From d29ff5332930afdd3b1d468e712891da62fab3ef Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?S=C3=A9bastien=20Crozet?= <developer@crozet.re>
Date: Sat, 20 Aug 2016 23:13:53 +0200
Subject: [PATCH] Implement traits from the algebra crate.

---
 CHANGELOG.md                    |   4 +
 Cargo.toml                      |   6 +-
 Makefile                        |   8 +-
 README.md                       |   3 +-
 src/lib.rs                      |   4 +
 src/structs/algebra/dummy.rs    |  17 +++
 src/structs/algebra/matrix.rs   |  25 +++++
 src/structs/algebra/mod.rs      |  13 +++
 src/structs/algebra/point.rs    |  34 ++++++
 src/structs/algebra/rotation.rs |  92 ++++++++++++++++
 src/structs/algebra/vector.rs   | 186 ++++++++++++++++++++++++++++++++
 src/structs/common_macros.rs    |  13 +++
 src/structs/matrix.rs           |   4 +
 src/structs/matrix_macros.rs    |   3 +
 src/structs/mod.rs              |   1 +
 src/structs/point.rs            |   4 +
 src/structs/point_macros.rs     |   2 +
 src/structs/quaternion.rs       |   2 +-
 src/structs/rotation.rs         |   3 +
 src/structs/rotation_macros.rs  |   3 +
 src/structs/unit.rs             |   4 +-
 src/structs/vector.rs           |  15 +--
 src/structs/vector_macros.rs    |  19 +---
 src/structs/vectorn.rs          |   2 +-
 24 files changed, 436 insertions(+), 31 deletions(-)
 create mode 100644 src/structs/algebra/dummy.rs
 create mode 100644 src/structs/algebra/matrix.rs
 create mode 100644 src/structs/algebra/mod.rs
 create mode 100644 src/structs/algebra/point.rs
 create mode 100644 src/structs/algebra/rotation.rs
 create mode 100644 src/structs/algebra/vector.rs

diff --git a/CHANGELOG.md b/CHANGELOG.md
index ab56eaa2..7dcecaeb 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -43,6 +43,10 @@ longer unsafe. Instead, their name end with `_unchecked`. In particular:
       well. It is clear that performing computations with floats requires
       approximate equality.
 
+Still WIP: add implementations of abstract algebra traits from the `algebra`
+crate for vectors, rotations and points. To enable them, activate the
+`abstract_algebra` feature.
+
 ## [0.8.0]
 ### Modified
   * Almost everything (types, methods, and traits) now use full names instead
diff --git a/Cargo.toml b/Cargo.toml
index 253be7f1..4cbd8a8f 100644
--- a/Cargo.toml
+++ b/Cargo.toml
@@ -19,12 +19,12 @@ path = "src/lib.rs"
 # Generate arbitrary instances of nalgebra types for testing with quickcheck
 arbitrary = [ "quickcheck" ]
 generic_sizes = [ "generic-array", "typenum" ]
+abstract_algebra = [ "algebra" ]
 
 [dependencies]
 rustc-serialize = "0.3.*"
 rand = "0.3.*"
 num  = "0.1.*"
-# algebra = "*"
 
 [dependencies.generic-array]
 optional = true
@@ -37,3 +37,7 @@ version  = "1.3.*"
 [dependencies.quickcheck]
 optional = true
 version  = "0.2.*"
+
+[dependencies.algebra]
+optional = true
+version = "0.2.*"
diff --git a/Makefile b/Makefile
index 87cc39fb..78a080ee 100644
--- a/Makefile
+++ b/Makefile
@@ -1,18 +1,18 @@
 tmp=_git_distcheck
 
 all:
-	cargo build --release --features "arbitrary generic_sizes"
+	cargo build --release --features "arbitrary generic_sizes abstract_algebra"
 
 test:
-	cargo test --features "arbitrary generic_sizes"
+	cargo test --features "arbitrary generic_sizes abstract_algebra"
 
 
 bench:
-	cargo bench --features "arbitrary generic_sizes"
+	cargo bench --features "arbitrary generic_sizes abstract_algebra"
 
 
 doc:
-	cargo doc --no-deps --features "arbitrary generic_sizes"
+	cargo doc --no-deps --features "arbitrary generic_sizes abstract_algebra"
 
 
 clean:
diff --git a/README.md b/README.md
index a5fca593..719d0b31 100644
--- a/README.md
+++ b/README.md
@@ -59,7 +59,8 @@ an optimized set of tools for computer graphics and physics. Those features incl
 * 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`, `UnitQuaternion`.
+* 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`.
diff --git a/src/lib.rs b/src/lib.rs
index df41c059..7e6c70d8 100644
--- a/src/lib.rs
+++ b/src/lib.rs
@@ -47,6 +47,7 @@ an optimized set of tools for computer graphics and physics. Those features incl
 * Square matrices with static sizes: `Matrix1`, `Matrix2`, `Matrix3`, `Matrix4`, `Matrix5`, `Matrix6 `.
 * Rotation matrices: `Rotation2`, `Rotation3`
 * Quaternions: `Quaternion`, `UnitQuaternion`.
+* 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: `Perspective3`, `PerspectiveMatrix3`, `Orthographic3`, `OrthographicMatrix3`.
@@ -85,6 +86,9 @@ extern crate generic_array;
 #[cfg(feature="arbitrary")]
 extern crate quickcheck;
 
+#[cfg(feature="abstract_algebra")]
+extern crate algebra;
+
 use std::cmp;
 use std::ops::{Neg, Mul};
 use num::{Zero, One};
diff --git a/src/structs/algebra/dummy.rs b/src/structs/algebra/dummy.rs
new file mode 100644
index 00000000..0f687775
--- /dev/null
+++ b/src/structs/algebra/dummy.rs
@@ -0,0 +1,17 @@
+#![macro_use]
+
+macro_rules! vector_space_impl(
+    ($t: ident, $dimension: expr, $($compN: ident),+) => { }
+);
+
+macro_rules! special_orthogonal_group_impl(
+    ($t: ident, $point: ident, $vector: ident) => { }
+);
+
+macro_rules! euclidean_space_impl(
+    ($t: ident, $vector: ident) => { }
+);
+
+macro_rules! matrix_group_approx_impl(
+    ($t: ident, $($compN: ident),+) => { }
+);
diff --git a/src/structs/algebra/matrix.rs b/src/structs/algebra/matrix.rs
new file mode 100644
index 00000000..68fc7fe7
--- /dev/null
+++ b/src/structs/algebra/matrix.rs
@@ -0,0 +1,25 @@
+#![macro_use]
+
+macro_rules! use_matrix_group_modules(
+    () => {
+        use algebra::cmp::ApproxEq as AlgebraApproxEq;
+    }
+);
+
+macro_rules! matrix_group_approx_impl(
+    ($t: ident, $($compN: ident),+) => {
+        impl<N: AlgebraApproxEq> AlgebraApproxEq for $t<N> {
+            type Eps = N::Eps;
+
+            #[inline]
+            fn default_epsilon() -> N::Eps {
+                N::default_epsilon()
+            }
+
+            #[inline]
+            fn approx_eq_eps(&self, other: &$t<N>, epsilon: &N::Eps) -> bool {
+                $(AlgebraApproxEq::approx_eq_eps(&self.$compN, &other.$compN, &epsilon))&&+
+            }
+        }
+    }
+);
diff --git a/src/structs/algebra/mod.rs b/src/structs/algebra/mod.rs
new file mode 100644
index 00000000..3bd34405
--- /dev/null
+++ b/src/structs/algebra/mod.rs
@@ -0,0 +1,13 @@
+#![macro_use]
+
+#[cfg(not(feature="abstract_algebra"))]
+mod dummy;
+
+#[cfg(feature="abstract_algebra")]
+mod vector;
+#[cfg(feature="abstract_algebra")]
+mod rotation;
+#[cfg(feature="abstract_algebra")]
+mod point;
+#[cfg(feature="abstract_algebra")]
+mod matrix;
diff --git a/src/structs/algebra/point.rs b/src/structs/algebra/point.rs
new file mode 100644
index 00000000..a273f022
--- /dev/null
+++ b/src/structs/algebra/point.rs
@@ -0,0 +1,34 @@
+#![macro_use]
+
+macro_rules! use_euclidean_space_modules(
+    () => {
+        use algebra::structure::{AffineSpaceApprox, EuclideanSpaceApprox,
+                                 FieldApprox, RealApprox};
+        use algebra::cmp::ApproxEq as AlgebraApproxEq;
+    }
+);
+
+
+macro_rules! euclidean_space_impl(
+    ($t: ident, $vector: ident) => {
+        impl<N> AffineSpaceApprox<N> for $t<N>
+            where N: Copy + Neg<Output = N> + Add<N, Output = N> +
+                     Sub<N, Output = N> + AlgebraApproxEq + FieldApprox {
+            type Translation = $vector<N>;
+            
+            #[inline]
+            fn translate_by(&self, vector: &Self::Translation) -> Self {
+                *self + *vector
+            }
+
+            #[inline]
+            fn subtract(&self, other: &Self) -> Self::Translation {
+                *self - *other
+            }
+        }
+
+        impl<N: RealApprox> EuclideanSpaceApprox<N> for $t<N> {
+            type Vector = $vector<N>;
+        }
+    }
+);
diff --git a/src/structs/algebra/rotation.rs b/src/structs/algebra/rotation.rs
new file mode 100644
index 00000000..786aae63
--- /dev/null
+++ b/src/structs/algebra/rotation.rs
@@ -0,0 +1,92 @@
+#![macro_use]
+
+macro_rules! use_special_orthogonal_group_modules(
+    () => {
+        use algebra::structure::{EuclideanGroupApprox,  SpecialEuclideanGroupApprox,
+                                 OrthogonalGroupApprox, SpecialOrthogonalGroupApprox,
+                                 GroupApprox, LoopApprox, MonoidApprox,
+                                 QuasigroupApprox, SemigroupApprox,
+                                 RealApprox};
+        use algebra::cmp::ApproxEq as AlgebraApproxEq;
+        use algebra::ident::Identity;
+        use algebra::ops::{Recip, Multiplicative};
+    }
+);
+
+macro_rules! special_orthogonal_group_impl(
+    ($t: ident, $point: ident, $vector: ident) => {
+        /*
+         * Operations.
+         */
+        impl<N: BaseNum> Identity<Multiplicative> for $t<N> {
+            #[inline]
+            fn id() -> Self {
+                ::one()
+            }
+        }
+
+        impl<N: Copy + AlgebraApproxEq<Eps = N>> AlgebraApproxEq for $t<N> {
+            type Eps = N;
+
+            #[inline]
+            fn default_epsilon() -> N {
+                <N as AlgebraApproxEq>::default_epsilon()
+            }
+
+            #[inline]
+            fn approx_eq_eps(&self, other: &$t<N>, epsilon: &N) -> bool {
+                AlgebraApproxEq::approx_eq_eps(&self.submatrix, &other.submatrix, &epsilon)
+            }
+        }
+
+        impl<N: Copy> Recip for $t<N> {
+            type Result = $t<N>;
+
+            #[inline]
+            fn recip(mut self) -> $t<N> {
+                self.inverse_mut();
+
+                self
+            }
+        }
+
+
+        /*
+         *
+         * Algebraic structures.
+         *
+         */
+        // FIXME: in the end, we will keep only RealApprox.
+        impl<N: BaseNum + RealApprox> GroupApprox<Multiplicative> for $t<N> { }
+        impl<N: BaseNum + RealApprox> LoopApprox<Multiplicative> for $t<N> { }
+        impl<N: BaseNum + RealApprox> MonoidApprox<Multiplicative> for $t<N> { }
+        impl<N: BaseNum + RealApprox> QuasigroupApprox<Multiplicative> for $t<N> { }
+        impl<N: BaseNum + RealApprox> SemigroupApprox<Multiplicative> for $t<N> { }
+
+        /*
+         *
+         * Matrix groups.
+         *
+         */
+        impl<N: BaseNum + RealApprox> EuclideanGroupApprox<N, $point<N>> for $t<N> {
+            #[inline]
+            fn transform_point(&self, pt: &$point<N>) -> $point<N> {
+                *self * *pt
+            }
+
+            #[inline]
+            fn transform_vector(&self, v: &$vector<N>) -> $vector<N> {
+                *self * *v
+            }
+        }
+
+        impl<N: BaseNum + RealApprox> SpecialEuclideanGroupApprox<N, $point<N>> for $t<N> {
+        }
+
+        impl<N: BaseNum + RealApprox> OrthogonalGroupApprox<N, $point<N>> for $t<N> {
+        }
+
+        impl<N: BaseNum + RealApprox> SpecialOrthogonalGroupApprox<N, $point<N>> for $t<N> {
+        }
+    }
+);
diff --git a/src/structs/algebra/vector.rs b/src/structs/algebra/vector.rs
new file mode 100644
index 00000000..ebcf2e30
--- /dev/null
+++ b/src/structs/algebra/vector.rs
@@ -0,0 +1,186 @@
+#![macro_use]
+
+macro_rules! use_vector_space_modules(
+    () => {
+        use algebra::structure::{FieldApprox, RingCommutativeApprox, GroupAbelianApprox,
+                                 GroupApprox, LoopApprox, MonoidApprox, QuasigroupApprox,
+                                 SemigroupApprox, VectorSpaceApprox, ModuleApprox,
+                                 NormedSpaceApprox, InnerSpaceApprox,
+                                 FiniteDimVectorSpaceApprox,
+                                 Field, RingCommutative, GroupAbelian,
+                                 Group, Loop, Monoid, Quasigroup,
+                                 Semigroup, VectorSpace, Module, RealApprox};
+        use algebra::cmp::ApproxEq as AlgebraApproxEq;
+        use algebra::ident::Identity;
+        use algebra::ops::Additive;
+    }
+);
+
+macro_rules! vector_space_impl(
+    ($t: ident, $dimension: expr, $($compN: ident),+) => {
+        /*
+         * Identity & ApproxEq
+         */
+        impl<N: Copy + Identity<Additive>> Identity<Additive> for $t<N> {
+            #[inline]
+            fn id() -> Self {
+                Repeat::repeat(Identity::id())
+            }
+        }
+
+        impl<N: AlgebraApproxEq> AlgebraApproxEq for $t<N> {
+            type Eps = N::Eps;
+
+            #[inline]
+            fn default_epsilon() -> N::Eps {
+                N::default_epsilon()
+            }
+
+            #[inline]
+            fn approx_eq_eps(&self, other: &$t<N>, epsilon: &N::Eps) -> bool {
+                $(AlgebraApproxEq::approx_eq_eps(&self.$compN, &other.$compN, &epsilon))&&+
+            }
+        }
+
+        /*
+         *
+         * Approximate algebraic structures.
+         *
+         */
+        product_space_inherit_structure!($t, GroupAbelianApprox<Additive>);
+        product_space_inherit_structure!($t, GroupApprox<Additive>);
+        product_space_inherit_structure!($t, LoopApprox<Additive>);
+        product_space_inherit_structure!($t, MonoidApprox<Additive>);
+        product_space_inherit_structure!($t, QuasigroupApprox<Additive>);
+        product_space_inherit_structure!($t, SemigroupApprox<Additive>);
+
+        /*
+         * Module.
+         */
+        impl<N> ModuleApprox<N> for $t<N> where N: Copy + Neg<Output = N> + Add<N, Output = N> +
+                     AlgebraApproxEq + RingCommutativeApprox
+            { }
+
+        /*
+         * Vector spaces.
+         */
+        impl<N> VectorSpaceApprox<N> for $t<N>
+            where N: Copy + Neg<Output = N> + Add<N, Output = N> +
+                     AlgebraApproxEq + FieldApprox { }
+
+        impl<N> FiniteDimVectorSpaceApprox<N> for $t<N>
+            where N: Copy + Zero + One + Neg<Output = N> + Add<N, Output = N> +
+                     AlgebraApproxEq + FieldApprox {
+            #[inline]
+            fn dimension() -> usize {
+                $dimension
+            }
+
+            #[inline]
+            fn canonical_basis<F: FnOnce(&[$t<N>])>(f: F) {
+                let basis = [
+                    $($t::$compN()),*
+                ];
+
+                f(&basis[..])
+            }
+
+            #[inline]
+            fn component(&self, i: usize) -> N {
+                self[i]
+            }
+
+            #[inline]
+            unsafe fn component_unchecked(&self, i: usize)  -> N {
+                self.at_fast(i)
+            }
+        }
+
+        impl<N: RealApprox> NormedSpaceApprox<N> for $t<N> {
+            #[inline]
+            fn norm_squared(&self) -> N {
+                self.inner_product(self)
+            }
+
+            #[inline]
+            fn norm(&self) -> N {
+                self.norm_squared().sqrt()
+            }
+
+            #[inline]
+            fn normalize(&self) -> Self {
+                *self / self.norm()
+            }
+
+            #[inline]
+            fn normalize_mut(&mut self) -> N {
+                let n = self.norm();
+                *self /= n;
+
+                n
+            }
+
+            #[inline]
+            fn try_normalize(&self, min_norm: &N) -> Option<Self> {
+                let n = self.norm();
+
+                if n <= *min_norm {
+                    None
+                }
+                else {
+                    Some(*self / n)
+                }
+            }
+
+            #[inline]
+            fn try_normalize_mut(&mut self, min_norm: &N) -> Option<N> {
+                let n = self.norm();
+
+                if n <= *min_norm {
+                    None
+                }
+                else {
+                    *self /= n;
+                    Some(n)
+                }
+            }
+        }
+
+        impl<N: RealApprox> InnerSpaceApprox<N> for $t<N> {
+            #[inline]
+            fn inner_product(&self, other: &Self) -> N {
+                fold_add!($(self.$compN * other.$compN ),+)
+            }
+        }
+
+        /*
+         *
+         * Exact algebraic structures.
+         *
+         */
+
+        product_space_inherit_structure!($t, GroupAbelian<Additive>);
+        product_space_inherit_structure!($t, Group<Additive>);
+        product_space_inherit_structure!($t, Loop<Additive>);
+        product_space_inherit_structure!($t, Monoid<Additive>);
+        product_space_inherit_structure!($t, Quasigroup<Additive>);
+        product_space_inherit_structure!($t, Semigroup<Additive>);
+
+        impl<N> VectorSpace<N> for $t<N>
+            where N: Copy + Neg<Output = N> + Add<N, Output = N> + AlgebraApproxEq + Field
+            { }
+
+        impl<N> Module<N> for $t<N>
+            where N: Copy + Neg<Output = N> + Add<N, Output = N> + AlgebraApproxEq + RingCommutative
+            { }
+    }
+);
+
+macro_rules! product_space_inherit_structure(
+    ($t: ident, $marker: ident<$operator: ident>) => {
+        impl<N> $marker<$operator> for $t<N>
+            where N: Copy + Neg<Output = N> + Add<N, Output = N> + AlgebraApproxEq +
+                     $marker<$operator>
+                 { }
+    }
+);
diff --git a/src/structs/common_macros.rs b/src/structs/common_macros.rs
index ac5438ad..78fabb5e 100644
--- a/src/structs/common_macros.rs
+++ b/src/structs/common_macros.rs
@@ -349,3 +349,16 @@ macro_rules! component_new(
         }
     );
 );
+
+
+macro_rules! fold_add(
+    // base case
+    ($x:expr) => {
+        $x
+    };
+    // `$x` followed by at least one `$y,`
+    ($x:expr, $($y:expr),+) => {
+        // call min! on the tail `$y`
+        Add::add($x, fold_add!($($y),+))
+    }
+);
diff --git a/src/structs/matrix.rs b/src/structs/matrix.rs
index 7e07c663..9b44fe31 100644
--- a/src/structs/matrix.rs
+++ b/src/structs/matrix.rs
@@ -18,9 +18,13 @@ use traits::structure::{Cast, Row, Column, Iterable, IterableMut, Dimension, Ind
 use traits::operations::{Absolute, Transpose, Inverse, Outer, EigenQR, Mean};
 use traits::geometry::{ToHomogeneous, FromHomogeneous, Origin};
 use linalg;
+
 #[cfg(feature="arbitrary")]
 use quickcheck::{Arbitrary, Gen};
 
+#[cfg(feature="abstract_algebra")]
+use_matrix_group_modules!();
+
 
 /// Special identity matrix. All its operation are no-ops.
 #[repr(C)]
diff --git a/src/structs/matrix_macros.rs b/src/structs/matrix_macros.rs
index 21404076..acfeb2cf 100644
--- a/src/structs/matrix_macros.rs
+++ b/src/structs/matrix_macros.rs
@@ -2,6 +2,9 @@
 
 macro_rules! matrix_impl(
     ($t: ident, $dimension: expr, $vector: ident, $dvector: ident, $($compN: ident),+) => (
+
+        matrix_group_approx_impl!($t, $($compN),+);
+
         impl<N> $t<N> {
             #[inline]
             pub fn new($($compN: N ),+) -> $t<N> {
diff --git a/src/structs/mod.rs b/src/structs/mod.rs
index 983b8ac5..befc49a7 100644
--- a/src/structs/mod.rs
+++ b/src/structs/mod.rs
@@ -17,6 +17,7 @@ pub use self::unit::Unit;
 pub use self::vectorn::VectorN;
 
 mod common_macros;
+mod algebra;
 mod dmatrix_macros;
 mod dmatrix;
 mod vectorn_macros;
diff --git a/src/structs/point.rs b/src/structs/point.rs
index 699d7aaa..4bb95b24 100644
--- a/src/structs/point.rs
+++ b/src/structs/point.rs
@@ -12,9 +12,13 @@ use traits::structure::{Cast, Dimension, Indexable, Iterable, IterableMut, Point
                         NumPoint, FloatPoint, BaseFloat, BaseNum, Bounded, Repeat};
 use traits::geometry::{Origin, FromHomogeneous, ToHomogeneous};
 use structs::vector::{Vector1, Vector2, Vector3, Vector4, Vector5, Vector6};
+
 #[cfg(feature="arbitrary")]
 use quickcheck::{Arbitrary, Gen};
 
+#[cfg(feature="abstract_algebra")]
+use_euclidean_space_modules!();
+
 
 /// Point of dimension 1.
 ///
diff --git a/src/structs/point_macros.rs b/src/structs/point_macros.rs
index 077255cb..0fdec197 100644
--- a/src/structs/point_macros.rs
+++ b/src/structs/point_macros.rs
@@ -3,6 +3,8 @@
 macro_rules! point_impl(
     ($t: ident, $tv: ident | $($compN: ident),+) => (
 
+        euclidean_space_impl!($t, $tv);
+
         /*
          *
          * Origin.
diff --git a/src/structs/quaternion.rs b/src/structs/quaternion.rs
index 737ec988..3d96e51a 100644
--- a/src/structs/quaternion.rs
+++ b/src/structs/quaternion.rs
@@ -17,7 +17,7 @@ use traits::geometry::{Norm, Rotation, RotationMatrix, Rotate, RotationTo, Trans
 use quickcheck::{Arbitrary, Gen};
 
 
-/// A quaternion. See `UnitQuaternion` for a quaternion that can be used as a rotation.
+/// A quaternion. See the `UnitQuaternion` type alias for a quaternion that can be used as a rotation.
 #[repr(C)]
 #[derive(Eq, PartialEq, RustcEncodable, RustcDecodable, Clone, Hash, Debug, Copy)]
 pub struct Quaternion<N> {
diff --git a/src/structs/rotation.rs b/src/structs/rotation.rs
index c8be3a11..c5354fba 100644
--- a/src/structs/rotation.rs
+++ b/src/structs/rotation.rs
@@ -11,9 +11,12 @@ use traits::operations::{Absolute, Inverse, Transpose, ApproxEq};
 use structs::vector::{Vector1, Vector2, Vector3};
 use structs::point::{Point2, Point3};
 use structs::matrix::{Matrix2, Matrix3, Matrix4};
+
 #[cfg(feature="arbitrary")]
 use quickcheck::{Arbitrary, Gen};
 
+#[cfg(feature="abstract_algebra")]
+use_special_orthogonal_group_modules!();
 
 /// Two dimensional rotation matrix.
 #[repr(C)]
diff --git a/src/structs/rotation_macros.rs b/src/structs/rotation_macros.rs
index 22307968..dfcdd835 100644
--- a/src/structs/rotation_macros.rs
+++ b/src/structs/rotation_macros.rs
@@ -2,6 +2,9 @@
 
 macro_rules! rotation_impl(
     ($t: ident, $submatrix: ident, $vector: ident, $rotvector: ident, $point: ident, $homogeneous: ident) => (
+
+        special_orthogonal_group_impl!($t, $point, $vector);
+
         impl<N> $t<N> {
             /// This rotation's underlying matrix.
             #[inline]
diff --git a/src/structs/unit.rs b/src/structs/unit.rs
index f6730e96..d48af8e1 100644
--- a/src/structs/unit.rs
+++ b/src/structs/unit.rs
@@ -2,6 +2,8 @@ use traits::geometry::Norm;
 
 
 /// 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, RustcEncodable, RustcDecodable, Clone, Hash, Debug, Copy)]
 pub struct Unit<T> {
@@ -47,7 +49,7 @@ impl<T: Norm> Unit<T> {
     /// Normalizes this value again. This is useful when repeated computations 
     /// might cause a drift in the norm because of float inaccuracies.
     ///
-    /// Returns the norm beform re-normalization (should be close to `1.0`).
+    /// Returns the norm before re-normalization (should be close to `1.0`).
     #[inline]
     pub fn renormalize(&mut self) -> T::NormType {
         self.v.normalize_mut()
diff --git a/src/structs/vector.rs b/src/structs/vector.rs
index 45ac1646..1c3fa4d3 100644
--- a/src/structs/vector.rs
+++ b/src/structs/vector.rs
@@ -17,6 +17,9 @@ use structs::point::{Point1, Point2, Point3, Point4, Point5, Point6};
 #[cfg(feature="arbitrary")]
 use quickcheck::{Arbitrary, Gen};
 
+#[cfg(feature="abstract_algebra")]
+use_vector_space_modules!();
+
 
 /// Vector of dimension 1.
 ///
@@ -29,7 +32,7 @@ pub struct Vector1<N> {
     pub x: N
 }
 
-vector_impl!(Vector1, Point1, x);
+vector_impl!(Vector1, Point1, 1, x);
 vectorlike_impl!(Vector1, 1, x);
 from_iterator_impl!(Vector1, iterator);
 // (specialized); basis_impl!(Vector1, 1);
@@ -51,7 +54,7 @@ pub struct Vector2<N> {
     pub y: N
 }
 
-vector_impl!(Vector2, Point2, x, y);
+vector_impl!(Vector2, Point2, 2, x, y);
 vectorlike_impl!(Vector2, 2, x, y);
 from_iterator_impl!(Vector2, iterator, iterator);
 // (specialized); basis_impl!(Vector2, 1);
@@ -75,7 +78,7 @@ pub struct Vector3<N> {
     pub z: N
 }
 
-vector_impl!(Vector3, Point3, x, y, z);
+vector_impl!(Vector3, Point3, 3, x, y, z);
 vectorlike_impl!(Vector3, 3, x, y, z);
 from_iterator_impl!(Vector3, iterator, iterator, iterator);
 // (specialized); basis_impl!(Vector3, 1);
@@ -100,7 +103,7 @@ pub struct Vector4<N> {
     pub w: N
 }
 
-vector_impl!(Vector4, Point4, x, y, z, w);
+vector_impl!(Vector4, Point4, 4, x, y, z, w);
 vectorlike_impl!(Vector4, 4, x, y, z, w);
 from_iterator_impl!(Vector4, iterator, iterator, iterator, iterator);
 basis_impl!(Vector4, 4);
@@ -128,7 +131,7 @@ pub struct Vector5<N> {
     pub a: N
 }
 
-vector_impl!(Vector5, Point5, x, y, z, w, a);
+vector_impl!(Vector5, Point5, 5, x, y, z, w, a);
 vectorlike_impl!(Vector5, 5, x, y, z, w, a);
 from_iterator_impl!(Vector5, iterator, iterator, iterator, iterator, iterator);
 basis_impl!(Vector5, 5);
@@ -157,7 +160,7 @@ pub struct Vector6<N> {
     pub b: N
 }
 
-vector_impl!(Vector6, Point6, x, y, z, w, a, b);
+vector_impl!(Vector6, Point6, 6, x, y, z, w, a, b);
 vectorlike_impl!(Vector6, 6, x, y, z, w, a, b);
 from_iterator_impl!(Vector6, iterator, iterator, iterator, iterator, iterator, iterator);
 
diff --git a/src/structs/vector_macros.rs b/src/structs/vector_macros.rs
index e2343261..7bd0e931 100644
--- a/src/structs/vector_macros.rs
+++ b/src/structs/vector_macros.rs
@@ -270,7 +270,7 @@ macro_rules! vectorlike_impl(
 );
 
 macro_rules! vector_impl(
-    ($t: ident, $tp: ident, $($compN: ident),+) => (
+    ($t: ident, $tp: ident, $dimension: expr, $($compN: ident),+) => (
         pointwise_add!($t, $($compN),+);
         pointwise_sub!($t, $($compN),+);
         pointwise_mul!($t, $($compN),+);
@@ -279,7 +279,7 @@ macro_rules! vector_impl(
         componentwise_one!($t, $($compN),+);
         componentwise_absolute!($t, $($compN),+);
         component_basis_element!($t, $($compN),+);
-
+        vector_space_impl!($t, $dimension, $($compN),+);
 
         /*
          *
@@ -289,11 +289,10 @@ macro_rules! vector_impl(
         impl<N: BaseNum> Dot<N> for $t<N> {
             #[inline]
             fn dot(&self, other: &$t<N>) -> N {
-                add!($(self.$compN * other.$compN ),+)
+                fold_add!($(self.$compN * other.$compN ),+)
             }
         }
 
-
         /*
          *
          * Norm
@@ -597,18 +596,6 @@ macro_rules! basis_impl(
 );
 
 
-macro_rules! add (
-    // base case
-    ($x:expr) => {
-        $x
-    };
-    // `$x` followed by at least one `$y,`
-    ($x:expr, $($y:expr),+) => {
-        // call min! on the tail `$y`
-        Add::add($x, add!($($y),+))
-    }
-);
-
 
 macro_rules! from_iterator_impl(
     ($t: ident, $param0: ident) => (
diff --git a/src/structs/vectorn.rs b/src/structs/vectorn.rs
index 9b0fec3e..c5433dce 100644
--- a/src/structs/vectorn.rs
+++ b/src/structs/vectorn.rs
@@ -11,7 +11,7 @@ use traits::structure::{Iterable, IterableMut, Indexable, Shape, BaseFloat, Base
 #[cfg(feature="arbitrary")]
 use quickcheck::{Arbitrary, Gen};
 
-/// A static array of arbitrary dimension.
+/// A stack-allocated vector of arbitrary dimension.
 #[repr(C)]
 #[derive(Eq, PartialEq, Debug)] //  FIXME: Hash, RustcEncodable, RustcDecodable
 pub struct VectorN<N, D: ArrayLength<N>> {