euclid/

rotation.rs

// Copyright 2013 The Servo Project Developers. See the COPYRIGHT
// file at the top-level directory of this distribution.
//
// Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
// http://www.apache.org/licenses/LICENSE-2.0> or the MIT license
// <LICENSE-MIT or http://opensource.org/licenses/MIT>, at your
// option. This file may not be copied, modified, or distributed
// except according to those terms.

use approxeq::ApproxEq;
use num_traits::{Float, FloatConst, One, Zero, NumCast};
use core::fmt;
use core::ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Neg, Rem, Sub, SubAssign};
use core::marker::PhantomData;
use trig::Trig;
use {TypedPoint2D, TypedPoint3D, TypedVector2D, TypedVector3D, Vector3D, point2, point3, vec3};
use {TypedTransform2D, TypedTransform3D, UnknownUnit};

/// An angle in radians
#[derive(Copy, Clone, Debug, PartialEq, Eq, PartialOrd, Hash)]
#[cfg_attr(feature = "serde", derive(Serialize, Deserialize))]
pub struct Angle<T> {
    pub radians: T,
}

impl<T> Angle<T> {
    #[inline]
    pub fn radians(radians: T) -> Self {
        Angle { radians }
    }

    #[inline]
    pub fn get(self) -> T {
        self.radians
    }
}

impl<T> Angle<T>
where
    T: Trig,
{
    #[inline]
    pub fn degrees(deg: T) -> Self {
        Angle {
            radians: T::degrees_to_radians(deg),
        }
    }

    #[inline]
    pub fn to_degrees(self) -> T {
        T::radians_to_degrees(self.radians)
    }
}

impl<T> Angle<T>
where
    T: Rem<Output = T> + Sub<Output = T> + Add<Output = T> + Zero + FloatConst + PartialOrd + Copy,
{
    /// Returns this angle in the [0..2*PI[ range.
    pub fn positive(&self) -> Self {
        let two_pi = T::PI() + T::PI();
        let mut a = self.radians % two_pi;
        if a < T::zero() {
            a = a + two_pi;
        }
        Angle::radians(a)
    }

    /// Returns this angle in the ]-PI..PI] range.
    pub fn signed(&self) -> Self {
        Angle::pi() - (Angle::pi() - *self).positive()
    }
}

impl<T> Angle<T>
where
    T: Float,
{
    /// Returns (sin(self), cos(self)).
    pub fn sin_cos(self) -> (T, T) {
        self.radians.sin_cos()
    }
}

impl<T> Angle<T>
where
    T: Zero,
{
    pub fn zero() -> Self {
        Angle::radians(T::zero())
    }
}

impl<T> Angle<T>
where
    T: FloatConst + Add<Output = T>,
{
    pub fn pi() -> Self {
        Angle::radians(T::PI())
    }

    pub fn two_pi() -> Self {
        Angle::radians(T::PI() + T::PI())
    }

    pub fn frac_pi_2() -> Self {
        Angle::radians(T::FRAC_PI_2())
    }

    pub fn frac_pi_3() -> Self {
        Angle::radians(T::FRAC_PI_3())
    }

    pub fn frac_pi_4() -> Self {
        Angle::radians(T::FRAC_PI_4())
    }
}

impl<T: Clone + Add<T, Output = T>> Add for Angle<T> {
    type Output = Angle<T>;
    fn add(self, other: Angle<T>) -> Angle<T> {
        Angle::radians(self.radians + other.radians)
    }
}

impl<T: Clone + AddAssign<T>> AddAssign for Angle<T> {
    fn add_assign(&mut self, other: Angle<T>) {
        self.radians += other.radians;
    }
}

impl<T: Clone + Sub<T, Output = T>> Sub<Angle<T>> for Angle<T> {
    type Output = Angle<T>;
    fn sub(self, other: Angle<T>) -> <Self as Sub>::Output {
        Angle::radians(self.radians - other.radians)
    }
}

impl<T: Clone + SubAssign<T>> SubAssign for Angle<T> {
    fn sub_assign(&mut self, other: Angle<T>) {
        self.radians -= other.radians;
    }
}

impl<T: Clone + Div<T, Output = T>> Div<Angle<T>> for Angle<T> {
    type Output = T;
    #[inline]
    fn div(self, other: Angle<T>) -> T {
        self.radians / other.radians
    }
}

impl<T: Clone + Div<T, Output = T>> Div<T> for Angle<T> {
    type Output = Angle<T>;
    #[inline]
    fn div(self, factor: T) -> Angle<T> {
        Angle::radians(self.radians / factor)
    }
}

impl<T: Clone + DivAssign<T>> DivAssign<T> for Angle<T> {
    fn div_assign(&mut self, factor: T) {
        self.radians /= factor;
    }
}

impl<T: Clone + Mul<T, Output = T>> Mul<T> for Angle<T> {
    type Output = Angle<T>;
    #[inline]
    fn mul(self, factor: T) -> Angle<T> {
        Angle::radians(self.radians * factor)
    }
}

impl<T: Clone + MulAssign<T>> MulAssign<T> for Angle<T> {
    fn mul_assign(&mut self, factor: T) {
        self.radians *= factor;
    }
}

impl<T: Neg<Output = T>> Neg for Angle<T> {
    type Output = Self;
    fn neg(self) -> Self {
        Angle::radians(-self.radians)
    }
}

/// A transform that can represent rotations in 2d, represented as an angle in radians.
#[derive(EuclidMatrix)]
#[repr(C)]
pub struct TypedRotation2D<T, Src, Dst> {
    pub angle : T,
    #[doc(hidden)]
    pub _unit: PhantomData<(Src, Dst)>,
}

/// The default 2d rotation type with no units.
pub type Rotation2D<T> = TypedRotation2D<T, UnknownUnit, UnknownUnit>;

impl<T, Src, Dst> TypedRotation2D<T, Src, Dst> {
    #[inline]
    /// Creates a rotation from an angle in radians.
    pub fn new(angle: Angle<T>) -> Self {
        TypedRotation2D {
            angle: angle.radians,
            _unit: PhantomData,
        }
    }

    pub fn radians(angle: T) -> Self {
        Self::new(Angle::radians(angle))
    }

    /// Creates the identity rotation.
    #[inline]
    pub fn identity() -> Self
    where
        T: Zero,
    {
        Self::radians(T::zero())
    }
}

impl<T, Src, Dst> TypedRotation2D<T, Src, Dst>
where
    T: Clone,
{
    /// Returns self.angle as a strongly typed `Angle<T>`.
    pub fn get_angle(&self) -> Angle<T> {
        Angle::radians(self.angle.clone())
    }
}

impl<T, Src, Dst> TypedRotation2D<T, Src, Dst>
where
    T: Copy
        + Clone
        + Add<T, Output = T>
        + Sub<T, Output = T>
        + Mul<T, Output = T>
        + Div<T, Output = T>
        + Neg<Output = T>
        + PartialOrd
        + Float
        + One
        + Zero,
{
    /// Creates a 3d rotation (around the z axis) from this 2d rotation.
    #[inline]
    pub fn to_3d(&self) -> TypedRotation3D<T, Src, Dst> {
        TypedRotation3D::around_z(self.get_angle())
    }

    /// Returns the inverse of this rotation.
    #[inline]
    pub fn inverse(&self) -> TypedRotation2D<T, Dst, Src> {
        TypedRotation2D::radians(-self.angle)
    }

    /// Returns a rotation representing the other rotation followed by this rotation.
    #[inline]
    pub fn pre_rotate<NewSrc>(
        &self,
        other: &TypedRotation2D<T, NewSrc, Src>,
    ) -> TypedRotation2D<T, NewSrc, Dst> {
        TypedRotation2D::radians(self.angle + other.angle)
    }

    /// Returns a rotation representing this rotation followed by the other rotation.
    #[inline]
    pub fn post_rotate<NewDst>(
        &self,
        other: &TypedRotation2D<T, Dst, NewDst>,
    ) -> TypedRotation2D<T, Src, NewDst> {
        other.pre_rotate(self)
    }

    /// Returns the given 2d point transformed by this rotation.
    ///
    /// The input point must be use the unit Src, and the returned point has the unit Dst.
    #[inline]
    pub fn transform_point(&self, point: &TypedPoint2D<T, Src>) -> TypedPoint2D<T, Dst> {
        let (sin, cos) = Float::sin_cos(self.angle);
        point2(point.x * cos - point.y * sin, point.y * cos + point.x * sin)
    }

    /// Returns the given 2d vector transformed by this rotation.
    ///
    /// The input point must be use the unit Src, and the returned point has the unit Dst.
    #[inline]
    pub fn transform_vector(&self, vector: &TypedVector2D<T, Src>) -> TypedVector2D<T, Dst> {
        self.transform_point(&vector.to_point()).to_vector()
    }
}

impl<T, Src, Dst> TypedRotation2D<T, Src, Dst>
where
    T: Copy
        + Clone
        + Add<T, Output = T>
        + Mul<T, Output = T>
        + Div<T, Output = T>
        + Sub<T, Output = T>
        + Trig
        + PartialOrd
        + One
        + Zero,
{
    /// Returns the matrix representation of this rotation.
    #[inline]
    pub fn to_transform(&self) -> TypedTransform2D<T, Src, Dst> {
        TypedTransform2D::create_rotation(self.get_angle())
    }
}

/// A transform that can represent rotations in 3d, represented as a quaternion.
///
/// Most methods expect the quaternion to be normalized.
/// When in doubt, use `unit_quaternion` instead of `quaternion` to create
/// a rotation as the former will ensure that its result is normalized.
///
/// Some people use the `x, y, z, w` (or `w, x, y, z`) notations. The equivalence is
/// as follows: `x -> i`, `y -> j`, `z -> k`, `w -> r`.
/// The memory layout of this type corresponds to the `x, y, z, w` notation
#[derive(EuclidMatrix)]
#[repr(C)]
pub struct TypedRotation3D<T, Src, Dst> {
    /// Component multiplied by the imaginary number `i`.
    pub i: T,
    /// Component multiplied by the imaginary number `j`.
    pub j: T,
    /// Component multiplied by the imaginary number `k`.
    pub k: T,
    /// The real part.
    pub r: T,
    #[doc(hidden)]
    pub _unit: PhantomData<(Src, Dst)>,
}

/// The default 3d rotation type with no units.
pub type Rotation3D<T> = TypedRotation3D<T, UnknownUnit, UnknownUnit>;

impl<T, Src, Dst> TypedRotation3D<T, Src, Dst> {
    /// Creates a rotation around from a quaternion representation.
    ///
    /// The parameters are a, b, c and r compose the quaternion `a*i + b*j + c*k + r`
    /// where `a`, `b` and `c` describe the vector part and the last parameter `r` is
    /// the real part.
    ///
    /// The resulting quaternion is not necessarily normalized. See `unit_quaternion`.
    #[inline]
    pub fn quaternion(a: T, b: T, c: T, r: T) -> Self {
        TypedRotation3D {
            i: a,
            j: b,
            k: c,
            r,
            _unit: PhantomData,
        }
    }
}

impl<T, Src, Dst> TypedRotation3D<T, Src, Dst>
where
    T: Copy,
{
    /// Returns the vector part (i, j, k) of this quaternion.
    #[inline]
    pub fn vector_part(&self) -> Vector3D<T> {
        vec3(self.i, self.j, self.k)
    }
}

impl<T, Src, Dst> TypedRotation3D<T, Src, Dst>
where
    T: Float,
{
    /// Creates the identity rotation.
    #[inline]
    pub fn identity() -> Self {
        let zero = T::zero();
        let one = T::one();
        Self::quaternion(zero, zero, zero, one)
    }

    /// Creates a rotation around from a quaternion representation and normalizes it.
    ///
    /// The parameters are a, b, c and r compose the quaternion `a*i + b*j + c*k + r`
    /// before normalization, where `a`, `b` and `c` describe the vector part and the
    /// last parameter `r` is the real part.
    #[inline]
    pub fn unit_quaternion(i: T, j: T, k: T, r: T) -> Self {
        Self::quaternion(i, j, k, r).normalize()
    }

    /// Creates a rotation around a given axis.
    pub fn around_axis(axis: TypedVector3D<T, Src>, angle: Angle<T>) -> Self {
        let axis = axis.normalize();
        let two = T::one() + T::one();
        let (sin, cos) = Angle::sin_cos(angle / two);
        Self::quaternion(axis.x * sin, axis.y * sin, axis.z * sin, cos)
    }

    /// Creates a rotation around the x axis.
    pub fn around_x(angle: Angle<T>) -> Self {
        let zero = Zero::zero();
        let two = T::one() + T::one();
        let (sin, cos) = Angle::sin_cos(angle / two);
        Self::quaternion(sin, zero, zero, cos)
    }

    /// Creates a rotation around the y axis.
    pub fn around_y(angle: Angle<T>) -> Self {
        let zero = Zero::zero();
        let two = T::one() + T::one();
        let (sin, cos) = Angle::sin_cos(angle / two);
        Self::quaternion(zero, sin, zero, cos)
    }

    /// Creates a rotation around the z axis.
    pub fn around_z(angle: Angle<T>) -> Self {
        let zero = Zero::zero();
        let two = T::one() + T::one();
        let (sin, cos) = Angle::sin_cos(angle / two);
        Self::quaternion(zero, zero, sin, cos)
    }

    /// Creates a rotation from Euler angles.
    ///
    /// The rotations are applied in roll then pitch then yaw order.
    ///
    ///  - Roll (also called bank) is a rotation around the x axis.
    ///  - Pitch (also called bearing) is a rotation around the y axis.
    ///  - Yaw (also called heading) is a rotation around the z axis.
    pub fn euler(roll: Angle<T>, pitch: Angle<T>, yaw: Angle<T>) -> Self {
        let half = T::one() / (T::one() + T::one());

        let (sy, cy) = Float::sin_cos(half * yaw.get());
        let (sp, cp) = Float::sin_cos(half * pitch.get());
        let (sr, cr) = Float::sin_cos(half * roll.get());

        Self::quaternion(
            cy * sr * cp - sy * cr * sp,
            cy * cr * sp + sy * sr * cp,
            sy * cr * cp - cy * sr * sp,
            cy * cr * cp + sy * sr * sp,
        )
    }

    /// Returns the inverse of this rotation.
    #[inline]
    pub fn inverse(&self) -> TypedRotation3D<T, Dst, Src> {
        TypedRotation3D::quaternion(-self.i, -self.j, -self.k, self.r)
    }

    /// Computes the norm of this quaternion
    #[inline]
    pub fn norm(&self) -> T {
        self.square_norm().sqrt()
    }

    #[inline]
    pub fn square_norm(&self) -> T {
        (self.i * self.i + self.j * self.j + self.k * self.k + self.r * self.r)
    }

    /// Returns a unit quaternion from this one.
    #[inline]
    pub fn normalize(&self) -> Self {
        self.mul(T::one() / self.norm())
    }

    #[inline]
    pub fn is_normalized(&self) -> bool
    where
        T: ApproxEq<T>,
    {
        let eps = NumCast::from(1.0e-5).unwrap();
        self.square_norm().approx_eq_eps(&T::one(), &eps)
    }

    /// Spherical linear interpolation between this rotation and another rotation.
    ///
    /// `t` is expected to be between zero and one.
    pub fn slerp(&self, other: &Self, t: T) -> Self
    where
        T: ApproxEq<T>,
    {
        debug_assert!(self.is_normalized());
        debug_assert!(other.is_normalized());

        let r1 = *self;
        let mut r2 = *other;

        let mut dot = r1.i * r2.i + r1.j * r2.j + r1.k * r2.k + r1.r * r2.r;

        let one = T::one();

        if dot.approx_eq(&T::one()) {
            // If the inputs are too close, linearly interpolate to avoid precision issues.
            return r1.lerp(&r2, t);
        }

        // If the dot product is negative, the quaternions
        // have opposite handed-ness and slerp won't take
        // the shorter path. Fix by reversing one quaternion.
        if dot < T::zero() {
            r2 = r2.mul(-T::one());
            dot = -dot;
        }

        // For robustness, stay within the domain of acos.
        dot = Float::min(dot, one);

        // Angle between r1 and the result.
        let theta = Float::acos(dot) * t;

        // r1 and r3 form an orthonormal basis.
        let r3 = r2.sub(r1.mul(dot)).normalize();
        let (sin, cos) = Float::sin_cos(theta);
        r1.mul(cos).add(r3.mul(sin))
    }

    /// Basic Linear interpolation between this rotation and another rotation.
    ///
    /// `t` is expected to be between zero and one.
    #[inline]
    pub fn lerp(&self, other: &Self, t: T) -> Self {
        let one_t = T::one() - t;
        self.mul(one_t).add(other.mul(t)).normalize()
    }

    /// Returns the given 3d point transformed by this rotation.
    ///
    /// The input point must be use the unit Src, and the returned point has the unit Dst.
    pub fn rotate_point3d(&self, point: &TypedPoint3D<T, Src>) -> TypedPoint3D<T, Dst>
    where
        T: ApproxEq<T>,
    {
        debug_assert!(self.is_normalized());

        let two = T::one() + T::one();
        let cross = self.vector_part().cross(point.to_vector().to_untyped()) * two;

        point3(
            point.x + self.r * cross.x + self.j * cross.z - self.k * cross.y,
            point.y + self.r * cross.y + self.k * cross.x - self.i * cross.z,
            point.z + self.r * cross.z + self.i * cross.y - self.j * cross.x,
        )
    }

    /// Returns the given 2d point transformed by this rotation then projected on the xy plane.
    ///
    /// The input point must be use the unit Src, and the returned point has the unit Dst.
    #[inline]
    pub fn rotate_point2d(&self, point: &TypedPoint2D<T, Src>) -> TypedPoint2D<T, Dst>
    where
        T: ApproxEq<T>,
    {
        self.rotate_point3d(&point.to_3d()).xy()
    }

    /// Returns the given 3d vector transformed by this rotation.
    ///
    /// The input vector must be use the unit Src, and the returned point has the unit Dst.
    #[inline]
    pub fn rotate_vector3d(&self, vector: &TypedVector3D<T, Src>) -> TypedVector3D<T, Dst>
    where
        T: ApproxEq<T>,
    {
        self.rotate_point3d(&vector.to_point()).to_vector()
    }

    /// Returns the given 2d vector transformed by this rotation then projected on the xy plane.
    ///
    /// The input vector must be use the unit Src, and the returned point has the unit Dst.
    #[inline]
    pub fn rotate_vector2d(&self, vector: &TypedVector2D<T, Src>) -> TypedVector2D<T, Dst>
    where
        T: ApproxEq<T>,
    {
        self.rotate_vector3d(&vector.to_3d()).xy()
    }

    /// Returns the matrix representation of this rotation.
    #[inline]
    pub fn to_transform(&self) -> TypedTransform3D<T, Src, Dst>
    where
        T: ApproxEq<T>,
    {
        debug_assert!(self.is_normalized());

        let i2 = self.i + self.i;
        let j2 = self.j + self.j;
        let k2 = self.k + self.k;
        let ii = self.i * i2;
        let ij = self.i * j2;
        let ik = self.i * k2;
        let jj = self.j * j2;
        let jk = self.j * k2;
        let kk = self.k * k2;
        let ri = self.r * i2;
        let rj = self.r * j2;
        let rk = self.r * k2;

        let one = T::one();
        let zero = T::zero();

        let m11 = one - (jj + kk);
        let m12 = ij + rk;
        let m13 = ik - rj;

        let m21 = ij - rk;
        let m22 = one - (ii + kk);
        let m23 = jk + ri;

        let m31 = ik + rj;
        let m32 = jk - ri;
        let m33 = one - (ii + jj);

        TypedTransform3D::row_major(
            m11,
            m12,
            m13,
            zero,
            m21,
            m22,
            m23,
            zero,
            m31,
            m32,
            m33,
            zero,
            zero,
            zero,
            zero,
            one,
        )
    }

    /// Returns a rotation representing the other rotation followed by this rotation.
    pub fn pre_rotate<NewSrc>(
        &self,
        other: &TypedRotation3D<T, NewSrc, Src>,
    ) -> TypedRotation3D<T, NewSrc, Dst>
    where
        T: ApproxEq<T>,
    {
        debug_assert!(self.is_normalized());
        TypedRotation3D::quaternion(
            self.i * other.r + self.r * other.i + self.j * other.k - self.k * other.j,
            self.j * other.r + self.r * other.j + self.k * other.i - self.i * other.k,
            self.k * other.r + self.r * other.k + self.i * other.j - self.j * other.i,
            self.r * other.r - self.i * other.i - self.j * other.j - self.k * other.k,
        )
    }

    /// Returns a rotation representing this rotation followed by the other rotation.
    #[inline]
    pub fn post_rotate<NewDst>(
        &self,
        other: &TypedRotation3D<T, Dst, NewDst>,
    ) -> TypedRotation3D<T, Src, NewDst>
    where
        T: ApproxEq<T>,
    {
        other.pre_rotate(self)
    }

    // add, sub and mul are used internally for intermediate computation but aren't public
    // because they don't carry real semantic meanings (I think?).

    #[inline]
    fn add(&self, other: Self) -> Self {
        Self::quaternion(
            self.i + other.i,
            self.j + other.j,
            self.k + other.k,
            self.r + other.r,
        )
    }

    #[inline]
    fn sub(&self, other: Self) -> Self {
        Self::quaternion(
            self.i - other.i,
            self.j - other.j,
            self.k - other.k,
            self.r - other.r,
        )
    }

    #[inline]
    fn mul(&self, factor: T) -> Self {
        Self::quaternion(
            self.i * factor,
            self.j * factor,
            self.k * factor,
            self.r * factor,
        )
    }
}

impl<T: fmt::Debug, Src, Dst> fmt::Debug for TypedRotation3D<T, Src, Dst> {
    fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
        write!(
            f,
            "Quat({:?}*i + {:?}*j + {:?}*k + {:?})",
            self.i, self.j, self.k, self.r
        )
    }
}

impl<T: fmt::Display, Src, Dst> fmt::Display for TypedRotation3D<T, Src, Dst> {
    fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
        write!(
            f,
            "Quat({}*i + {}*j + {}*k + {})",
            self.i, self.j, self.k, self.r
        )
    }
}

impl<T, Src, Dst> ApproxEq<T> for TypedRotation3D<T, Src, Dst>
where
    T: Copy + Neg<Output = T> + ApproxEq<T>,
{
    fn approx_epsilon() -> T {
        T::approx_epsilon()
    }

    fn approx_eq(&self, other: &Self) -> bool {
        self.approx_eq_eps(other, &Self::approx_epsilon())
    }

    fn approx_eq_eps(&self, other: &Self, eps: &T) -> bool {
        (self.i.approx_eq_eps(&other.i, eps) && self.j.approx_eq_eps(&other.j, eps)
            && self.k.approx_eq_eps(&other.k, eps) && self.r.approx_eq_eps(&other.r, eps))
            || (self.i.approx_eq_eps(&-other.i, eps) && self.j.approx_eq_eps(&-other.j, eps)
                && self.k.approx_eq_eps(&-other.k, eps)
                && self.r.approx_eq_eps(&-other.r, eps))
    }
}

#[test]
fn simple_rotation_2d() {
    use core::f32::consts::{FRAC_PI_2, PI};
    let ri = Rotation2D::identity();
    let r90 = Rotation2D::radians(FRAC_PI_2);
    let rm90 = Rotation2D::radians(-FRAC_PI_2);
    let r180 = Rotation2D::radians(PI);

    assert!(
        ri.transform_point(&point2(1.0, 2.0))
            .approx_eq(&point2(1.0, 2.0))
    );
    assert!(
        r90.transform_point(&point2(1.0, 2.0))
            .approx_eq(&point2(-2.0, 1.0))
    );
    assert!(
        rm90.transform_point(&point2(1.0, 2.0))
            .approx_eq(&point2(2.0, -1.0))
    );
    assert!(
        r180.transform_point(&point2(1.0, 2.0))
            .approx_eq(&point2(-1.0, -2.0))
    );

    assert!(
        r90.inverse()
            .inverse()
            .transform_point(&point2(1.0, 2.0))
            .approx_eq(&r90.transform_point(&point2(1.0, 2.0)))
    );
}

#[test]
fn simple_rotation_3d_in_2d() {
    use core::f32::consts::{FRAC_PI_2, PI};
    let ri = Rotation3D::identity();
    let r90 = Rotation3D::around_z(Angle::radians(FRAC_PI_2));
    let rm90 = Rotation3D::around_z(Angle::radians(-FRAC_PI_2));
    let r180 = Rotation3D::around_z(Angle::radians(PI));

    assert!(
        ri.rotate_point2d(&point2(1.0, 2.0))
            .approx_eq(&point2(1.0, 2.0))
    );
    assert!(
        r90.rotate_point2d(&point2(1.0, 2.0))
            .approx_eq(&point2(-2.0, 1.0))
    );
    assert!(
        rm90.rotate_point2d(&point2(1.0, 2.0))
            .approx_eq(&point2(2.0, -1.0))
    );
    assert!(
        r180.rotate_point2d(&point2(1.0, 2.0))
            .approx_eq(&point2(-1.0, -2.0))
    );

    assert!(
        r90.inverse()
            .inverse()
            .rotate_point2d(&point2(1.0, 2.0))
            .approx_eq(&r90.rotate_point2d(&point2(1.0, 2.0)))
    );
}

#[test]
fn pre_post() {
    use core::f32::consts::FRAC_PI_2;
    let r1 = Rotation3D::around_x(Angle::radians(FRAC_PI_2));
    let r2 = Rotation3D::around_y(Angle::radians(FRAC_PI_2));
    let r3 = Rotation3D::around_z(Angle::radians(FRAC_PI_2));

    let t1 = r1.to_transform();
    let t2 = r2.to_transform();
    let t3 = r3.to_transform();

    let p = point3(1.0, 2.0, 3.0);

    // Check that the order of transformations is correct (corresponds to what
    // we do in Transform3D).
    let p1 = r1.post_rotate(&r2).post_rotate(&r3).rotate_point3d(&p);
    let p2 = t1.post_mul(&t2).post_mul(&t3).transform_point3d(&p);

    assert!(p1.approx_eq(&p2.unwrap()));

    // Check that changing the order indeed matters.
    let p3 = t3.post_mul(&t1).post_mul(&t2).transform_point3d(&p);
    assert!(!p1.approx_eq(&p3.unwrap()));
}

#[test]
fn to_transform3d() {
    use core::f32::consts::{FRAC_PI_2, PI};
    let rotations = [
        Rotation3D::identity(),
        Rotation3D::around_x(Angle::radians(FRAC_PI_2)),
        Rotation3D::around_x(Angle::radians(-FRAC_PI_2)),
        Rotation3D::around_x(Angle::radians(PI)),
        Rotation3D::around_y(Angle::radians(FRAC_PI_2)),
        Rotation3D::around_y(Angle::radians(-FRAC_PI_2)),
        Rotation3D::around_y(Angle::radians(PI)),
        Rotation3D::around_z(Angle::radians(FRAC_PI_2)),
        Rotation3D::around_z(Angle::radians(-FRAC_PI_2)),
        Rotation3D::around_z(Angle::radians(PI)),
    ];

    let points = [
        point3(0.0, 0.0, 0.0),
        point3(1.0, 2.0, 3.0),
        point3(-5.0, 3.0, -1.0),
        point3(-0.5, -1.0, 1.5),
    ];

    for rotation in &rotations {
        for point in &points {
            let p1 = rotation.rotate_point3d(point);
            let p2 = rotation.to_transform().transform_point3d(point);
            assert!(p1.approx_eq(&p2.unwrap()));
        }
    }
}

#[test]
fn slerp() {
    let q1 = Rotation3D::quaternion(1.0, 0.0, 0.0, 0.0);
    let q2 = Rotation3D::quaternion(0.0, 1.0, 0.0, 0.0);
    let q3 = Rotation3D::quaternion(0.0, 0.0, -1.0, 0.0);

    // The values below can be obtained with a python program:
    // import numpy
    // import quaternion
    // q1 = numpy.quaternion(1, 0, 0, 0)
    // q2 = numpy.quaternion(0, 1, 0, 0)
    // quaternion.slerp_evaluate(q1, q2, 0.2)

    assert!(q1.slerp(&q2, 0.0).approx_eq(&q1));
    assert!(q1.slerp(&q2, 0.2).approx_eq(&Rotation3D::quaternion(
        0.951056516295154,
        0.309016994374947,
        0.0,
        0.0
    )));
    assert!(q1.slerp(&q2, 0.4).approx_eq(&Rotation3D::quaternion(
        0.809016994374947,
        0.587785252292473,
        0.0,
        0.0
    )));
    assert!(q1.slerp(&q2, 0.6).approx_eq(&Rotation3D::quaternion(
        0.587785252292473,
        0.809016994374947,
        0.0,
        0.0
    )));
    assert!(q1.slerp(&q2, 0.8).approx_eq(&Rotation3D::quaternion(
        0.309016994374947,
        0.951056516295154,
        0.0,
        0.0
    )));
    assert!(q1.slerp(&q2, 1.0).approx_eq(&q2));

    assert!(q1.slerp(&q3, 0.0).approx_eq(&q1));
    assert!(q1.slerp(&q3, 0.2).approx_eq(&Rotation3D::quaternion(
        0.951056516295154,
        0.0,
        -0.309016994374947,
        0.0
    )));
    assert!(q1.slerp(&q3, 0.4).approx_eq(&Rotation3D::quaternion(
        0.809016994374947,
        0.0,
        -0.587785252292473,
        0.0
    )));
    assert!(q1.slerp(&q3, 0.6).approx_eq(&Rotation3D::quaternion(
        0.587785252292473,
        0.0,
        -0.809016994374947,
        0.0
    )));
    assert!(q1.slerp(&q3, 0.8).approx_eq(&Rotation3D::quaternion(
        0.309016994374947,
        0.0,
        -0.951056516295154,
        0.0
    )));
    assert!(q1.slerp(&q3, 1.0).approx_eq(&q3));
}

#[test]
fn around_axis() {
    use core::f32::consts::{FRAC_PI_2, PI};

    // Two sort of trivial cases:
    let r1 = Rotation3D::around_axis(vec3(1.0, 1.0, 0.0), Angle::radians(PI));
    let r2 = Rotation3D::around_axis(vec3(1.0, 1.0, 0.0), Angle::radians(FRAC_PI_2));
    assert!(
        r1.rotate_point3d(&point3(1.0, 2.0, 0.0))
            .approx_eq(&point3(2.0, 1.0, 0.0))
    );
    assert!(
        r2.rotate_point3d(&point3(1.0, 0.0, 0.0))
            .approx_eq(&point3(0.5, 0.5, -0.5.sqrt()))
    );

    // A more arbitrary test (made up with numpy):
    let r3 = Rotation3D::around_axis(vec3(0.5, 1.0, 2.0), Angle::radians(2.291288));
    assert!(r3.rotate_point3d(&point3(1.0, 0.0, 0.0)).approx_eq(&point3(
        -0.58071821,
        0.81401868,
        -0.01182979
    )));
}

#[test]
fn from_euler() {
    use core::f32::consts::FRAC_PI_2;

    // First test simple separate yaw pitch and roll rotations, because it is easy to come
    // up with the corresponding quaternion.
    // Since several quaternions can represent the same transformation we compare the result
    // of transforming a point rather than the values of each quaternions.
    let p = point3(1.0, 2.0, 3.0);

    let angle = Angle::radians(FRAC_PI_2);
    let zero = Angle::radians(0.0);

    // roll
    let roll_re = Rotation3D::euler(angle, zero, zero);
    let roll_rq = Rotation3D::around_x(angle);
    let roll_pe = roll_re.rotate_point3d(&p);
    let roll_pq = roll_rq.rotate_point3d(&p);

    // pitch
    let pitch_re = Rotation3D::euler(zero, angle, zero);
    let pitch_rq = Rotation3D::around_y(angle);
    let pitch_pe = pitch_re.rotate_point3d(&p);
    let pitch_pq = pitch_rq.rotate_point3d(&p);

    // yaw
    let yaw_re = Rotation3D::euler(zero, zero, angle);
    let yaw_rq = Rotation3D::around_z(angle);
    let yaw_pe = yaw_re.rotate_point3d(&p);
    let yaw_pq = yaw_rq.rotate_point3d(&p);

    assert!(roll_pe.approx_eq(&roll_pq));
    assert!(pitch_pe.approx_eq(&pitch_pq));
    assert!(yaw_pe.approx_eq(&yaw_pq));

    // Now check that the yaw pitch and roll transformations when combined are applied in
    // the proper order: roll -> pitch -> yaw.
    let ypr_e = Rotation3D::euler(angle, angle, angle);
    let ypr_q = roll_rq.post_rotate(&pitch_rq).post_rotate(&yaw_rq);
    let ypr_pe = ypr_e.rotate_point3d(&p);
    let ypr_pq = ypr_q.rotate_point3d(&p);

    assert!(ypr_pe.approx_eq(&ypr_pq));
}

#[test]
fn wrap_angles() {
    use core::f32::consts::{FRAC_PI_2, PI};
    assert!(Angle::radians(0.0).positive().radians.approx_eq(&0.0));
    assert!(
        Angle::radians(FRAC_PI_2)
            .positive()
            .radians
            .approx_eq(&FRAC_PI_2)
    );
    assert!(
        Angle::radians(-FRAC_PI_2)
            .positive()
            .radians
            .approx_eq(&(3.0 * FRAC_PI_2))
    );
    assert!(
        Angle::radians(3.0 * FRAC_PI_2)
            .positive()
            .radians
            .approx_eq(&(3.0 * FRAC_PI_2))
    );
    assert!(
        Angle::radians(5.0 * FRAC_PI_2)
            .positive()
            .radians
            .approx_eq(&FRAC_PI_2)
    );
    assert!(Angle::radians(2.0 * PI).positive().radians.approx_eq(&0.0));
    assert!(Angle::radians(-2.0 * PI).positive().radians.approx_eq(&0.0));
    assert!(Angle::radians(PI).positive().radians.approx_eq(&PI));
    assert!(Angle::radians(-PI).positive().radians.approx_eq(&PI));

    assert!(
        Angle::radians(FRAC_PI_2)
            .signed()
            .radians
            .approx_eq(&FRAC_PI_2)
    );
    assert!(
        Angle::radians(3.0 * FRAC_PI_2)
            .signed()
            .radians
            .approx_eq(&-FRAC_PI_2)
    );
    assert!(
        Angle::radians(5.0 * FRAC_PI_2)
            .signed()
            .radians
            .approx_eq(&FRAC_PI_2)
    );
    assert!(Angle::radians(2.0 * PI).signed().radians.approx_eq(&0.0));
    assert!(Angle::radians(-2.0 * PI).signed().radians.approx_eq(&0.0));
    assert!(Angle::radians(-PI).signed().radians.approx_eq(&PI));
    assert!(Angle::radians(PI).signed().radians.approx_eq(&PI));
}