nalgebra/src/lib.rs

/*!
# nalgebra

**nalgebra** is a 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](https://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]
// TODO: replace the * by the latest version.
nalgebra = "*"
```


Most useful functionalities of **nalgebra** are grouped in the root module `nalgebra::`.

However, the recommended way to use **nalgebra** is to import types and traits
explicitly, and call free-functions using the `na::` prefix:

```
#[macro_use]
extern crate approx; // For the macro relative_eq!
extern crate nalgebra as na;
use na::{Vector3, Rotation3};

fn main() {
    let axis  = Vector3::x_axis();
    let angle = 1.57;
    let b     = Rotation3::from_axis_angle(&axis, angle);

    relative_eq!(b.axis().unwrap(), axis);
    relative_eq!(b.angle(), angle);
}
```


## 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:

* A single parametrizable type [`Matrix`](Matrix) for vectors, (square or rectangular) matrices, and
  slices with dimensions known either at compile-time (using type-level integers) or at runtime.
* Matrices and vectors with compile-time sizes are statically allocated while dynamic ones are
  allocated on the heap.
* Convenient aliases for low-dimensional matrices and vectors: [`Vector1`](Vector1) to
  [`Vector6`](Vector6) and [`Matrix1x1`](Matrix1) to [`Matrix6x6`](Matrix6), including rectangular
  matrices like [`Matrix2x5`](Matrix2x5).
* Points sizes known at compile time, and convenience aliases: [`Point1`](Point1) to
  [`Point6`](Point6).
* Translation (seen as a transformation that composes by multiplication):
  [`Translation2`](Translation2), [`Translation3`](Translation3).
* Rotation matrices: [`Rotation2`](Rotation2), [`Rotation3`](Rotation3).
* Quaternions: [`Quaternion`](Quaternion), [`UnitQuaternion`](UnitQuaternion) (for 3D rotation).
* Unit complex numbers can be used for 2D rotation: [`UnitComplex`](UnitComplex).
* Algebraic entities with a norm equal to one: [`Unit<T>`](Unit), e.g., `Unit<Vector3<f32>>`.
* Isometries (translation ⨯ rotation): [`Isometry2`](Isometry2), [`Isometry3`](Isometry3)
* Similarity transformations (translation ⨯ rotation ⨯ uniform scale):
  [`Similarity2`](Similarity2), [`Similarity3`](Similarity3).
* Affine transformations stored as a homogeneous matrix:
  [`Affine2`](Affine2), [`Affine3`](Affine3).
* Projective (i.e. invertible) transformations stored as a homogeneous matrix:
  [`Projective2`](Projective2), [`Projective3`](Projective3).
* General transformations that does not have to be invertible, stored as a homogeneous matrix:
  [`Transform2`](Transform2), [`Transform3`](Transform3).
* 3D projections for computer graphics: [`Perspective3`](Perspective3),
  [`Orthographic3`](Orthographic3).
* Matrix factorizations: [`Cholesky`](Cholesky), [`QR`](QR), [`LU`](LU), [`FullPivLU`](FullPivLU),
  [`SVD`](SVD), [`Schur`](Schur), [`Hessenberg`](Hessenberg), [`SymmetricEigen`](SymmetricEigen).
* Insertion and removal of rows of columns of a matrix.
*/

#![deny(
    missing_docs,
    nonstandard_style,
    unused_variables,
    unused_mut,
    unused_parens,
    unused_qualifications,
    rust_2018_idioms,
    rust_2018_compatibility,
    future_incompatible,
    missing_copy_implementations
)]
#![cfg_attr(feature = "rkyv-serialize-no-std", warn(unused_results))] // TODO: deny this once bytecheck stops generating warnings.
#![cfg_attr(not(feature = "rkyv-serialize-no-std"), deny(unused_results))]
#![doc(
    html_favicon_url = "https://nalgebra.org/img/favicon.ico",
    html_root_url = "https://docs.rs/nalgebra/0.25.0"
)]
#![cfg_attr(not(feature = "std"), no_std)]

#[cfg(feature = "rand-no-std")]
extern crate rand_package as rand;

#[cfg(feature = "serde-serialize-no-std")]
#[macro_use]
extern crate serde;

#[macro_use]
extern crate approx;
extern crate num_traits as num;

#[cfg(all(feature = "alloc", not(feature = "std")))]
#[cfg_attr(test, macro_use)]
extern crate alloc;

#[cfg(not(feature = "std"))]
extern crate core as std;

#[macro_use]
#[cfg(feature = "io")]
extern crate pest_derive;

pub mod base;
#[cfg(feature = "debug")]
pub mod debug;
pub mod geometry;
#[cfg(feature = "io")]
pub mod io;
pub mod linalg;
#[cfg(feature = "proptest-support")]
pub mod proptest;
#[cfg(feature = "sparse")]
pub mod sparse;
mod third_party;

pub use crate::base::*;
pub use crate::geometry::*;
pub use crate::linalg::*;
#[cfg(feature = "sparse")]
pub use crate::sparse::*;
#[cfg(feature = "std")]
#[deprecated(
    note = "The 'core' module is being renamed to 'base' to avoid conflicts with the 'core' crate."
)]
pub use base as core;

#[cfg(feature = "macros")]
pub use nalgebra_macros::{dmatrix, dvector, matrix, point, vector};

use simba::scalar::SupersetOf;
use std::cmp::{self, Ordering, PartialOrd};

use num::{One, Signed, Zero};

use base::allocator::Allocator;
pub use num_complex::Complex;
pub use simba::scalar::{
    ClosedAdd, ClosedDiv, ClosedMul, ClosedSub, ComplexField, Field, RealField,
};
pub use simba::simd::{SimdBool, SimdComplexField, SimdPartialOrd, SimdRealField, SimdValue};

/// Gets the multiplicative identity element.
///
/// # See also:
///
/// * [`origin`](../nalgebra/fn.origin.html)
/// * [`zero`](fn.zero.html)
#[inline]
pub fn one<T: One>() -> T {
    T::one()
}

/// Gets the additive identity element.
///
/// # See also:
///
/// * [`one`](fn.one.html)
/// * [`origin`](../nalgebra/fn.origin.html)
#[inline]
pub fn zero<T: Zero>() -> T {
    T::zero()
}

/*
 *
 * Ordering
 *
 */
// XXX: this is very naive and could probably be optimized for specific types.
// XXX: also, we might just want to use divisions, but assuming `val` is usually not far from `min`
// or `max`, would it still be more efficient?
/// Wraps `val` into the range `[min, max]` using modular arithmetics.
///
/// The range must not be empty.
#[must_use]
#[inline]
pub fn wrap<T>(mut val: T, min: T, max: T) -> T
where
    T: Copy + PartialOrd + ClosedAdd + ClosedSub,
{
    assert!(min < max, "Invalid wrapping bounds.");
    let width = max - min;

    if val < min {
        val += width;

        while val < min {
            val += width
        }
    } else if val > max {
        val -= width;

        while val > max {
            val -= width
        }
    }

    val
}

/// Returns a reference to the input value clamped to the interval `[min, max]`.
///
/// In particular:
///     * If `min < val < max`, this returns `val`.
///     * If `val <= min`, this returns `min`.
///     * If `val >= max`, this returns `max`.
#[must_use]
#[inline]
pub fn clamp<T: PartialOrd>(val: T, min: T, max: T) -> T {
    if val > min {
        if val < max {
            val
        } else {
            max
        }
    } else {
        min
    }
}

/// Same as `cmp::max`.
#[inline]
pub fn max<T: Ord>(a: T, b: T) -> T {
    cmp::max(a, b)
}

/// Same as `cmp::min`.
#[inline]
pub fn min<T: Ord>(a: T, b: T) -> T {
    cmp::min(a, b)
}

/// The absolute value of `a`.
///
/// Deprecated: Use [`Matrix::abs`] or [`ComplexField::abs`] instead.
#[deprecated(note = "use the inherent method `Matrix::abs` or `ComplexField::abs` instead")]
#[inline]
pub fn abs<T: Signed>(a: &T) -> T {
    a.abs()
}

/// Returns the infimum of `a` and `b`.
#[deprecated(note = "use the inherent method `Matrix::inf` instead")]
#[inline]
pub fn inf<T, R: Dim, C: Dim>(a: &OMatrix<T, R, C>, b: &OMatrix<T, R, C>) -> OMatrix<T, R, C>
where
    T: Scalar + SimdPartialOrd,
    DefaultAllocator: Allocator<T, R, C>,
{
    a.inf(b)
}

/// Returns the supremum of `a` and `b`.
#[deprecated(note = "use the inherent method `Matrix::sup` instead")]
#[inline]
pub fn sup<T, R: Dim, C: Dim>(a: &OMatrix<T, R, C>, b: &OMatrix<T, R, C>) -> OMatrix<T, R, C>
where
    T: Scalar + SimdPartialOrd,
    DefaultAllocator: Allocator<T, R, C>,
{
    a.sup(b)
}

/// Returns simultaneously the infimum and supremum of `a` and `b`.
#[deprecated(note = "use the inherent method `Matrix::inf_sup` instead")]
#[inline]
pub fn inf_sup<T, R: Dim, C: Dim>(
    a: &OMatrix<T, R, C>,
    b: &OMatrix<T, R, C>,
) -> (OMatrix<T, R, C>, OMatrix<T, R, C>)
where
    T: Scalar + SimdPartialOrd,
    DefaultAllocator: Allocator<T, R, C>,
{
    a.inf_sup(b)
}

/// Compare `a` and `b` using a partial ordering relation.
#[inline]
pub fn partial_cmp<T: PartialOrd>(a: &T, b: &T) -> Option<Ordering> {
    a.partial_cmp(b)
}

/// Returns `true` iff `a` and `b` are comparable and `a < b`.
#[inline]
pub fn partial_lt<T: PartialOrd>(a: &T, b: &T) -> bool {
    a.lt(b)
}

/// Returns `true` iff `a` and `b` are comparable and `a <= b`.
#[inline]
pub fn partial_le<T: PartialOrd>(a: &T, b: &T) -> bool {
    a.le(b)
}

/// Returns `true` iff `a` and `b` are comparable and `a > b`.
#[inline]
pub fn partial_gt<T: PartialOrd>(a: &T, b: &T) -> bool {
    a.gt(b)
}

/// Returns `true` iff `a` and `b` are comparable and `a >= b`.
#[inline]
pub fn partial_ge<T: PartialOrd>(a: &T, b: &T) -> bool {
    a.ge(b)
}

/// Return the minimum of `a` and `b` if they are comparable.
#[inline]
pub fn partial_min<'a, T: PartialOrd>(a: &'a T, b: &'a T) -> Option<&'a T> {
    if let Some(ord) = a.partial_cmp(b) {
        match ord {
            Ordering::Greater => Some(b),
            _ => Some(a),
        }
    } else {
        None
    }
}

/// Return the maximum of `a` and `b` if they are comparable.
#[inline]
pub fn partial_max<'a, T: PartialOrd>(a: &'a T, b: &'a T) -> Option<&'a T> {
    if let Some(ord) = a.partial_cmp(b) {
        match ord {
            Ordering::Less => Some(b),
            _ => Some(a),
        }
    } else {
        None
    }
}

/// Clamp `value` between `min` and `max`. Returns `None` if `value` is not comparable to
/// `min` or `max`.
#[inline]
pub fn partial_clamp<'a, T: PartialOrd>(value: &'a T, min: &'a T, max: &'a T) -> Option<&'a T> {
    if let (Some(cmp_min), Some(cmp_max)) = (value.partial_cmp(min), value.partial_cmp(max)) {
        if cmp_min == Ordering::Less {
            Some(min)
        } else if cmp_max == Ordering::Greater {
            Some(max)
        } else {
            Some(value)
        }
    } else {
        None
    }
}

/// Sorts two values in increasing order using a partial ordering.
#[inline]
pub fn partial_sort2<'a, T: PartialOrd>(a: &'a T, b: &'a T) -> Option<(&'a T, &'a T)> {
    if let Some(ord) = a.partial_cmp(b) {
        match ord {
            Ordering::Less => Some((a, b)),
            _ => Some((b, a)),
        }
    } else {
        None
    }
}

/*
 *
 * Point operations.
 *
 */
/// The center of two points.
///
/// # See also:
///
/// * [distance](fn.distance.html)
/// * [`distance_squared`](fn.distance_squared.html)
#[inline]
pub fn center<T: SimdComplexField, const D: usize>(
    p1: &Point<T, D>,
    p2: &Point<T, D>,
) -> Point<T, D> {
    ((&p1.coords + &p2.coords) * convert::<_, T>(0.5)).into()
}

/// The distance between two points.
///
/// # See also:
///
/// * [center](fn.center.html)
/// * [`distance_squared`](fn.distance_squared.html)
#[inline]
pub fn distance<T: SimdComplexField, const D: usize>(
    p1: &Point<T, D>,
    p2: &Point<T, D>,
) -> T::SimdRealField {
    (&p2.coords - &p1.coords).norm()
}

/// The squared distance between two points.
///
/// # See also:
///
/// * [center](fn.center.html)
/// * [distance](fn.distance.html)
#[inline]
pub fn distance_squared<T: SimdComplexField, const D: usize>(
    p1: &Point<T, D>,
    p2: &Point<T, D>,
) -> T::SimdRealField {
    (&p2.coords - &p1.coords).norm_squared()
}

/*
 * Cast
 */
/// Converts an object from one type to an equivalent or more general one.
///
/// See also [`try_convert`](fn.try_convert.html) for conversion to more specific types.
///
/// # See also:
///
/// * [`convert_ref`](fn.convert_ref.html)
/// * [`convert_ref_unchecked`](fn.convert_ref_unchecked.html)
/// * [`is_convertible`](../nalgebra/fn.is_convertible.html)
/// * [`try_convert`](fn.try_convert.html)
/// * [`try_convert_ref`](fn.try_convert_ref.html)
#[inline]
pub fn convert<From, To: SupersetOf<From>>(t: From) -> To {
    To::from_subset(&t)
}

/// Attempts to convert an object to a more specific one.
///
/// See also [`convert`](fn.convert.html) for conversion to more general types.
///
/// # See also:
///
/// * [convert](fn.convert.html)
/// * [`convert_ref`](fn.convert_ref.html)
/// * [`convert_ref_unchecked`](fn.convert_ref_unchecked.html)
/// * [`is_convertible`](../nalgebra/fn.is_convertible.html)
/// * [`try_convert_ref`](fn.try_convert_ref.html)
#[inline]
pub fn try_convert<From: SupersetOf<To>, To>(t: From) -> Option<To> {
    t.to_subset()
}

/// Indicates if [`try_convert`](fn.try_convert.html) will succeed without
/// actually performing the conversion.
///
/// # See also:
///
/// * [convert](fn.convert.html)
/// * [`convert_ref`](fn.convert_ref.html)
/// * [`convert_ref_unchecked`](fn.convert_ref_unchecked.html)
/// * [`try_convert`](fn.try_convert.html)
/// * [`try_convert_ref`](fn.try_convert_ref.html)
#[inline]
pub fn is_convertible<From: SupersetOf<To>, To>(t: &From) -> bool {
    t.is_in_subset()
}

/// Use with care! Same as [`try_convert`](fn.try_convert.html) but
/// without any property checks.
///
/// # See also:
///
/// * [convert](fn.convert.html)
/// * [`convert_ref`](fn.convert_ref.html)
/// * [`convert_ref_unchecked`](fn.convert_ref_unchecked.html)
/// * [`is_convertible`](../nalgebra/fn.is_convertible.html)
/// * [`try_convert`](fn.try_convert.html)
/// * [`try_convert_ref`](fn.try_convert_ref.html)
#[inline]
pub fn convert_unchecked<From: SupersetOf<To>, To>(t: From) -> To {
    t.to_subset_unchecked()
}

/// Converts an object from one type to an equivalent or more general one.
///
/// # See also:
///
/// * [convert](fn.convert.html)
/// * [`convert_ref_unchecked`](fn.convert_ref_unchecked.html)
/// * [`is_convertible`](../nalgebra/fn.is_convertible.html)
/// * [`try_convert`](fn.try_convert.html)
/// * [`try_convert_ref`](fn.try_convert_ref.html)
#[inline]
pub fn convert_ref<From, To: SupersetOf<From>>(t: &From) -> To {
    To::from_subset(t)
}

/// Attempts to convert an object to a more specific one.
///
/// # See also:
///
/// * [convert](fn.convert.html)
/// * [`convert_ref`](fn.convert_ref.html)
/// * [`convert_ref_unchecked`](fn.convert_ref_unchecked.html)
/// * [`is_convertible`](../nalgebra/fn.is_convertible.html)
/// * [`try_convert`](fn.try_convert.html)
#[inline]
pub fn try_convert_ref<From: SupersetOf<To>, To>(t: &From) -> Option<To> {
    t.to_subset()
}

/// Use with care! Same as [`try_convert`](fn.try_convert.html) but
/// without any property checks.
///
/// # See also:
///
/// * [convert](fn.convert.html)
/// * [`convert_ref`](fn.convert_ref.html)
/// * [`is_convertible`](../nalgebra/fn.is_convertible.html)
/// * [`try_convert`](fn.try_convert.html)
/// * [`try_convert_ref`](fn.try_convert_ref.html)
#[inline]
pub fn convert_ref_unchecked<From: SupersetOf<To>, To>(t: &From) -> To {
    t.to_subset_unchecked()
}