_impl!(
+            $Op, $op;
+            ($R1, $C1),($R2, $C2) $(for $Dims: $DimsBound),*;
+            $lhs: $Lhs, $rhs: $Rhs, Output = $Output;
+            $action_val_val; );
+
+        similarity_from_composition_impl!(
+            $Op, $op;
+            ($R1, $C1),($R2, $C2) $(for $Dims: $DimsBound),*;
+            $lhs: &'a $Lhs, $rhs: $Rhs, Output = $Output;
+            $action_ref_val; 'a);
+
+        similarity_from_composition_impl!(
+            $Op, $op;
+            ($R1, $C1),($R2, $C2) $(for $Dims: $DimsBound),*;
+            $lhs: $Lhs, $rhs: &'b $Rhs, Output = $Output;
+            $action_val_ref; 'b);
+
+        similarity_from_composition_impl!(
+            $Op, $op;
+            ($R1, $C1),($R2, $C2) $(for $Dims: $DimsBound),*;
+            $lhs: &'a $Lhs, $rhs: &'b $Rhs, Output = $Output;
+            $action_ref_ref; 'a, 'b);
+    }
+);
+
+
+// RotationBase × SimilarityBase
+similarity_from_composition_impl_all!(
+    Mul, mul;
+    (D, D), (D, U1) for D: DimName;
+    self: RotationBase<N, D, SA>, right: SimilarityBase<N, D, SB, RotationBase<N, D, SA>>,
+    Output = SimilarityBase<N, D, SB, RotationBase<N, D, SA>>;
+    [val val] => &self * &right;
+    [ref val] =>  self * &right;
+    [val ref] => &self *  right;
+    [ref ref] => SimilarityBase::from_isometry(self * &right.isometry, right.scaling());
+);
+
+
+// RotationBase ÷ SimilarityBase
+similarity_from_composition_impl_all!(
+    Div, div;
+    (D, D), (D, U1) for D: DimName;
+    self: RotationBase<N, D, SA>, right: SimilarityBase<N, D, SB, RotationBase<N, D, SA>>,
+    Output = SimilarityBase<N, D, SB, RotationBase<N, D, SA>>;
+    // FIXME: don't call iverse explicitly?
+    [val val] => self * right.inverse();
+    [ref val] => self * right.inverse();
+    [val ref] => self * right.inverse();
+    [ref ref] => self * right.inverse();
+);
+
+
+// UnitQuaternion × SimilarityBase
+similarity_from_composition_impl_all!(
+    Mul, mul;
+    (U4, U1), (U3, U1);
+    self: UnitQuaternionBase<N, SA>, right: SimilarityBase<N, U3, SB, UnitQuaternionBase<N, SA>>,
+    Output = SimilarityBase<N, U3, SB, UnitQuaternionBase<N, SA>>;
+    [val val] => &self * &right;
+    [ref val] =>  self * &right;
+    [val ref] => &self *  right;
+    [ref ref] => SimilarityBase::from_isometry(self * &right.isometry, right.scaling());
+);
+
+
+// UnitQuaternion ÷ SimilarityBase
+similarity_from_composition_impl_all!(
+    Div, div;
+    (U4, U1), (U3, U1);
+    self: UnitQuaternionBase<N, SA>, right: SimilarityBase<N, U3, SB, UnitQuaternionBase<N, SA>>,
+    Output = SimilarityBase<N, U3, SB, UnitQuaternionBase<N, SA>>;
+    // FIXME: don't call inverse explicitly?
+    [val val] => self * right.inverse();
+    [ref val] => self * right.inverse();
+    [val ref] => self * right.inverse();
+    [ref ref] => self * right.inverse();
+);

src/geometry/transform.rs

@@ -0,0 +1,310 @@
+use std::any::Any;
+use std::fmt::Debug;
+use std::marker::PhantomData;
+use approx::ApproxEq;
+
+use alga::general::Field;
+
+use core::{Scalar, SquareMatrix, OwnedSquareMatrix};
+use core::dimension::{DimName, DimNameAdd, DimNameSum, U1};
+use core::storage::{Storage, StorageMut};
+use core::allocator::Allocator;
+
+/// Trait implemented by phantom types identifying the projective transformation type.
+///
+/// NOTE: this trait is not intended to be implementable outside of the `nalgebra` crate.
+pub trait TCategory: Any + Debug + Copy + PartialEq + Send {
+    /// Indicates whether a `Transform` with the category `Self` has a bottom-row different from
+    /// `0 0 .. 1`.
+    #[inline]
+    fn has_normalizer() -> bool {
+        true
+    }
+
+    /// Checks that the given matrix is a valid homogeneous representation of an element of the
+    /// category `Self`.
+    fn check_homogeneous_invariants<N, D, S>(mat: &SquareMatrix<N, D, S>) -> bool
+        where N: Scalar + Field + ApproxEq,
+              D: DimName,
+              S: Storage<N, D, D>,
+              N::Epsilon: Copy;
+}
+
+/// Traits that gives the `Transform` category that is compatible with the result of the
+/// multiplication of transformations with categories `Self` and `Other`.
+pub trait TCategoryMul<Other: TCategory>: TCategory {
+    /// The transform category that results from the multiplication of a `Transform<Self>` to a
+    /// `Transform<Other>`. This is usually equal to `Self` or `Other`, whichever is the most
+    /// general category.
+    type Representative: TCategory;
+}
+
+/// Indicates that `Self` is a more general `Transform` category than `Other`.
+pub trait SuperTCategoryOf<Other: TCategory>: TCategory { }
+
+/// Indicates that `Self` is a more specific `Transform` category than `Other`.
+///
+/// Automatically implemented based on `SuperTCategoryOf`.
+pub trait SubTCategoryOf<Other: TCategory>: TCategory { }
+impl<T1, T2> SubTCategoryOf<T2> for T1
+where T1: TCategory,
+      T2: SuperTCategoryOf<T1> {
+}
+
+/// Tag representing the most general (not necessarily inversible) `Transform` type.
+#[derive(Debug, Copy, Clone, Hash, PartialEq, Eq)]
+#[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
+pub enum TGeneral { }
+
+/// Tag representing the most general inversible `Transform` type.
+#[derive(Debug, Copy, Clone, Hash, PartialEq, Eq)]
+#[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
+pub enum TProjective { }
+
+/// Tag representing an affine `Transform`. Its bottom-row is equal to `(0, 0 ... 0, 1)`.
+#[derive(Debug, Copy, Clone, Hash, PartialEq, Eq)]
+#[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
+pub enum TAffine { }
+
+impl TCategory for TGeneral {
+    #[inline]
+    fn check_homogeneous_invariants<N, D, S>(_: &SquareMatrix<N, D, S>) -> bool
+        where N: Scalar + Field + ApproxEq,
+              D: DimName,
+              S: Storage<N, D, D>,
+              N::Epsilon: Copy {
+        true
+    }
+}
+
+impl TCategory for TProjective {
+    #[inline]
+    fn check_homogeneous_invariants<N, D, S>(mat: &SquareMatrix<N, D, S>) -> bool
+        where N: Scalar + Field + ApproxEq,
+              D: DimName,
+              S: Storage<N, D, D>,
+              N::Epsilon: Copy {
+        mat.is_invertible()
+    }
+}
+
+impl TCategory for TAffine {
+    #[inline]
+    fn has_normalizer() -> bool {
+        false
+    }
+
+    #[inline]
+    fn check_homogeneous_invariants<N, D, S>(mat: &SquareMatrix<N, D, S>) -> bool
+        where N: Scalar + Field + ApproxEq,
+              D: DimName,
+              S: Storage<N, D, D>,
+              N::Epsilon: Copy {
+        mat.is_invertible()                   &&
+        mat[(D::dim(), D::dim())] == N::one() &&
+        (0 .. D::dim()).all(|i| mat[(D::dim(), i)].is_zero())
+    }
+}
+
+macro_rules! category_mul_impl(
+    ($($a: ident * $b: ident => $c: ty);* $(;)*) => {$(
+        impl TCategoryMul<$a> for $b {
+            type Representative = $c;
+        }
+    )*}
+);
+
+// We require stability uppon multiplication.
+impl<T: TCategory> TCategoryMul<T> for T {
+    type Representative = T;
+}
+
+category_mul_impl!(
+    // TGeneral * TGeneral     => TGeneral;
+    TGeneral * TProjective  => TGeneral;
+    TGeneral * TAffine      => TGeneral;
+
+    TProjective * TGeneral     => TGeneral;
+    // TProjective * TProjective  => TProjective;
+    TProjective * TAffine      => TProjective;
+
+    TAffine * TGeneral     => TGeneral;
+    TAffine * TProjective  => TProjective;
+    // TAffine * TAffine      => TAffine;
+);
+
+macro_rules! super_tcategory_impl(
+    ($($a: ident >= $b: ident);* $(;)*) => {$(
+        impl SuperTCategoryOf<$b> for $a { }
+    )*}
+);
+
+impl<T: TCategory> SuperTCategoryOf<T> for T { }
+
+super_tcategory_impl!(
+    TGeneral >= TProjective;
+    TGeneral >= TAffine;
+    TProjective >= TAffine;
+);
+
+
+/// A transformation matrix that owns its data.
+pub type OwnedTransform<N, D, A, C>
+    = TransformBase<N, D, <A as Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1>>>::Buffer, C>;
+
+
+/// A transformation matrix in homogeneous coordinates.
+///
+/// It is stored as a matrix with dimensions `(D + 1, D + 1)`, e.g., it stores a 4x4 matrix for a
+/// 3D transformation.
+#[repr(C)]
+#[derive(Debug, Clone, Copy)] // FIXME: Hash
+#[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
+pub struct TransformBase<N: Scalar, D: DimNameAdd<U1>, S, C: TCategory> {
+    matrix:   SquareMatrix<N, DimNameSum<D, U1>, S>,
+
+    #[cfg_attr(feature = "serde-serialize", serde(skip_serializing, skip_deserializing))]
+    _phantom: PhantomData<C>
+}
+
+// XXX: for some reasons, implementing Clone and Copy manually causes an ICE…
+
+impl<N, D, S, C: TCategory> Eq for TransformBase<N, D, S, C>
+    where N: Scalar + Eq,
+          D: DimNameAdd<U1>,
+          S: Storage<N, DimNameSum<D, U1>, DimNameSum<D, U1>> { }
+
+impl<N, D, S, C: TCategory> PartialEq for TransformBase<N, D, S, C>
+    where N: Scalar,
+          D: DimNameAdd<U1>,
+          S: Storage<N, DimNameSum<D, U1>, DimNameSum<D, U1>> {
+    #[inline]
+    fn eq(&self, right: &Self) -> bool {
+        self.matrix == right.matrix
+    }
+}
+
+impl<N, D, S, C: TCategory> TransformBase<N, D, S, C>
+    where N: Scalar,
+          D: DimNameAdd<U1>,
+          S: Storage<N, DimNameSum<D, U1>, DimNameSum<D, U1>> {
+    /// Creates a new transformation from the given homogeneous matrix. The transformation category
+    /// of `Self` is not checked to be verified by the given matrix.
+    #[inline]
+    pub fn from_matrix_unchecked(matrix: SquareMatrix<N, DimNameSum<D, U1>, S>) -> Self {
+        TransformBase {
+            matrix:   matrix,
+            _phantom: PhantomData
+        }
+    }
+
+    /// Moves this transform into one that owns its data.
+    #[inline]
+    pub fn into_owned(self) -> OwnedTransform<N, D, S::Alloc, C> {
+        TransformBase::from_matrix_unchecked(self.matrix.into_owned())
+    }
+
+    /// Clones this transform into one that owns its data.
+    #[inline]
+    pub fn clone_owned(&self) -> OwnedTransform<N, D, S::Alloc, C> {
+        TransformBase::from_matrix_unchecked(self.matrix.clone_owned())
+    }
+
+    /// The underlying matrix.
+    #[inline]
+    pub fn unwrap(self) -> SquareMatrix<N, DimNameSum<D, U1>, S> {
+        self.matrix
+    }
+
+    /// A reference to the underlynig matrix.
+    #[inline]
+    pub fn matrix(&self) -> &SquareMatrix<N, DimNameSum<D, U1>, S> {
+        &self.matrix
+    }
+
+    /// A mutable reference to the underlying matrix.
+    ///
+    /// It is `_unchecked` because direct modifications of this matrix may break invariants
+    /// identified by this transformation category.
+    #[inline]
+    pub fn matrix_mut_unchecked(&mut self) -> &mut SquareMatrix<N, DimNameSum<D, U1>, S> {
+        &mut self.matrix
+    }
+
+    /// Sets the category of this transform.
+    ///
+    /// This can be done only if the new category is more general than the current one, e.g., a
+    /// transform with category `TProjective` cannot be converted to a transform with category
+    /// `TAffine` because not all projective transformations are affine (the other way-round is
+    /// valid though).
+    #[inline]
+    pub fn set_category<CNew: SuperTCategoryOf<C>>(self) -> TransformBase<N, D, S, CNew> {
+        TransformBase::from_matrix_unchecked(self.matrix)
+    }
+
+    /// Converts this transform into its equivalent homogeneous transformation matrix.
+    #[inline]
+    pub fn to_homogeneous(&self) -> OwnedSquareMatrix<N, DimNameSum<D, U1>, S::Alloc> {
+        self.matrix().clone_owned()
+    }
+}
+
+impl<N, D, S, C> TransformBase<N, D, S, C>
+    where N: Scalar + Field + ApproxEq,
+          D: DimNameAdd<U1>,
+          C: TCategory,
+          S: Storage<N, DimNameSum<D, U1>, DimNameSum<D, U1>> {
+    /// Attempts to invert this transformation. You may use `.inverse` instead of this
+    /// transformation has a subcategory of `TProjective`.
+    #[inline]
+    pub fn try_inverse(self) -> Option<OwnedTransform<N, D, S::Alloc, C>> {
+        if let Some(m) = self.matrix.try_inverse() {
+            Some(TransformBase::from_matrix_unchecked(m))
+        }
+        else {
+            None
+        }
+    }
+
+    /// Inverts this transformation. Use `.try_inverse` if this transform has the `TGeneral`
+    /// category (it may not be invertible).
+    #[inline]
+    pub fn inverse(self) -> OwnedTransform<N, D, S::Alloc, C>
+        where C: SubTCategoryOf<TProjective> {
+        // FIXME: specialize for TAffine?
+        TransformBase::from_matrix_unchecked(self.matrix.try_inverse().unwrap())
+    }
+}
+
+impl<N, D, S, C> TransformBase<N, D, S, C>
+    where N: Scalar + Field + ApproxEq,
+          D: DimNameAdd<U1>,
+          C: TCategory,
+          S: StorageMut<N, DimNameSum<D, U1>, DimNameSum<D, U1>> {
+    /// Attempts to invert this transformation in-place. You may use `.inverse_mut` instead of this
+    /// transformation has a subcategory of `TProjective`.
+    #[inline]
+    pub fn try_inverse_mut(&mut self) -> bool {
+        self.matrix.try_inverse_mut()
+    }
+
+    /// Inverts this transformation in-place. Use `.try_inverse_mut` if this transform has the
+    /// `TGeneral` category (it may not be invertible).
+    #[inline]
+    pub fn inverse_mut(&mut self)
+        where C: SubTCategoryOf<TProjective> {
+        let _ = self.matrix.try_inverse_mut();
+    }
+}
+
+impl<N, D, S> TransformBase<N, D, S, TGeneral>
+    where N: Scalar,
+          D: DimNameAdd<U1>,
+          S: Storage<N, DimNameSum<D, U1>, DimNameSum<D, U1>> {
+    /// A mutable reference to underlying matrix. Use `.matrix_mut_unchecked` instead if this
+    /// transformation category is not `TGeneral`.
+    #[inline]
+    pub fn matrix_mut(&mut self) -> &mut SquareMatrix<N, DimNameSum<D, U1>, S> {
+        self.matrix_mut_unchecked()
+    }
+}

src/geometry/transform_alga.rs

@@ -0,0 +1,157 @@
+use approx::ApproxEq;
+
+use alga::general::{AbstractMagma, AbstractGroup, AbstractLoop, AbstractMonoid, AbstractQuasigroup,
+                    AbstractSemigroup, Field, Real, Inverse, Multiplicative, Identity};
+use alga::linear::{Transformation, ProjectiveTransformation};
+
+use core::{Scalar, ColumnVector};
+use core::dimension::{DimNameSum, DimNameAdd, U1};
+use core::storage::OwnedStorage;
+use core::allocator::{Allocator, OwnedAllocator};
+
+use geometry::{PointBase, TransformBase, TCategory, SubTCategoryOf, TProjective};
+
+
+/*
+ *
+ * Algebraic structures.
+ *
+ */
+impl<N, D: DimNameAdd<U1>, S, C> Identity<Multiplicative> for TransformBase<N, D, S, C>
+    where N: Scalar + Field,
+          S: OwnedStorage<N, DimNameSum<D, U1>, DimNameSum<D, U1>>,
+          C: TCategory,
+          S::Alloc: OwnedAllocator<N, DimNameSum<D, U1>, DimNameSum<D, U1>, S> {
+    #[inline]
+    fn identity() -> Self {
+        Self::identity()
+    }
+}
+
+impl<N, D: DimNameAdd<U1>, S, C> Inverse<Multiplicative> for TransformBase<N, D, S, C>
+    where N: Scalar + Field + ApproxEq,
+          S: OwnedStorage<N, DimNameSum<D, U1>, DimNameSum<D, U1>>,
+          C: SubTCategoryOf<TProjective>,
+          S::Alloc: OwnedAllocator<N, DimNameSum<D, U1>, DimNameSum<D, U1>, S> {
+    #[inline]
+    fn inverse(&self) -> Self {
+        self.clone().inverse()
+    }
+
+    #[inline]
+    fn inverse_mut(&mut self) {
+        self.inverse_mut()
+    }
+}
+
+impl<N, D: DimNameAdd<U1>, S, C> AbstractMagma<Multiplicative> for TransformBase<N, D, S, C>
+    where N: Scalar + Field,
+          S: OwnedStorage<N, DimNameSum<D, U1>, DimNameSum<D, U1>>,
+          C: TCategory,
+          S::Alloc: OwnedAllocator<N, DimNameSum<D, U1>, DimNameSum<D, U1>, S> {
+    #[inline]
+    fn operate(&self, rhs: &Self) -> Self {
+        self * rhs
+    }
+}
+
+macro_rules! impl_multiplicative_structures(
+    ($($marker: ident<$operator: ident>),* $(,)*) => {$(
+        impl<N, D: DimNameAdd<U1>, S, C> $marker<$operator> for TransformBase<N, D, S, C>
+            where N: Scalar + Field,
+                  S: OwnedStorage<N, DimNameSum<D, U1>, DimNameSum<D, U1>>,
+                  C: TCategory,
+                  S::Alloc: OwnedAllocator<N, DimNameSum<D, U1>, DimNameSum<D, U1>, S> { }
+    )*}
+);
+
+macro_rules! impl_inversible_multiplicative_structures(
+    ($($marker: ident<$operator: ident>),* $(,)*) => {$(
+        impl<N, D: DimNameAdd<U1>, S, C> $marker<$operator> for TransformBase<N, D, S, C>
+            where N: Scalar + Field + ApproxEq,
+                  S: OwnedStorage<N, DimNameSum<D, U1>, DimNameSum<D, U1>>,
+                  C: SubTCategoryOf<TProjective>,
+                  S::Alloc: OwnedAllocator<N, DimNameSum<D, U1>, DimNameSum<D, U1>, S> { }
+    )*}
+);
+
+impl_multiplicative_structures!(
+    AbstractSemigroup<Multiplicative>,
+    AbstractMonoid<Multiplicative>,
+);
+
+impl_inversible_multiplicative_structures!(
+    AbstractQuasigroup<Multiplicative>,
+    AbstractLoop<Multiplicative>,
+    AbstractGroup<Multiplicative>
+);
+
+/*
+ *
+ * Transformation groups.
+ *
+ */
+impl<N, D: DimNameAdd<U1>, SA, SB, C> Transformation<PointBase<N, D, SB>> for TransformBase<N, D, SA, C>
+    where N:  Real,
+          SA: OwnedStorage<N, DimNameSum<D, U1>, DimNameSum<D, U1>>,
+          SB: OwnedStorage<N, D, U1, Alloc = SA::Alloc>,
+          C:  TCategory,
+          SA::Alloc: OwnedAllocator<N, DimNameSum<D, U1>, DimNameSum<D, U1>, SA> +
+                     Allocator<N, D, D>  +
+                     Allocator<N, D, U1> +
+                     Allocator<N, U1, D>,
+          SB::Alloc: OwnedAllocator<N, D, U1, SB> {
+    #[inline]
+    fn transform_point(&self, pt: &PointBase<N, D, SB>) -> PointBase<N, D, SB> {
+        self * pt
+    }
+
+    #[inline]
+    fn transform_vector(&self, v: &ColumnVector<N, D, SB>) -> ColumnVector<N, D, SB> {
+        self * v
+    }
+}
+
+impl<N, D: DimNameAdd<U1>, SA, SB, C> ProjectiveTransformation<PointBase<N, D, SB>> for TransformBase<N, D, SA, C>
+    where N:  Real,
+          SA: OwnedStorage<N, DimNameSum<D, U1>, DimNameSum<D, U1>>,
+          SB: OwnedStorage<N, D, U1, Alloc = SA::Alloc>,
+          C:  SubTCategoryOf<TProjective>,
+          SA::Alloc: OwnedAllocator<N, DimNameSum<D, U1>, DimNameSum<D, U1>, SA> +
+                     Allocator<N, D, D>  +
+                     Allocator<N, D, U1> +
+                     Allocator<N, U1, D>,
+          SB::Alloc: OwnedAllocator<N, D, U1, SB> {
+    #[inline]
+    fn inverse_transform_point(&self, pt: &PointBase<N, D, SB>) -> PointBase<N, D, SB> {
+        self.inverse() * pt
+    }
+
+    #[inline]
+    fn inverse_transform_vector(&self, v: &ColumnVector<N, D, SB>) -> ColumnVector<N, D, SB> {
+        self.inverse() * v
+    }
+}
+
+// FIXME: we need to implement an SVD for this.
+//
+// impl<N, D: DimNameAdd<U1>, SA, SB, C> AffineTransformation<PointBase<N, D, SB>> for TransformBase<N, D, SA, C>
+//     where N:  Real,
+//           SA: OwnedStorage<N, DimNameSum<D, U1>, DimNameSum<D, U1>>,
+//           SB: OwnedStorage<N, D, U1, Alloc = SA::Alloc>,
+//           C: SubTCategoryOf<TAffine>,
+//           SA::Alloc: OwnedAllocator<N, DimNameSum<D, U1>, DimNameSum<D, U1>, SA> +
+//                      Allocator<N, D, D>  +
+//                      Allocator<N, D, U1> +
+//                      Allocator<N, U1, D>,
+//           SB::Alloc: OwnedAllocator<N, D, U1, SB> {
+//     type PreRotation       = OwnedRotation<N, D, SA::Alloc>;
+//     type NonUniformScaling = OwnedColumnVector<N, D, SA::Alloc>;
+//     type PostRotation      = OwnedRotation<N, D, SA::Alloc>;
+//     type Translation       = OwnedTranslation<N, D, SA::Alloc>;
+// 
+//     #[inline]
+//     fn decompose(&self) -> (Self::Translation, Self::PostRotation, Self::NonUniformScaling, Self::PreRotation) {
+//         unimplemented!()
+//     }
+// }

src/geometry/transform_alias.rs

@@ -0,0 +1,29 @@
+use core::MatrixArray;
+use core::dimension::{U1, U2, U3, DimNameSum};
+
+use geometry::{TransformBase, TGeneral, TProjective, TAffine};
+
+/// A `D`-dimensional general transformation that may not be inversible. Stored as an homogeneous
+/// `(D + 1) × (D + 1)` matrix.
+pub type Transform<N, D> = TransformBase<N, D, MatrixArray<N, DimNameSum<D, U1>, DimNameSum<D, U1>>, TGeneral>;
+
+/// An inversible `D`-dimensional general transformation. Stored as an homogeneous
+/// `(D + 1) × (D + 1)` matrix.
+pub type Projective<N, D> = TransformBase<N, D, MatrixArray<N, DimNameSum<D, U1>, DimNameSum<D, U1>>, TProjective>;
+
+/// A `D`-dimensional affine transformation. Stored as an homogeneous `(D + 1) × (D + 1)` matrix.
+pub type Affine<N, D> = TransformBase<N, D, MatrixArray<N, DimNameSum<D, U1>, DimNameSum<D, U1>>, TAffine>;
+
+/// A 2D general transformation that may not be inversible. Stored as an homogeneous 3x3 matrix.
+pub type Transform2<N> = Transform<N, U2>;
+/// An inversible 2D general transformation. Stored as an homogeneous 3x3 matrix.
+pub type Projective2<N> = Projective<N, U2>;
+/// A 2D affine transformation. Stored as an homogeneous 3x3 matrix.
+pub type Affine2<N> = Affine<N, U2>;
+
+/// A 3D general transformation that may not be inversible. Stored as an homogeneous 4x4 matrix.
+pub type Transform3<N> = Transform<N, U3>;
+/// An inversible 3D general transformation. Stored as an homogeneous 4x4 matrix.
+pub type Projective3<N> = Projective<N, U3>;
+/// A 3D affine transformation. Stored as an homogeneous 4x4 matrix.
+pub type Affine3<N> = Affine<N, U3>;

src/geometry/transform_construction.rs

@@ -0,0 +1,36 @@
+use num::{Zero, One};
+
+use alga::general::Field;
+
+use core::{Scalar, OwnedSquareMatrix};
+use core::dimension::{DimNameAdd, DimNameSum, U1};
+use core::storage::OwnedStorage;
+use core::allocator::OwnedAllocator;
+
+use geometry::{TransformBase, TCategory};
+
+
+impl<N, D, S, C: TCategory> TransformBase<N, D, S, C>
+    where N: Scalar + Zero + One,
+          D: DimNameAdd<U1>,
+          S: OwnedStorage<N, DimNameSum<D, U1>, DimNameSum<D, U1>>,
+          S::Alloc: OwnedAllocator<N, DimNameSum<D, U1>, DimNameSum<D, U1>, S> {
+    /// Creates a new identity transform.
+    #[inline]
+    pub fn identity() -> Self {
+        Self::from_matrix_unchecked(OwnedSquareMatrix::<N, _, S::Alloc>::identity())
+    }
+}
+
+impl<N, D, S, C: TCategory> One for TransformBase<N, D, S, C>
+    where N: Scalar + Field,
+          D: DimNameAdd<U1>,
+          S: OwnedStorage<N, DimNameSum<D, U1>, DimNameSum<D, U1>>,
+          S::Alloc: OwnedAllocator<N, DimNameSum<D, U1>, DimNameSum<D, U1>, S> {
+    /// Creates a new identity transform.
+    #[inline]
+    fn one() -> Self {
+        Self::identity()
+    }
+}
+

src/geometry/transform_conversion.rs

@@ -0,0 +1,67 @@
+use approx::ApproxEq;
+
+use alga::general::{SubsetOf, Field};
+
+use core::{Scalar, SquareMatrix};
+use core::dimension::{DimName, DimNameAdd, DimNameSum, U1};
+use core::storage::OwnedStorage;
+use core::allocator::OwnedAllocator;
+
+use geometry::{TransformBase, TCategory, SuperTCategoryOf};
+
+
+impl<N1, N2, D: DimName, SA, SB, C1, C2> SubsetOf<TransformBase<N2, D, SB, C2>> for TransformBase<N1, D, SA, C1>
+    where N1: Scalar + Field + ApproxEq + SubsetOf<N2>,
+          N2: Scalar + Field + ApproxEq,
+          C1: TCategory,
+          C2: SuperTCategoryOf<C1>,
+          D: DimNameAdd<U1>,
+          SA: OwnedStorage<N1, DimNameSum<D, U1>, DimNameSum<D, U1>>,
+          SB: OwnedStorage<N2, DimNameSum<D, U1>, DimNameSum<D, U1>>,
+          SA::Alloc: OwnedAllocator<N1, DimNameSum<D, U1>, DimNameSum<D, U1>, SA>,
+          SB::Alloc: OwnedAllocator<N2, DimNameSum<D, U1>, DimNameSum<D, U1>, SB>,
+          N1::Epsilon: Copy,
+          N2::Epsilon: Copy {
+    #[inline]
+    fn to_superset(&self) -> TransformBase<N2, D, SB, C2> {
+        TransformBase::from_matrix_unchecked(self.to_homogeneous().to_superset())
+    }
+
+    #[inline]
+    fn is_in_subset(t: &TransformBase<N2, D, SB, C2>) -> bool {
+        <Self as SubsetOf<_>>::is_in_subset(t.matrix())
+    }
+
+    #[inline]
+    unsafe fn from_superset_unchecked(t: &TransformBase<N2, D, SB, C2>) -> Self {
+        Self::from_superset_unchecked(t.matrix())
+    }
+}
+
+
+impl<N1, N2, D: DimName, SA, SB, C> SubsetOf<SquareMatrix<N2, DimNameSum<D, U1>, SB>> for TransformBase<N1, D, SA, C>
+    where N1: Scalar + Field + ApproxEq + SubsetOf<N2>,
+          N2: Scalar + Field + ApproxEq,
+          C: TCategory,
+          D: DimNameAdd<U1>,
+          SA: OwnedStorage<N1, DimNameSum<D, U1>, DimNameSum<D, U1>>,
+          SB: OwnedStorage<N2, DimNameSum<D, U1>, DimNameSum<D, U1>>,
+          SA::Alloc: OwnedAllocator<N1, DimNameSum<D, U1>, DimNameSum<D, U1>, SA>,
+          SB::Alloc: OwnedAllocator<N2, DimNameSum<D, U1>, DimNameSum<D, U1>, SB>,
+          N1::Epsilon: Copy,
+          N2::Epsilon: Copy {
+    #[inline]
+    fn to_superset(&self) -> SquareMatrix<N2, DimNameSum<D, U1>, SB> {
+        self.matrix().to_superset()
+    }
+
+    #[inline]
+    fn is_in_subset(m: &SquareMatrix<N2, DimNameSum<D, U1>, SB>) -> bool {
+        C::check_homogeneous_invariants(m)
+    }
+
+    #[inline]
+    unsafe fn from_superset_unchecked(m: &SquareMatrix<N2, DimNameSum<D, U1>, SB>) -> Self {
+        TransformBase::from_matrix_unchecked(::convert_ref_unchecked(m))
+    }
+}

src/geometry/transform_ops.rs

@@ -0,0 +1,615 @@
+use num::{Zero, One};
+use std::ops::{Index, IndexMut, Mul, MulAssign, Div, DivAssign};
+use approx::ApproxEq;
+
+use alga::general::{Field, Real, ClosedAdd, ClosedMul, ClosedNeg, SubsetOf};
+
+use core::{Scalar, ColumnVector, OwnedColumnVector, OwnedSquareMatrix};
+use core::storage::{Storage, StorageMut, OwnedStorage};
+use core::allocator::{Allocator, OwnedAllocator};
+use core::dimension::{DimName, DimNameAdd, DimNameSum, U1, U3, U4};
+
+use geometry::{PointBase, OwnedPoint, TransformBase, OwnedTransform, TCategory, TCategoryMul,
+               SubTCategoryOf, SuperTCategoryOf, TGeneral, TProjective, TAffine, RotationBase,
+               UnitQuaternionBase, IsometryBase, SimilarityBase, TranslationBase};
+
+/*
+ *
+ * In the following, we provide:
+ * =========================
+ *
+ * Index<(usize, usize)>
+ * IndexMut<(usize, usize)> (where TCategory == TGeneral)
+ *
+ * (Operators)
+ *
+ * TransformBase × IsometryBase
+ * TransformBase × RotationBase
+ * TransformBase × SimilarityBase
+ * TransformBase × TransformBase
+ * TransformBase × UnitQuaternion
+ * FIXME: TransformBase × UnitComplex
+ * TransformBase × TranslationBase
+ * TransformBase × ColumnVector
+ * TransformBase × PointBase
+ *
+ * IsometryBase       × TransformBase
+ * RotationBase       × TransformBase
+ * SimilarityBase     × TransformBase
+ * TranslationBase    × TransformBase
+ * UnitQuaternionBase × TransformBase
+ * FIXME: UnitComplex × TransformBase
+ *
+ * FIXME: TransformBase ÷ IsometryBase
+ * TransformBase ÷ RotationBase
+ * FIXME: TransformBase ÷ SimilarityBase
+ * TransformBase ÷ TransformBase
+ * TransformBase ÷ UnitQuaternion
+ * TransformBase ÷ TranslationBase
+ *
+ * FIXME: IsometryBase       ÷ TransformBase
+ * RotationBase       ÷ TransformBase
+ * FIXME: SimilarityBase     ÷ TransformBase
+ * TranslationBase    ÷ TransformBase
+ * UnitQuaternionBase ÷ TransformBase
+ * FIXME: UnitComplex ÷ TransformBase
+ *
+ *
+ * (Assignment Operators)
+ *
+ *
+ * TransformBase ×= TransformBase
+ * TransformBase ×= SimilarityBase
+ * TransformBase ×= IsometryBase
+ * TransformBase ×= RotationBase
+ * TransformBase ×= UnitQuaternionBase
+ * FIXME: TransformBase ×= UnitComplex
+ * TransformBase ×= TranslationBase
+ *
+ * TransformBase ÷= TransformBase
+ * FIXME: TransformBase ÷= SimilarityBase
+ * FIXME: TransformBase ÷= IsometryBase
+ * TransformBase ÷= RotationBase
+ * TransformBase ÷= UnitQuaternionBase
+ * FIXME: TransformBase ÷= UnitComplex
+ *
+ */
+
+/*
+ *
+ * Indexing.
+ *
+ */
+impl<N, D, S, C: TCategory> Index<(usize, usize)> for TransformBase<N, D, S, C>
+    where N: Scalar,
+          D: DimName + DimNameAdd<U1>,
+          S: Storage<N, DimNameSum<D, U1>, DimNameSum<D, U1>> {
+    type Output = N;
+
+    #[inline]
+    fn index(&self, ij: (usize, usize)) -> &N {
+        self.matrix().index(ij)
+    }
+}
+
+// Only general transformations are mutably indexable.
+impl<N, D, S> IndexMut<(usize, usize)> for TransformBase<N, D, S, TGeneral>
+    where N: Scalar,
+          D: DimName + DimNameAdd<U1>,
+          S: StorageMut<N, DimNameSum<D, U1>, DimNameSum<D, U1>> {
+    #[inline]
+    fn index_mut(&mut self, ij: (usize, usize)) -> &mut N {
+        self.matrix_mut().index_mut(ij)
+    }
+}
+
+
+// TransformBase × ColumnVector
+md_impl_all!(
+    Mul, mul where N: Field;
+    (DimNameSum<D, U1>, DimNameSum<D, U1>), (D, U1) for D: DimNameAdd<U1>, C: TCategory
+    where SA::Alloc: Allocator<N, D, D>
+    where SA::Alloc: Allocator<N, D, U1>
+    where SA::Alloc: Allocator<N, U1, D>;
+    self: TransformBase<N, D, SA, C>, rhs: ColumnVector<N, D, SB>, Output = OwnedColumnVector<N, D, SA::Alloc>;
+    [val val] => &self * &rhs;
+    [ref val] =>  self * &rhs;
+    [val ref] => &self *  rhs;
+    [ref ref] => {
+        let transform = self.matrix().fixed_slice::<D, D>(0, 0);
+
+        if C::has_normalizer() {
+            let normalizer = self.matrix().fixed_slice::<U1, D>(D::dim(), 0);
+            let n = normalizer.tr_dot(&rhs);
+
+            if !n.is_zero() {
+                return transform * (rhs / n);
+            }
+        }
+
+        transform * rhs
+    };
+);
+
+
+// TransformBase × PointBase
+md_impl_all!(
+    Mul, mul where N: Field;
+    (DimNameSum<D, U1>, DimNameSum<D, U1>), (D, U1) for D: DimNameAdd<U1>, C: TCategory
+    where SA::Alloc: Allocator<N, D, D>
+    where SA::Alloc: Allocator<N, D, U1>
+    where SA::Alloc: Allocator<N, U1, D>;
+    self: TransformBase<N, D, SA, C>, rhs: PointBase<N, D, SB>, Output = OwnedPoint<N, D, SA::Alloc>;
+    [val val] => &self * &rhs;
+    [ref val] =>  self * &rhs;
+    [val ref] => &self *  rhs;
+    [ref ref] => {
+        let transform   = self.matrix().fixed_slice::<D, D>(0, 0);
+        let translation = self.matrix().fixed_slice::<D, U1>(0, D::dim());
+
+        if C::has_normalizer() {
+            let normalizer = self.matrix().fixed_slice::<U1, D>(D::dim(), 0);
+            let n = normalizer.tr_dot(&rhs.coords) + unsafe { *self.matrix().get_unchecked(D::dim(), D::dim()) };
+
+            if !n.is_zero() {
+                return transform * (rhs / n) + translation;
+            }
+        }
+
+        transform * rhs + translation
+    };
+);
+
+
+// TransformBase × TransformBase
+md_impl_all!(
+    Mul, mul;
+    (DimNameSum<D, U1>, DimNameSum<D, U1>), (DimNameSum<D, U1>, DimNameSum<D, U1>) for D: DimNameAdd<U1>, CA: TCategoryMul<CB>, CB: TCategory;
+    self: TransformBase<N, D, SA, CA>, rhs: TransformBase<N, D, SB, CB>, Output = OwnedTransform<N, D, SA::Alloc, CA::Representative>;
+    [val val] => Self::Output::from_matrix_unchecked(self.unwrap() * rhs.unwrap());
+    [ref val] => Self::Output::from_matrix_unchecked(self.matrix() * rhs.unwrap());
+    [val ref] => Self::Output::from_matrix_unchecked(self.unwrap() * rhs.matrix());
+    [ref ref] => Self::Output::from_matrix_unchecked(self.matrix() * rhs.matrix());
+);
+
+
+// TransformBase × RotationBase
+md_impl_all!(
+    Mul, mul where N: One;
+    (DimNameSum<D, U1>, DimNameSum<D, U1>), (D, D) for D: DimNameAdd<U1>, C: TCategoryMul<TAffine>
+    where SB::Alloc: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1> >;
+    self: TransformBase<N, D, SA, C>, rhs: RotationBase<N, D, SB>, Output = OwnedTransform<N, D, SA::Alloc, C::Representative>;
+    [val val] => Self::Output::from_matrix_unchecked(self.unwrap() * rhs.to_homogeneous());
+    [ref val] => Self::Output::from_matrix_unchecked(self.matrix() * rhs.to_homogeneous());
+    [val ref] => Self::Output::from_matrix_unchecked(self.unwrap() * rhs.to_homogeneous());
+    [ref ref] => Self::Output::from_matrix_unchecked(self.matrix() * rhs.to_homogeneous());
+);
+
+
+// RotationBase × TransformBase
+md_impl_all!(
+    Mul, mul where N: One;
+    (D, D), (DimNameSum<D, U1>, DimNameSum<D, U1>) for D: DimNameAdd<U1>, C: TCategoryMul<TAffine>
+    where SA::Alloc: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1> >;
+    self: RotationBase<N, D, SA>, rhs: TransformBase<N, D, SB, C>, Output = OwnedTransform<N, D, SA::Alloc, C::Representative>;
+    [val val] => Self::Output::from_matrix_unchecked(self.to_homogeneous() * rhs.unwrap());
+    [ref val] => Self::Output::from_matrix_unchecked(self.to_homogeneous() * rhs.unwrap());
+    [val ref] => Self::Output::from_matrix_unchecked(self.to_homogeneous() * rhs.matrix());
+    [ref ref] => Self::Output::from_matrix_unchecked(self.to_homogeneous() * rhs.matrix());
+);
+
+
+// TransformBase × UnitQuaternionBase
+md_impl_all!(
+    Mul, mul where N: Real;
+    (U4, U4), (U4, U1) for C: TCategoryMul<TAffine>
+    where SB::Alloc: Allocator<N, U3, U3>
+    where SB::Alloc: Allocator<N, U4, U4>;
+    self: TransformBase<N, U3, SA, C>, rhs: UnitQuaternionBase<N, SB>, Output = OwnedTransform<N, U3, SA::Alloc, C::Representative>;
+    [val val] => Self::Output::from_matrix_unchecked(self.unwrap() * rhs.to_homogeneous());
+    [ref val] => Self::Output::from_matrix_unchecked(self.matrix() * rhs.to_homogeneous());
+    [val ref] => Self::Output::from_matrix_unchecked(self.unwrap() * rhs.to_homogeneous());
+    [ref ref] => Self::Output::from_matrix_unchecked(self.matrix() * rhs.to_homogeneous());
+);
+
+
+// UnitQuaternionBase × TransformBase
+md_impl_all!(
+    Mul, mul where N: Real;
+    (U4, U1), (U4, U4) for C: TCategoryMul<TAffine>
+    where SA::Alloc: Allocator<N, U3, U3>
+    where SA::Alloc: Allocator<N, U4, U4>;
+    self: UnitQuaternionBase<N, SA>, rhs: TransformBase<N, U3, SB, C>, Output = OwnedTransform<N, U3, SA::Alloc, C::Representative>;
+    [val val] => Self::Output::from_matrix_unchecked(self.to_homogeneous() * rhs.unwrap());
+    [ref val] => Self::Output::from_matrix_unchecked(self.to_homogeneous() * rhs.unwrap());
+    [val ref] => Self::Output::from_matrix_unchecked(self.to_homogeneous() * rhs.matrix());
+    [ref ref] => Self::Output::from_matrix_unchecked(self.to_homogeneous() * rhs.matrix());
+);
+
+
+
+// TransformBase × IsometryBase
+md_impl_all!(
+    Mul, mul where N: Real;
+    (DimNameSum<D, U1>, DimNameSum<D, U1>), (D, U1)
+    for D: DimNameAdd<U1>, C: TCategoryMul<TAffine>, R: SubsetOf<OwnedSquareMatrix<N, DimNameSum<D, U1>, SB::Alloc> >
+    where SB::Alloc: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1> >;
+    self: TransformBase<N, D, SA, C>, rhs: IsometryBase<N, D, SB, R>, Output = OwnedTransform<N, D, SA::Alloc, C::Representative>;
+    [val val] => Self::Output::from_matrix_unchecked(self.unwrap() * rhs.to_homogeneous());
+    [ref val] => Self::Output::from_matrix_unchecked(self.matrix() * rhs.to_homogeneous());
+    [val ref] => Self::Output::from_matrix_unchecked(self.unwrap() * rhs.to_homogeneous());
+    [ref ref] => Self::Output::from_matrix_unchecked(self.matrix() * rhs.to_homogeneous());
+);
+
+// IsometryBase × TransformBase
+md_impl_all!(
+    Mul, mul where N: Real;
+    (D, U1), (DimNameSum<D, U1>, DimNameSum<D, U1>)
+    for D: DimNameAdd<U1>, C: TCategoryMul<TAffine>, R: SubsetOf<OwnedSquareMatrix<N, DimNameSum<D, U1>, SA::Alloc> >
+    where SA::Alloc: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1> >;
+    self: IsometryBase<N, D, SA, R>, rhs: TransformBase<N, D, SB, C>, Output = OwnedTransform<N, D, SA::Alloc, C::Representative>;
+    [val val] => Self::Output::from_matrix_unchecked(self.to_homogeneous() * rhs.unwrap());
+    [ref val] => Self::Output::from_matrix_unchecked(self.to_homogeneous() * rhs.unwrap());
+    [val ref] => Self::Output::from_matrix_unchecked(self.to_homogeneous() * rhs.matrix());
+    [ref ref] => Self::Output::from_matrix_unchecked(self.to_homogeneous() * rhs.matrix());
+);
+
+
+// TransformBase × SimilarityBase
+md_impl_all!(
+    Mul, mul where N: Real;
+    (DimNameSum<D, U1>, DimNameSum<D, U1>), (D, U1)
+    for D: DimNameAdd<U1>, C: TCategoryMul<TAffine>, R: SubsetOf<OwnedSquareMatrix<N, DimNameSum<D, U1>, SB::Alloc> >
+    where SB::Alloc: Allocator<N, D, D >
+    where SB::Alloc: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1> >;
+    self: TransformBase<N, D, SA, C>, rhs: SimilarityBase<N, D, SB, R>, Output = OwnedTransform<N, D, SA::Alloc, C::Representative>;
+    [val val] => Self::Output::from_matrix_unchecked(self.unwrap() * rhs.to_homogeneous());
+    [ref val] => Self::Output::from_matrix_unchecked(self.matrix() * rhs.to_homogeneous());
+    [val ref] => Self::Output::from_matrix_unchecked(self.unwrap() * rhs.to_homogeneous());
+    [ref ref] => Self::Output::from_matrix_unchecked(self.matrix() * rhs.to_homogeneous());
+);
+
+// SimilarityBase × TransformBase
+md_impl_all!(
+    Mul, mul where N: Real;
+    (D, U1), (DimNameSum<D, U1>, DimNameSum<D, U1>)
+    for D: DimNameAdd<U1>, C: TCategoryMul<TAffine>, R: SubsetOf<OwnedSquareMatrix<N, DimNameSum<D, U1>, SA::Alloc> >
+    where SA::Alloc: Allocator<N, D, D >
+    where SA::Alloc: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1> >;
+    self: SimilarityBase<N, D, SA, R>, rhs: TransformBase<N, D, SB, C>, Output = OwnedTransform<N, D, SA::Alloc, C::Representative>;
+    [val val] => Self::Output::from_matrix_unchecked(self.to_homogeneous() * rhs.unwrap());
+    [ref val] => Self::Output::from_matrix_unchecked(self.to_homogeneous() * rhs.unwrap());
+    [val ref] => Self::Output::from_matrix_unchecked(self.to_homogeneous() * rhs.matrix());
+    [ref ref] => Self::Output::from_matrix_unchecked(self.to_homogeneous() * rhs.matrix());
+);
+
+
+
+/*
+ *
+ * FIXME: don't explicitly build the homogeneous translation matrix.
+ * Directly apply the translation, just as in `Matrix::{append,prepend}_translation`. This has not
+ * been done yet because of the `DimNameDiff` requirement (which is not automatically deduced from
+ * `DimNameAdd` requirement).
+ *
+ */
+// TransformBase × TranslationBase
+md_impl_all!(
+    Mul, mul where N: Real;
+    (DimNameSum<D, U1>, DimNameSum<D, U1>), (D, U1) for D: DimNameAdd<U1>, C: TCategoryMul<TAffine>
+    where SB::Alloc: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1> >;
+    self: TransformBase<N, D, SA, C>, rhs: TranslationBase<N, D, SB>, Output = OwnedTransform<N, D, SA::Alloc, C::Representative>;
+    [val val] => Self::Output::from_matrix_unchecked(self.unwrap() * rhs.to_homogeneous());
+    [ref val] => Self::Output::from_matrix_unchecked(self.matrix() * rhs.to_homogeneous());
+    [val ref] => Self::Output::from_matrix_unchecked(self.unwrap() * rhs.to_homogeneous());
+    [ref ref] => Self::Output::from_matrix_unchecked(self.matrix() * rhs.to_homogeneous());
+);
+
+// TranslationBase × TransformBase
+md_impl_all!(
+    Mul, mul where N: Real;
+    (D, U1), (DimNameSum<D, U1>, DimNameSum<D, U1>)
+    for D: DimNameAdd<U1>, C: TCategoryMul<TAffine>
+    where SA::Alloc: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1> >;
+    self: TranslationBase<N, D, SA>, rhs: TransformBase<N, D, SB, C>, Output = OwnedTransform<N, D, SA::Alloc, C::Representative>;
+    [val val] => Self::Output::from_matrix_unchecked(self.to_homogeneous() * rhs.unwrap());
+    [ref val] => Self::Output::from_matrix_unchecked(self.to_homogeneous() * rhs.unwrap());
+    [val ref] => Self::Output::from_matrix_unchecked(self.to_homogeneous() * rhs.matrix());
+    [ref ref] => Self::Output::from_matrix_unchecked(self.to_homogeneous() * rhs.matrix());
+);
+
+
+
+// TransformBase ÷ TransformBase
+md_impl_all!(
+    Div, div where N: ApproxEq, Field;
+    (DimNameSum<D, U1>, DimNameSum<D, U1>), (DimNameSum<D, U1>, DimNameSum<D, U1>) for D: DimNameAdd<U1>, CA: TCategoryMul<CB>, CB: SubTCategoryOf<TProjective>;
+    self: TransformBase<N, D, SA, CA>, rhs: TransformBase<N, D, SB, CB>, Output = OwnedTransform<N, D, SA::Alloc, CA::Representative>;
+    [val val] => self * rhs.inverse();
+    [ref val] => self * rhs.inverse();
+    [val ref] => self * rhs.clone_owned().inverse();
+    [ref ref] => self * rhs.clone_owned().inverse();
+);
+
+// TransformBase ÷ RotationBase
+md_impl_all!(
+    Div, div where N: One;
+    (DimNameSum<D, U1>, DimNameSum<D, U1>), (D, D) for D: DimNameAdd<U1>, C: TCategoryMul<TAffine>
+    where SB::Alloc: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1> >;
+    self: TransformBase<N, D, SA, C>, rhs: RotationBase<N, D, SB>, Output = OwnedTransform<N, D, SA::Alloc, C::Representative>;
+    [val val] => self * rhs.inverse();
+    [ref val] => self * rhs.inverse();
+    [val ref] => self * rhs.inverse();
+    [ref ref] => self * rhs.inverse();
+);
+
+
+// RotationBase ÷ TransformBase
+md_impl_all!(
+    Div, div where N: One;
+    (D, D), (DimNameSum<D, U1>, DimNameSum<D, U1>) for D: DimNameAdd<U1>, C: TCategoryMul<TAffine>
+    where SA::Alloc: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1> >;
+    self: RotationBase<N, D, SA>, rhs: TransformBase<N, D, SB, C>, Output = OwnedTransform<N, D, SA::Alloc, C::Representative>;
+    [val val] => self.inverse() * rhs;
+    [ref val] => self.inverse() * rhs;
+    [val ref] => self.inverse() * rhs;
+    [ref ref] => self.inverse() * rhs;
+);
+
+
+// TransformBase ÷ UnitQuaternionBase
+md_impl_all!(
+    Div, div where N: Real;
+    (U4, U4), (U4, U1) for C: TCategoryMul<TAffine>
+    where SB::Alloc: Allocator<N, U3, U3>
+    where SB::Alloc: Allocator<N, U4, U4>;
+    self: TransformBase<N, U3, SA, C>, rhs: UnitQuaternionBase<N, SB>, Output = OwnedTransform<N, U3, SA::Alloc, C::Representative>;
+    [val val] => self * rhs.inverse();
+    [ref val] => self * rhs.inverse();
+    [val ref] => self * rhs.inverse();
+    [ref ref] => self * rhs.inverse();
+);
+
+
+// UnitQuaternionBase ÷ TransformBase
+md_impl_all!(
+    Div, div where N: Real;
+    (U4, U1), (U4, U4) for C: TCategoryMul<TAffine>
+    where SA::Alloc: Allocator<N, U3, U3>
+    where SA::Alloc: Allocator<N, U4, U4>;
+    self: UnitQuaternionBase<N, SA>, rhs: TransformBase<N, U3, SB, C>, Output = OwnedTransform<N, U3, SA::Alloc, C::Representative>;
+    [val val] => self.inverse() * rhs;
+    [ref val] => self.inverse() * rhs;
+    [val ref] => self.inverse() * rhs;
+    [ref ref] => self.inverse() * rhs;
+);
+
+
+
+//      // TransformBase ÷ IsometryBase
+//      md_impl_all!(
+//          Div, div where N: Real;
+//          (DimNameSum<D, U1>, DimNameSum<D, U1>), (D, U1)
+//          for D: DimNameAdd<U1>, C: TCategoryMul<TAffine>, R: SubsetOf<OwnedSquareMatrix<N, DimNameSum<D, U1>, SB::Alloc> >
+//          where SB::Alloc: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1> >;
+//          self: TransformBase<N, D, SA, C>, rhs: IsometryBase<N, D, SB, R>, Output = OwnedTransform<N, D, SA::Alloc, C::Representative>;
+//          [val val] => Self::Output::from_matrix_unchecked(self.unwrap() * rhs.inverse().to_homogeneous());
+//          [ref val] => Self::Output::from_matrix_unchecked(self.matrix() * rhs.inverse().to_homogeneous());
+//          [val ref] => Self::Output::from_matrix_unchecked(self.unwrap() * rhs.inverse().to_homogeneous());
+//          [ref ref] => Self::Output::from_matrix_unchecked(self.matrix() * rhs.inverse().to_homogeneous());
+//      );
+
+//      // IsometryBase ÷ TransformBase
+//      md_impl_all!(
+//          Div, div where N: Real;
+//          (D, U1), (DimNameSum<D, U1>, DimNameSum<D, U1>)
+//          for D: DimNameAdd<U1>, C: TCategoryMul<TAffine>, R: SubsetOf<OwnedSquareMatrix<N, DimNameSum<D, U1>, SA::Alloc> >
+//          where SA::Alloc: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1> >;
+//          self: IsometryBase<N, D, SA, R>, rhs: TransformBase<N, D, SB, C>, Output = OwnedTransform<N, D, SA::Alloc, C::Representative>;
+//          [val val] => Self::Output::from_matrix_unchecked(self.to_homogeneous() * rhs.unwrap());
+//          [ref val] => Self::Output::from_matrix_unchecked(self.to_homogeneous() * rhs.unwrap());
+//          [val ref] => Self::Output::from_matrix_unchecked(self.to_homogeneous() * rhs.matrix());
+//          [ref ref] => Self::Output::from_matrix_unchecked(self.to_homogeneous() * rhs.matrix());
+//      );
+
+
+//      // TransformBase ÷ SimilarityBase
+//      md_impl_all!(
+//          Div, div where N: Real;
+//          (DimNameSum<D, U1>, DimNameSum<D, U1>), (D, U1)
+//          for D: DimNameAdd<U1>, C: TCategoryMul<TAffine>, R: SubsetOf<OwnedSquareMatrix<N, DimNameSum<D, U1>, SB::Alloc> >
+//          where SB::Alloc: Allocator<N, D, D >
+//          where SB::Alloc: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1> >;
+//          self: TransformBase<N, D, SA, C>, rhs: SimilarityBase<N, D, SB, R>, Output = OwnedTransform<N, D, SA::Alloc, C::Representative>;
+//          [val val] => Self::Output::from_matrix_unchecked(self.unwrap() * rhs.to_homogeneous());
+//          [ref val] => Self::Output::from_matrix_unchecked(self.matrix() * rhs.to_homogeneous());
+//          [val ref] => Self::Output::from_matrix_unchecked(self.unwrap() * rhs.to_homogeneous());
+//          [ref ref] => Self::Output::from_matrix_unchecked(self.matrix() * rhs.to_homogeneous());
+//      );
+
+//      // SimilarityBase ÷ TransformBase
+//      md_impl_all!(
+//          Div, div where N: Real;
+//          (D, U1), (DimNameSum<D, U1>, DimNameSum<D, U1>)
+//          for D: DimNameAdd<U1>, C: TCategoryMul<TAffine>, R: SubsetOf<OwnedSquareMatrix<N, DimNameSum<D, U1>, SA::Alloc> >
+//          where SA::Alloc: Allocator<N, D, D >
+//          where SA::Alloc: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1> >;
+//          self: SimilarityBase<N, D, SA, R>, rhs: TransformBase<N, D, SB, C>, Output = OwnedTransform<N, D, SA::Alloc, C::Representative>;
+//          [val val] => Self::Output::from_matrix_unchecked(self.to_homogeneous() * rhs.unwrap());
+//          [ref val] => Self::Output::from_matrix_unchecked(self.to_homogeneous() * rhs.unwrap());
+//          [val ref] => Self::Output::from_matrix_unchecked(self.to_homogeneous() * rhs.matrix());
+//          [ref ref] => Self::Output::from_matrix_unchecked(self.to_homogeneous() * rhs.matrix());
+//      );
+
+
+
+// TransformBase ÷ TranslationBase
+md_impl_all!(
+    Div, div where N: Real;
+    (DimNameSum<D, U1>, DimNameSum<D, U1>), (D, U1) for D: DimNameAdd<U1>, C: TCategoryMul<TAffine>
+    where SB::Alloc: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1> >;
+    self: TransformBase<N, D, SA, C>, rhs: TranslationBase<N, D, SB>, Output = OwnedTransform<N, D, SA::Alloc, C::Representative>;
+    [val val] => self * rhs.inverse();
+    [ref val] => self * rhs.inverse();
+    [val ref] => self * rhs.inverse();
+    [ref ref] => self * rhs.inverse();
+);
+
+// TranslationBase ÷ TransformBase
+md_impl_all!(
+    Div, div where N: Real;
+    (D, U1), (DimNameSum<D, U1>, DimNameSum<D, U1>)
+    for D: DimNameAdd<U1>, C: TCategoryMul<TAffine>
+    where SA::Alloc: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1> >;
+    self: TranslationBase<N, D, SA>, rhs: TransformBase<N, D, SB, C>, Output = OwnedTransform<N, D, SA::Alloc, C::Representative>;
+    [val val] => self.inverse() * rhs;
+    [ref val] => self.inverse() * rhs;
+    [val ref] => self.inverse() * rhs;
+    [ref ref] => self.inverse() * rhs;
+);
+
+
+// TransformBase ×= TransformBase
+md_assign_impl_all!(
+    MulAssign, mul_assign;
+    (DimNameSum<D, U1>, DimNameSum<D, U1>), (DimNameSum<D, U1>, DimNameSum<D, U1>) for D: DimNameAdd<U1>, CA: TCategory, CB: SubTCategoryOf<CA>;
+    self: TransformBase<N, D, SA, CA>, rhs: TransformBase<N, D, SB, CB>;
+    [val] => *self.matrix_mut_unchecked() *= rhs.unwrap();
+    [ref] => *self.matrix_mut_unchecked() *= rhs.matrix();
+);
+
+
+// TransformBase ×= SimilarityBase
+md_assign_impl_all!(
+    MulAssign, mul_assign;
+    (DimNameSum<D, U1>, DimNameSum<D, U1>), (D, U1)
+    for D: DimNameAdd<U1>, C: TCategory, R: SubsetOf<OwnedSquareMatrix<N, DimNameSum<D, U1>, SB::Alloc> >
+    where SB::Alloc: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1> >
+    where SB::Alloc: Allocator<N, D, D>;
+    self: TransformBase<N, D, SA, C>, rhs: SimilarityBase<N, D, SB, R>;
+    [val] => *self.matrix_mut_unchecked() *= rhs.to_homogeneous();
+    [ref] => *self.matrix_mut_unchecked() *= rhs.to_homogeneous();
+);
+
+
+// TransformBase ×= IsometryBase
+md_assign_impl_all!(
+    MulAssign, mul_assign;
+    (DimNameSum<D, U1>, DimNameSum<D, U1>), (D, U1)
+    for D: DimNameAdd<U1>, C: TCategory, R: SubsetOf<OwnedSquareMatrix<N, DimNameSum<D, U1>, SB::Alloc> >
+    where SB::Alloc: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1> >;
+    self: TransformBase<N, D, SA, C>, rhs: IsometryBase<N, D, SB, R>;
+    [val] => *self.matrix_mut_unchecked() *= rhs.to_homogeneous();
+    [ref] => *self.matrix_mut_unchecked() *= rhs.to_homogeneous();
+);
+
+/*
+ *
+ * FIXME: don't explicitly build the homogeneous translation matrix.
+ * Directly apply the translation, just as in `Matrix::{append,prepend}_translation`. This has not
+ * been done yet because of the `DimNameDiff` requirement (which is not automatically deduced from
+ * `DimNameAdd` requirement).
+ *
+ */
+// TransformBase ×= TranslationBase
+md_assign_impl_all!(
+    MulAssign, mul_assign where N: One;
+    (DimNameSum<D, U1>, DimNameSum<D, U1>), (D, U1) for D: DimNameAdd<U1>, C: TCategory
+    where SB::Alloc: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1> >;
+    self: TransformBase<N, D, SA, C>, rhs: TranslationBase<N, D, SB>;
+    [val] => *self.matrix_mut_unchecked() *= rhs.to_homogeneous();
+    [ref] => *self.matrix_mut_unchecked() *= rhs.to_homogeneous();
+);
+
+
+// TransformBase ×= RotationBase
+md_assign_impl_all!(
+    MulAssign, mul_assign where N: One;
+    (DimNameSum<D, U1>, DimNameSum<D, U1>), (D, D) for D: DimNameAdd<U1>, C: TCategory
+    where SB::Alloc: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1> >;
+    self: TransformBase<N, D, SA, C>, rhs: RotationBase<N, D, SB>;
+    [val] => *self.matrix_mut_unchecked() *= rhs.to_homogeneous();
+    [ref] => *self.matrix_mut_unchecked() *= rhs.to_homogeneous();
+);
+
+
+// TransformBase ×= UnitQuaternionBase
+md_assign_impl_all!(
+    MulAssign, mul_assign where N: Real;
+    (U4, U4), (U4, U1) for C: TCategory
+    where SB::Alloc: Allocator<N, U3, U3>
+    where SB::Alloc: Allocator<N, U4, U4>;
+    self: TransformBase<N, U3, SA, C>, rhs: UnitQuaternionBase<N, SB>;
+    [val] => *self.matrix_mut_unchecked() *= rhs.to_homogeneous();
+    [ref] => *self.matrix_mut_unchecked() *= rhs.to_homogeneous();
+);
+
+
+// TransformBase ÷= TransformBase
+md_assign_impl_all!(
+    DivAssign, div_assign where N: Field, ApproxEq;
+    (DimNameSum<D, U1>, DimNameSum<D, U1>), (DimNameSum<D, U1>, DimNameSum<D, U1>)
+    for D: DimNameAdd<U1>, CA: SuperTCategoryOf<CB>, CB: SubTCategoryOf<TProjective>;
+    self: TransformBase<N, D, SA, CA>, rhs: TransformBase<N, D, SB, CB>;
+    [val] => *self *= rhs.clone_owned().inverse();
+    [ref] => *self *= rhs.clone_owned().inverse();
+);
+
+
+//      // TransformBase ÷= SimilarityBase
+//      md_assign_impl_all!(
+//          DivAssign, div_assign;
+//          (DimNameSum<D, U1>, DimNameSum<D, U1>), (D, U1)
+//          for D: DimNameAdd<U1>, C: TCategory, R: SubsetOf<OwnedSquareMatrix<N, DimNameSum<D, U1>, SB::Alloc> >
+//          where SB::Alloc: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1> >
+//          where SB::Alloc: Allocator<N, D, D>;
+//          self: TransformBase<N, D, SA, C>, rhs: SimilarityBase<N, D, SB, R>;
+//          [val] => *self *= rhs.inverse();
+//          [ref] => *self *= rhs.inverse();
+//      );
+//      
+//      
+//      // TransformBase ÷= IsometryBase
+//      md_assign_impl_all!(
+//          DivAssign, div_assign;
+//          (DimNameSum<D, U1>, DimNameSum<D, U1>), (D, U1)
+//          for D: DimNameAdd<U1>, C: TCategory, R: SubsetOf<OwnedSquareMatrix<N, DimNameSum<D, U1>, SB::Alloc> >
+//          where SB::Alloc: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1> >;
+//          self: TransformBase<N, D, SA, C>, rhs: IsometryBase<N, D, SB, R>;
+//          [val] => *self *= rhs.inverse();
+//          [ref] => *self *= rhs.inverse();
+//      );
+
+
+// TransformBase ÷= TranslationBase
+md_assign_impl_all!(
+    DivAssign, div_assign where N: One, ClosedNeg;
+    (DimNameSum<D, U1>, DimNameSum<D, U1>), (D, U1) for D: DimNameAdd<U1>, C: TCategory
+    where SB::Alloc: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1> >;
+    self: TransformBase<N, D, SA, C>, rhs: TranslationBase<N, D, SB>;
+    [val] => *self *= rhs.inverse();
+    [ref] => *self *= rhs.inverse();
+);
+
+
+// TransformBase ÷= RotationBase
+md_assign_impl_all!(
+    DivAssign, div_assign where N: One;
+    (DimNameSum<D, U1>, DimNameSum<D, U1>), (D, D) for D: DimNameAdd<U1>, C: TCategory
+    where SB::Alloc: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1> >;
+    self: TransformBase<N, D, SA, C>, rhs: RotationBase<N, D, SB>;
+    [val] => *self *= rhs.inverse();
+    [ref] => *self *= rhs.inverse();
+);
+
+
+// TransformBase ÷= UnitQuaternionBase
+md_assign_impl_all!(
+    DivAssign, div_assign where N: Real;
+    (U4, U4), (U4, U1) for C: TCategory
+    where SB::Alloc: Allocator<N, U3, U3>
+    where SB::Alloc: Allocator<N, U4, U4>;
+    self: TransformBase<N, U3, SA, C>, rhs: UnitQuaternionBase<N, SB>;
+    [val] => *self *= rhs.inverse();
+    [ref] => *self *= rhs.inverse();
+);

src/geometry/translation.rs

@@ -0,0 +1,145 @@
+use num::{Zero, One};
+use std::fmt;
+use approx::ApproxEq;
+
+use alga::general::{Real, ClosedNeg};
+
+use core::{Scalar, ColumnVector, OwnedSquareMatrix};
+use core::dimension::{DimName, DimNameSum, DimNameAdd, U1};
+use core::storage::{Storage, StorageMut, Owned};
+use core::allocator::Allocator;
+
+/// A translation with an owned vector storage.
+pub type OwnedTranslation<N, D, S> = TranslationBase<N, D, Owned<N, D, U1, <S as Storage<N, D, U1>>::Alloc>>;
+
+/// A translation.
+#[repr(C)]
+#[derive(Hash, Debug, Clone, Copy)]
+#[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
+pub struct TranslationBase<N: Scalar, D: DimName, S/*: Storage<N, D, U1>*/> {
+    /// The translation coordinates, i.e., how much is added to a point's coordinates when it is
+    /// translated.
+    pub vector: ColumnVector<N, D, S>
+}
+
+impl<N, D: DimName, S> TranslationBase<N, D, S>
+    where N: Scalar,
+          S: Storage<N, D, U1> {
+    /// Creates a new translation from the given vector.
+    #[inline]
+    pub fn from_vector(vector: ColumnVector<N, D, S>) -> TranslationBase<N, D, S> {
+        TranslationBase {
+            vector: vector
+        }
+    }
+
+    /// Inverts `self`.
+    #[inline]
+    pub fn inverse(&self) -> OwnedTranslation<N, D, S>
+        where N: ClosedNeg {
+        TranslationBase::from_vector(-&self.vector)
+    }
+
+    /// Converts this translation into its equivalent homogeneous transformation matrix.
+    #[inline]
+    pub fn to_homogeneous(&self) -> OwnedSquareMatrix<N, DimNameSum<D, U1>, S::Alloc>
+        where N: Zero + One,
+              D: DimNameAdd<U1>,
+              S::Alloc: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1>> {
+        let mut res = OwnedSquareMatrix::<N, _, S::Alloc>::identity();
+        res.fixed_slice_mut::<D, U1>(0, D::dim()).copy_from(&self.vector);
+
+        res
+    }
+}
+
+
+impl<N, D: DimName, S> TranslationBase<N, D, S>
+    where N: Scalar + ClosedNeg,
+          S: StorageMut<N, D, U1> {
+    /// Inverts `self` in-place.
+    #[inline]
+    pub fn inverse_mut(&mut self) {
+        self.vector.neg_mut()
+    }
+}
+
+impl<N, D: DimName, S> Eq for TranslationBase<N, D, S>
+    where N: Scalar + Eq,
+          S: Storage<N, D, U1> {
+}
+
+impl<N, D: DimName, S> PartialEq for TranslationBase<N, D, S>
+    where N: Scalar + PartialEq,
+          S: Storage<N, D, U1> {
+    #[inline]
+    fn eq(&self, right: &TranslationBase<N, D, S>) -> bool {
+        self.vector == right.vector
+    }
+}
+
+impl<N, D: DimName, S> ApproxEq for TranslationBase<N, D, S>
+    where N: Scalar + ApproxEq,
+          S: Storage<N, D, U1>,
+          N::Epsilon: Copy {
+    type Epsilon = N::Epsilon;
+
+    #[inline]
+    fn default_epsilon() -> Self::Epsilon {
+        N::default_epsilon()
+    }
+
+    #[inline]
+    fn default_max_relative() -> Self::Epsilon {
+        N::default_max_relative()
+    }
+
+    #[inline]
+    fn default_max_ulps() -> u32 {
+        N::default_max_ulps()
+    }
+
+    #[inline]
+    fn relative_eq(&self, other: &Self, epsilon: Self::Epsilon, max_relative: Self::Epsilon) -> bool {
+        self.vector.relative_eq(&other.vector, epsilon, max_relative)
+    }
+
+    #[inline]
+    fn ulps_eq(&self, other: &Self, epsilon: Self::Epsilon, max_ulps: u32) -> bool {
+        self.vector.ulps_eq(&other.vector, epsilon, max_ulps)
+    }
+}
+
+/*
+ *
+ * Display
+ *
+ */
+impl<N, D: DimName, S> fmt::Display for TranslationBase<N, D, S>
+    where N: Real + fmt::Display,
+          S: Storage<N, D, U1>,
+          S::Alloc: Allocator<usize, D, U1> {
+    fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
+        let precision = f.precision().unwrap_or(3);
+
+        try!(writeln!(f, "TranslationBase {{"));
+        try!(write!(f, "{:.*}", precision, self.vector));
+        writeln!(f, "}}")
+    }
+}
+
+
+//          //         /*
+//          //          *
+//          //          * Absolute
+//          //          *
+//          //          */
+//          //         impl<N: Absolute> Absolute for $t<N> {
+//          //             type AbsoluteValue = $submatrix<N::AbsoluteValue>;
+//          //
+//          //             #[inline]
+//          //             fn abs(m: &$t<N>) -> $submatrix<N::AbsoluteValue> {
+//          //                 Absolute::abs(&m.submatrix)
+//          //             }
+//          //         }
+//          */

src/geometry/translation_alga.rs

@@ -0,0 +1,209 @@
+use alga::general::{AbstractMagma, AbstractGroup, AbstractLoop, AbstractMonoid, AbstractQuasigroup,
+                    AbstractSemigroup, Real, Inverse, Multiplicative, Identity, Id};
+use alga::linear::{Transformation, ProjectiveTransformation, Similarity, AffineTransformation,
+                   Isometry, DirectIsometry, Translation};
+
+use core::ColumnVector;
+use core::dimension::{DimName, U1};
+use core::storage::OwnedStorage;
+use core::allocator::OwnedAllocator;
+
+use geometry::{TranslationBase, PointBase};
+
+
+/*
+ *
+ * Algebraic structures.
+ *
+ */
+impl<N, D: DimName, S> Identity<Multiplicative> for TranslationBase<N, D, S>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    #[inline]
+    fn identity() -> Self {
+        Self::identity()
+    }
+}
+
+impl<N, D: DimName, S> Inverse<Multiplicative> for TranslationBase<N, D, S>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    #[inline]
+    fn inverse(&self) -> Self {
+        self.inverse()
+    }
+
+    #[inline]
+    fn inverse_mut(&mut self) {
+        self.inverse_mut()
+    }
+}
+
+impl<N, D: DimName, S> AbstractMagma<Multiplicative> for TranslationBase<N, D, S>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    #[inline]
+    fn operate(&self, rhs: &Self) -> Self {
+        self * rhs
+    }
+}
+
+macro_rules! impl_multiplicative_structures(
+    ($($marker: ident<$operator: ident>),* $(,)*) => {$(
+        impl<N, D: DimName, S> $marker<$operator> for TranslationBase<N, D, S>
+            where N: Real,
+                  S: OwnedStorage<N, D, U1>,
+                  S::Alloc: OwnedAllocator<N, D, U1, S> { }
+    )*}
+);
+
+impl_multiplicative_structures!(
+    AbstractSemigroup<Multiplicative>,
+    AbstractMonoid<Multiplicative>,
+    AbstractQuasigroup<Multiplicative>,
+    AbstractLoop<Multiplicative>,
+    AbstractGroup<Multiplicative>
+);
+
+/*
+ *
+ * Transformation groups.
+ *
+ */
+impl<N, D: DimName, S> Transformation<PointBase<N, D, S>> for TranslationBase<N, D, S>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    #[inline]
+    fn transform_point(&self, pt: &PointBase<N, D, S>) -> PointBase<N, D, S> {
+        pt + &self.vector
+    }
+
+    #[inline]
+    fn transform_vector(&self, v: &ColumnVector<N, D, S>) -> ColumnVector<N, D, S> {
+        v.clone()
+    }
+}
+
+impl<N, D: DimName, S> ProjectiveTransformation<PointBase<N, D, S>> for TranslationBase<N, D, S>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    #[inline]
+    fn inverse_transform_point(&self, pt: &PointBase<N, D, S>) -> PointBase<N, D, S> {
+        pt - &self.vector
+    }
+
+    #[inline]
+    fn inverse_transform_vector(&self, v: &ColumnVector<N, D, S>) -> ColumnVector<N, D, S> {
+        v.clone()
+    }
+}
+
+impl<N, D: DimName, S> AffineTransformation<PointBase<N, D, S>> for TranslationBase<N, D, S>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    type Rotation          = Id;
+    type NonUniformScaling = Id;
+    type Translation       = Self;
+
+    #[inline]
+    fn decompose(&self) -> (Self, Id, Id, Id) {
+        (self.clone(), Id::new(), Id::new(), Id::new())
+    }
+
+    #[inline]
+    fn append_translation(&self, t: &Self::Translation) -> Self {
+        t * self
+    }
+
+    #[inline]
+    fn prepend_translation(&self, t: &Self::Translation) -> Self {
+        self * t
+    }
+
+    #[inline]
+    fn append_rotation(&self, _: &Self::Rotation) -> Self {
+        self.clone()
+    }
+
+    #[inline]
+    fn prepend_rotation(&self, _: &Self::Rotation) -> Self {
+        self.clone()
+    }
+
+    #[inline]
+    fn append_scaling(&self, _: &Self::NonUniformScaling) -> Self {
+        self.clone()
+    }
+
+    #[inline]
+    fn prepend_scaling(&self, _: &Self::NonUniformScaling) -> Self {
+        self.clone()
+    }
+}
+
+
+impl<N, D: DimName, S> Similarity<PointBase<N, D, S>> for TranslationBase<N, D, S>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    type Scaling = Id;
+
+    #[inline]
+    fn translation(&self) -> Self {
+        self.clone()
+    }
+
+    #[inline]
+    fn rotation(&self) -> Id {
+        Id::new()
+    }
+
+    #[inline]
+    fn scaling(&self) -> Id {
+        Id::new()
+    }
+}
+
+macro_rules! marker_impl(
+    ($($Trait: ident),*) => {$(
+        impl<N, D: DimName, S> $Trait<PointBase<N, D, S>> for TranslationBase<N, D, S>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> { }
+    )*}
+);
+
+marker_impl!(Isometry, DirectIsometry);
+
+
+/// Subgroups of the n-dimensional translation group `T(n)`.
+impl<N, D: DimName, S> Translation<PointBase<N, D, S>> for TranslationBase<N, D, S>
+    where N: Real,
+          S: OwnedStorage<N, D, U1>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    #[inline]
+    fn to_vector(&self) -> ColumnVector<N, D, S> {
+        self.vector.clone()
+    }
+
+    #[inline]
+    fn from_vector(v: ColumnVector<N, D, S>) -> Option<Self> {
+        Some(Self::from_vector(v))
+    }
+
+    #[inline]
+    fn powf(&self, n: N) -> Option<Self> {
+        Some(Self::from_vector(&self.vector * n))
+    }
+
+    #[inline]
+    fn translation_between(a: &PointBase<N, D, S>, b: &PointBase<N, D, S>) -> Option<Self> {
+        Some(Self::from_vector(b - a))
+    }
+}

src/geometry/translation_alias.rs

@@ -0,0 +1,13 @@
+use core::MatrixArray;
+use core::dimension::{U1, U2, U3};
+
+use geometry::TranslationBase;
+
+/// A D-dimensional translation.
+pub type Translation<N, D> = TranslationBase<N, D, MatrixArray<N, D, U1>>;
+
+/// A 2-dimensional translation.
+pub type Translation2<N> = Translation<N, U2>;
+
+/// A 3-dimensional translation.
+pub type Translation3<N> = Translation<N, U3>;

src/geometry/translation_construction.rs

@@ -0,0 +1,86 @@
+#[cfg(feature = "arbitrary")]
+use quickcheck::{Arbitrary, Gen};
+
+use num::{Zero, One};
+use rand::{Rng, Rand};
+
+use alga::general::ClosedAdd;
+
+use core::{ColumnVector, Scalar};
+use core::dimension::{DimName, U1, U2, U3, U4, U5, U6};
+use core::storage::OwnedStorage;
+use core::allocator::OwnedAllocator;
+
+use geometry::TranslationBase;
+
+impl<N, D: DimName, S> TranslationBase<N, D, S>
+    where N: Scalar + Zero,
+          S: OwnedStorage<N, D, U1>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    /// Creates a new square identity rotation of the given `dimension`.
+    #[inline]
+    pub fn identity() -> TranslationBase<N, D, S> {
+        Self::from_vector(ColumnVector::<N, D, S>::from_element(N::zero()))
+    }
+}
+
+impl<N, D: DimName, S> One for TranslationBase<N, D, S>
+    where N: Scalar + Zero + ClosedAdd,
+          S: OwnedStorage<N, D, U1>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    #[inline]
+    fn one() -> Self {
+        Self::identity()
+    }
+}
+
+impl<N, D: DimName, S> Rand for TranslationBase<N, D, S>
+    where N: Scalar + Rand,
+          S: OwnedStorage<N, D, U1>,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    #[inline]
+    fn rand<G: Rng>(rng: &mut G) -> Self {
+        Self::from_vector(rng.gen())
+    }
+}
+
+
+#[cfg(feature = "arbitrary")]
+impl<N, D: DimName, S> Arbitrary for TranslationBase<N, D, S>
+    where N: Scalar + Arbitrary + Send,
+          S: OwnedStorage<N, D, U1> + Send,
+          S::Alloc: OwnedAllocator<N, D, U1, S> {
+    #[inline]
+    fn arbitrary<G: Gen>(rng: &mut G) -> Self {
+        Self::from_vector(Arbitrary::arbitrary(rng))
+    }
+}
+
+/*
+ *
+ * Small translation construction from components.
+ *
+ */
+macro_rules! componentwise_constructors_impl(
+    ($($D: ty, $($args: ident:$irow: expr),*);* $(;)*) => {$(
+        impl<N, S> TranslationBase<N, $D, S>
+            where N: Scalar,
+                  S: OwnedStorage<N, $D, U1>,
+                  S::Alloc: OwnedAllocator<N, $D, U1, S> {
+            /// Initializes this matrix from its components.
+            #[inline]
+            pub fn new($($args: N),*) -> Self {
+                Self::from_vector(ColumnVector::<N, $D, S>::new($($args),*))
+            }
+        }
+    )*}
+);
+
+componentwise_constructors_impl!(
+    U1, x:0;
+    U2, x:0, y:1;
+    U3, x:0, y:1, z:2;
+    U4, x:0, y:1, z:2, w:3;
+    U5, x:0, y:1, z:2, w:3, a:4;
+    U6, x:0, y:1, z:2, w:3, a:4, b:5;
+);

src/geometry/translation_conversion.rs

@@ -0,0 +1,159 @@
+use alga::general::{SubsetOf, SupersetOf, Real};
+use alga::linear::Rotation;
+
+use core::{Scalar, ColumnVector, SquareMatrix};
+use core::dimension::{DimName, DimNameAdd, DimNameSum, U1};
+use core::storage::OwnedStorage;
+use core::allocator::{Allocator, OwnedAllocator};
+
+use geometry::{PointBase, TranslationBase, IsometryBase, SimilarityBase, TransformBase, SuperTCategoryOf, TAffine};
+
+/*
+ * This file provides the following conversions:
+ * =============================================
+ *
+ * TranslationBase -> TranslationBase
+ * TranslationBase -> IsometryBase
+ * TranslationBase -> SimilarityBase
+ * TranslationBase -> TransformBase
+ * TranslationBase -> Matrix (homogeneous)
+ */
+
+impl<N1, N2, D: DimName, SA, SB> SubsetOf<TranslationBase<N2, D, SB>> for TranslationBase<N1, D, SA>
+    where N1: Scalar,
+          N2: Scalar + SupersetOf<N1>,
+          SA: OwnedStorage<N1, D, U1>,
+          SB: OwnedStorage<N2, D, U1>,
+          SA::Alloc: OwnedAllocator<N1, D, U1, SA>,
+          SB::Alloc: OwnedAllocator<N2, D, U1, SB> {
+    #[inline]
+    fn to_superset(&self) -> TranslationBase<N2, D, SB> {
+        TranslationBase::from_vector(self.vector.to_superset())
+    }
+
+    #[inline]
+    fn is_in_subset(rot: &TranslationBase<N2, D, SB>) -> bool {
+        ::is_convertible::<_, ColumnVector<N1, D, SA>>(&rot.vector)
+    }
+
+    #[inline]
+    unsafe fn from_superset_unchecked(rot: &TranslationBase<N2, D, SB>) -> Self {
+        TranslationBase::from_vector(rot.vector.to_subset_unchecked())
+    }
+}
+
+
+impl<N1, N2, D: DimName, SA, SB, R> SubsetOf<IsometryBase<N2, D, SB, R>> for TranslationBase<N1, D, SA>
+    where N1: Real,
+          N2: Real + SupersetOf<N1>,
+          SA: OwnedStorage<N1, D, U1>,
+          SB: OwnedStorage<N2, D, U1>,
+          R:  Rotation<PointBase<N2, D, SB>>,
+          SA::Alloc: OwnedAllocator<N1, D, U1, SA>,
+          SB::Alloc: OwnedAllocator<N2, D, U1, SB> {
+    #[inline]
+    fn to_superset(&self) -> IsometryBase<N2, D, SB, R> {
+        IsometryBase::from_parts(self.to_superset(), R::identity())
+    }
+
+    #[inline]
+    fn is_in_subset(iso: &IsometryBase<N2, D, SB, R>) -> bool {
+        iso.rotation == R::identity()
+    }
+
+    #[inline]
+    unsafe fn from_superset_unchecked(iso: &IsometryBase<N2, D, SB, R>) -> Self {
+        Self::from_superset_unchecked(&iso.translation)
+    }
+}
+
+
+impl<N1, N2, D: DimName, SA, SB, R> SubsetOf<SimilarityBase<N2, D, SB, R>> for TranslationBase<N1, D, SA>
+    where N1: Real,
+          N2: Real + SupersetOf<N1>,
+          SA: OwnedStorage<N1, D, U1>,
+          SB: OwnedStorage<N2, D, U1>,
+          R:  Rotation<PointBase<N2, D, SB>>,
+          SA::Alloc: OwnedAllocator<N1, D, U1, SA>,
+          SB::Alloc: OwnedAllocator<N2, D, U1, SB> {
+    #[inline]
+    fn to_superset(&self) -> SimilarityBase<N2, D, SB, R> {
+        SimilarityBase::from_parts(self.to_superset(), R::identity(), N2::one())
+    }
+
+    #[inline]
+    fn is_in_subset(sim: &SimilarityBase<N2, D, SB, R>) -> bool {
+        sim.isometry.rotation == R::identity() &&
+        sim.scaling() == N2::one()
+    }
+
+    #[inline]
+    unsafe fn from_superset_unchecked(sim: &SimilarityBase<N2, D, SB, R>) -> Self {
+        Self::from_superset_unchecked(&sim.isometry.translation)
+    }
+}
+
+
+impl<N1, N2, D, SA, SB, C> SubsetOf<TransformBase<N2, D, SB, C>> for TranslationBase<N1, D, SA>
+    where N1: Real,
+          N2: Real + SupersetOf<N1>,
+          SA: OwnedStorage<N1, D, U1>,
+          SB: OwnedStorage<N2, DimNameSum<D, U1>, DimNameSum<D, U1>>,
+          C:  SuperTCategoryOf<TAffine>,
+          D:  DimNameAdd<U1>,
+          SA::Alloc: OwnedAllocator<N1, D, U1, SA> +
+                     Allocator<N1, DimNameSum<D, U1>, DimNameSum<D, U1>>,
+          SB::Alloc: OwnedAllocator<N2, DimNameSum<D, U1>, DimNameSum<D, U1>, SB> +
+                     Allocator<N2, D, U1> +
+                     Allocator<N2, DimNameSum<D, U1>, D> {
+    #[inline]
+    fn to_superset(&self) -> TransformBase<N2, D, SB, C> {
+        TransformBase::from_matrix_unchecked(self.to_homogeneous().to_superset())
+    }
+
+    #[inline]
+    fn is_in_subset(t: &TransformBase<N2, D, SB, C>) -> bool {
+        <Self as SubsetOf<_>>::is_in_subset(t.matrix())
+    }
+
+    #[inline]
+    unsafe fn from_superset_unchecked(t: &TransformBase<N2, D, SB, C>) -> Self {
+        Self::from_superset_unchecked(t.matrix())
+    }
+}
+
+
+impl<N1, N2, D, SA, SB> SubsetOf<SquareMatrix<N2, DimNameSum<D, U1>, SB>> for TranslationBase<N1, D, SA>
+    where N1: Real,
+          N2: Real + SupersetOf<N1>,
+          SA: OwnedStorage<N1, D, U1>,
+          SB: OwnedStorage<N2, DimNameSum<D, U1>, DimNameSum<D, U1>>,
+          D:  DimNameAdd<U1>,
+          SA::Alloc: OwnedAllocator<N1, D, U1, SA> +
+                     Allocator<N1, DimNameSum<D, U1>, DimNameSum<D, U1>>,
+          SB::Alloc: OwnedAllocator<N2, DimNameSum<D, U1>, DimNameSum<D, U1>, SB> +
+                     Allocator<N2, D, U1> +
+                     Allocator<N2, DimNameSum<D, U1>, D> {
+    #[inline]
+    fn to_superset(&self) -> SquareMatrix<N2, DimNameSum<D, U1>, SB> {
+        self.to_homogeneous().to_superset()
+    }
+
+    #[inline]
+    fn is_in_subset(m: &SquareMatrix<N2, DimNameSum<D, U1>, SB>) -> bool {
+        let id = m.fixed_slice::<DimNameSum<D, U1>, D>(0, 0);
+
+        // Scalar types agree.
+        m.iter().all(|e| SupersetOf::<N1>::is_in_subset(e)) &&
+        // The block part does nothing.
+        id.is_identity(N2::zero()) &&
+        // The normalization factor is one.
+        m[(D::dim(), D::dim())] == N2::one()
+    }
+
+    #[inline]
+    unsafe fn from_superset_unchecked(m: &SquareMatrix<N2, DimNameSum<D, U1>, SB>) -> Self {
+        let t = m.fixed_slice::<D, U1>(0, D::dim());
+        Self::from_vector(::convert_unchecked(t.into_owned()))
+    }
+}