978 lines
33 KiB
Rust
978 lines
33 KiB
Rust
#[cfg(feature = "arbitrary")]
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use crate::base::storage::Owned;
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#[cfg(feature = "arbitrary")]
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use quickcheck::{Arbitrary, Gen};
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use num::Zero;
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#[cfg(feature = "rand-no-std")]
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use rand::{
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distributions::{Distribution, OpenClosed01, Standard, Uniform, uniform::SampleUniform},
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Rng,
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};
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use simba::scalar::RealField;
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use simba::simd::{SimdBool, SimdRealField};
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use std::ops::Neg;
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use crate::base::dimension::{U1, U2, U3};
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use crate::base::storage::Storage;
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use crate::base::{Matrix2, Matrix3, MatrixN, Unit, Vector, Vector1, Vector2, Vector3, VectorN};
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use crate::geometry::{Rotation2, Rotation3, UnitComplex, UnitQuaternion};
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/*
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*
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* 2D Rotation matrix.
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*
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*/
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/// # Construction from a 2D rotation angle
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impl<N: SimdRealField> Rotation2<N> {
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/// Builds a 2 dimensional rotation matrix from an angle in radian.
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///
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/// # Example
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///
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/// ```
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/// # #[macro_use] extern crate approx;
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/// # use std::f32;
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/// # use nalgebra::{Rotation2, Point2};
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/// let rot = Rotation2::new(f32::consts::FRAC_PI_2);
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///
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/// assert_relative_eq!(rot * Point2::new(3.0, 4.0), Point2::new(-4.0, 3.0));
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/// ```
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pub fn new(angle: N) -> Self {
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let (sia, coa) = angle.simd_sin_cos();
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Self::from_matrix_unchecked(Matrix2::new(coa, -sia, sia, coa))
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}
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/// Builds a 2 dimensional rotation matrix from an angle in radian wrapped in a 1-dimensional vector.
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///
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///
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/// This is generally used in the context of generic programming. Using
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/// the `::new(angle)` method instead is more common.
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#[inline]
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pub fn from_scaled_axis<SB: Storage<N, U1>>(axisangle: Vector<N, U1, SB>) -> Self {
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Self::new(axisangle[0])
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}
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}
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/// # Construction from an existing 2D matrix or rotations
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impl<N: SimdRealField> Rotation2<N> {
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/// Builds a rotation from a basis assumed to be orthonormal.
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///
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/// In order to get a valid unit-quaternion, the input must be an
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/// orthonormal basis, i.e., all vectors are normalized, and the are
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/// all orthogonal to each other. These invariants are not checked
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/// by this method.
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pub fn from_basis_unchecked(basis: &[Vector2<N>; 2]) -> Self {
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let mat = Matrix2::from_columns(&basis[..]);
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Self::from_matrix_unchecked(mat)
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}
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/// Builds a rotation matrix by extracting the rotation part of the given transformation `m`.
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///
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/// This is an iterative method. See `.from_matrix_eps` to provide mover
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/// convergence parameters and starting solution.
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/// This implements "A Robust Method to Extract the Rotational Part of Deformations" by Müller et al.
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pub fn from_matrix(m: &Matrix2<N>) -> Self
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where
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N: RealField,
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{
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Self::from_matrix_eps(m, N::default_epsilon(), 0, Self::identity())
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}
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/// Builds a rotation matrix by extracting the rotation part of the given transformation `m`.
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///
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/// This implements "A Robust Method to Extract the Rotational Part of Deformations" by Müller et al.
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///
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/// # Parameters
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///
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/// * `m`: the matrix from which the rotational part is to be extracted.
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/// * `eps`: the angular errors tolerated between the current rotation and the optimal one.
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/// * `max_iter`: the maximum number of iterations. Loops indefinitely until convergence if set to `0`.
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/// * `guess`: an estimate of the solution. Convergence will be significantly faster if an initial solution close
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/// to the actual solution is provided. Can be set to `Rotation2::identity()` if no other
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/// guesses come to mind.
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pub fn from_matrix_eps(m: &Matrix2<N>, eps: N, mut max_iter: usize, guess: Self) -> Self
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where
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N: RealField,
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{
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if max_iter == 0 {
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max_iter = usize::max_value();
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}
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let mut rot = guess.into_inner();
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for _ in 0..max_iter {
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let axis = rot.column(0).perp(&m.column(0)) + rot.column(1).perp(&m.column(1));
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let denom = rot.column(0).dot(&m.column(0)) + rot.column(1).dot(&m.column(1));
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let angle = axis / (denom.abs() + N::default_epsilon());
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if angle.abs() > eps {
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rot = Self::new(angle) * rot;
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} else {
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break;
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}
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}
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Self::from_matrix_unchecked(rot)
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}
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/// The rotation matrix required to align `a` and `b` but with its angle.
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///
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/// This is the rotation `R` such that `(R * a).angle(b) == 0 && (R * a).dot(b).is_positive()`.
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///
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/// # Example
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/// ```
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/// # #[macro_use] extern crate approx;
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/// # use nalgebra::{Vector2, Rotation2};
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/// let a = Vector2::new(1.0, 2.0);
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/// let b = Vector2::new(2.0, 1.0);
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/// let rot = Rotation2::rotation_between(&a, &b);
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/// assert_relative_eq!(rot * a, b);
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/// assert_relative_eq!(rot.inverse() * b, a);
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/// ```
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#[inline]
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pub fn rotation_between<SB, SC>(a: &Vector<N, U2, SB>, b: &Vector<N, U2, SC>) -> Self
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where
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N: RealField,
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SB: Storage<N, U2>,
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SC: Storage<N, U2>,
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{
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crate::convert(UnitComplex::rotation_between(a, b).to_rotation_matrix())
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}
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/// The smallest rotation needed to make `a` and `b` collinear and point toward the same
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/// direction, raised to the power `s`.
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///
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/// # Example
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/// ```
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/// # #[macro_use] extern crate approx;
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/// # use nalgebra::{Vector2, Rotation2};
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/// let a = Vector2::new(1.0, 2.0);
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/// let b = Vector2::new(2.0, 1.0);
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/// let rot2 = Rotation2::scaled_rotation_between(&a, &b, 0.2);
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/// let rot5 = Rotation2::scaled_rotation_between(&a, &b, 0.5);
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/// assert_relative_eq!(rot2 * rot2 * rot2 * rot2 * rot2 * a, b, epsilon = 1.0e-6);
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/// assert_relative_eq!(rot5 * rot5 * a, b, epsilon = 1.0e-6);
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/// ```
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#[inline]
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pub fn scaled_rotation_between<SB, SC>(
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a: &Vector<N, U2, SB>,
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b: &Vector<N, U2, SC>,
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s: N,
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) -> Self
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where
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N: RealField,
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SB: Storage<N, U2>,
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SC: Storage<N, U2>,
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{
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crate::convert(UnitComplex::scaled_rotation_between(a, b, s).to_rotation_matrix())
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}
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/// The rotation matrix needed to make `self` and `other` coincide.
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///
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/// The result is such that: `self.rotation_to(other) * self == other`.
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///
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/// # Example
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/// ```
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/// # #[macro_use] extern crate approx;
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/// # use nalgebra::Rotation2;
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/// let rot1 = Rotation2::new(0.1);
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/// let rot2 = Rotation2::new(1.7);
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/// let rot_to = rot1.rotation_to(&rot2);
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///
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/// assert_relative_eq!(rot_to * rot1, rot2);
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/// assert_relative_eq!(rot_to.inverse() * rot2, rot1);
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/// ```
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#[inline]
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pub fn rotation_to(&self, other: &Self) -> Self {
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other * self.inverse()
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}
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/// Ensure this rotation is an orthonormal rotation matrix. This is useful when repeated
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/// computations might cause the matrix from progressively not being orthonormal anymore.
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#[inline]
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pub fn renormalize(&mut self)
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where
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N: RealField,
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{
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let mut c = UnitComplex::from(*self);
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let _ = c.renormalize();
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*self = Self::from_matrix_eps(self.matrix(), N::default_epsilon(), 0, c.into())
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}
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/// Raise the quaternion to a given floating power, i.e., returns the rotation with the angle
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/// of `self` multiplied by `n`.
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///
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/// # Example
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/// ```
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/// # #[macro_use] extern crate approx;
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/// # use nalgebra::Rotation2;
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/// let rot = Rotation2::new(0.78);
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/// let pow = rot.powf(2.0);
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/// assert_relative_eq!(pow.angle(), 2.0 * 0.78);
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/// ```
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#[inline]
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pub fn powf(&self, n: N) -> Self {
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Self::new(self.angle() * n)
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}
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}
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/// # 2D angle extraction
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impl<N: SimdRealField> Rotation2<N> {
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/// The rotation angle.
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///
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/// # Example
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/// ```
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/// # #[macro_use] extern crate approx;
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/// # use nalgebra::Rotation2;
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/// let rot = Rotation2::new(1.78);
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/// assert_relative_eq!(rot.angle(), 1.78);
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/// ```
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#[inline]
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pub fn angle(&self) -> N {
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self.matrix()[(1, 0)].simd_atan2(self.matrix()[(0, 0)])
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}
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/// The rotation angle needed to make `self` and `other` coincide.
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///
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/// # Example
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/// ```
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/// # #[macro_use] extern crate approx;
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/// # use nalgebra::Rotation2;
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/// let rot1 = Rotation2::new(0.1);
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/// let rot2 = Rotation2::new(1.7);
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/// assert_relative_eq!(rot1.angle_to(&rot2), 1.6);
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/// ```
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#[inline]
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pub fn angle_to(&self, other: &Self) -> N {
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self.rotation_to(other).angle()
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}
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/// The rotation angle returned as a 1-dimensional vector.
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///
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/// This is generally used in the context of generic programming. Using
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/// the `.angle()` method instead is more common.
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#[inline]
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pub fn scaled_axis(&self) -> VectorN<N, U1> {
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Vector1::new(self.angle())
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}
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}
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#[cfg(feature = "rand-no-std")]
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impl<N: SimdRealField> Distribution<Rotation2<N>> for Standard
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where
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N::Element: SimdRealField,
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N: SampleUniform,
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{
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/// Generate a uniformly distributed random rotation.
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#[inline]
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fn sample<'a, R: Rng + ?Sized>(&self, rng: &'a mut R) -> Rotation2<N> {
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let twopi = Uniform::new(N::zero(), N::simd_two_pi());
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Rotation2::new(rng.sample(twopi))
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}
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}
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#[cfg(feature = "arbitrary")]
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impl<N: SimdRealField + Arbitrary> Arbitrary for Rotation2<N>
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where
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N::Element: SimdRealField,
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Owned<N, U2, U2>: Send,
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{
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#[inline]
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fn arbitrary(g: &mut Gen) -> Self {
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Self::new(N::arbitrary(g))
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}
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}
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/*
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*
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* 3D Rotation matrix.
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*
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*/
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/// # Construction from a 3D axis and/or angles
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impl<N: SimdRealField> Rotation3<N>
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where
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N::Element: SimdRealField,
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{
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/// Builds a 3 dimensional rotation matrix from an axis and an angle.
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///
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/// # Arguments
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/// * `axisangle` - A vector representing the rotation. Its magnitude is the amount of rotation
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/// in radian. Its direction is the axis of rotation.
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///
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/// # Example
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/// ```
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/// # #[macro_use] extern crate approx;
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/// # use std::f32;
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/// # use nalgebra::{Rotation3, Point3, Vector3};
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/// let axisangle = Vector3::y() * f32::consts::FRAC_PI_2;
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/// // Point and vector being transformed in the tests.
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/// let pt = Point3::new(4.0, 5.0, 6.0);
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/// let vec = Vector3::new(4.0, 5.0, 6.0);
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/// let rot = Rotation3::new(axisangle);
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///
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/// assert_relative_eq!(rot * pt, Point3::new(6.0, 5.0, -4.0), epsilon = 1.0e-6);
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/// assert_relative_eq!(rot * vec, Vector3::new(6.0, 5.0, -4.0), epsilon = 1.0e-6);
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///
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/// // A zero vector yields an identity.
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/// assert_eq!(Rotation3::new(Vector3::<f32>::zeros()), Rotation3::identity());
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/// ```
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pub fn new<SB: Storage<N, U3>>(axisangle: Vector<N, U3, SB>) -> Self {
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let axisangle = axisangle.into_owned();
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let (axis, angle) = Unit::new_and_get(axisangle);
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Self::from_axis_angle(&axis, angle)
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}
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/// Builds a 3D rotation matrix from an axis scaled by the rotation angle.
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///
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/// This is the same as `Self::new(axisangle)`.
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///
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/// # Example
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/// ```
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/// # #[macro_use] extern crate approx;
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/// # use std::f32;
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/// # use nalgebra::{Rotation3, Point3, Vector3};
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/// let axisangle = Vector3::y() * f32::consts::FRAC_PI_2;
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/// // Point and vector being transformed in the tests.
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/// let pt = Point3::new(4.0, 5.0, 6.0);
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/// let vec = Vector3::new(4.0, 5.0, 6.0);
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/// let rot = Rotation3::new(axisangle);
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///
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/// assert_relative_eq!(rot * pt, Point3::new(6.0, 5.0, -4.0), epsilon = 1.0e-6);
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/// assert_relative_eq!(rot * vec, Vector3::new(6.0, 5.0, -4.0), epsilon = 1.0e-6);
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///
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/// // A zero vector yields an identity.
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/// assert_eq!(Rotation3::from_scaled_axis(Vector3::<f32>::zeros()), Rotation3::identity());
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/// ```
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pub fn from_scaled_axis<SB: Storage<N, U3>>(axisangle: Vector<N, U3, SB>) -> Self {
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Self::new(axisangle)
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}
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/// Builds a 3D rotation matrix from an axis and a rotation angle.
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///
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/// # Example
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/// ```
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/// # #[macro_use] extern crate approx;
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/// # use std::f32;
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/// # use nalgebra::{Rotation3, Point3, Vector3};
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/// let axis = Vector3::y_axis();
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/// let angle = f32::consts::FRAC_PI_2;
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/// // Point and vector being transformed in the tests.
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/// let pt = Point3::new(4.0, 5.0, 6.0);
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/// let vec = Vector3::new(4.0, 5.0, 6.0);
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/// let rot = Rotation3::from_axis_angle(&axis, angle);
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///
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/// assert_eq!(rot.axis().unwrap(), axis);
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/// assert_eq!(rot.angle(), angle);
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/// assert_relative_eq!(rot * pt, Point3::new(6.0, 5.0, -4.0), epsilon = 1.0e-6);
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/// assert_relative_eq!(rot * vec, Vector3::new(6.0, 5.0, -4.0), epsilon = 1.0e-6);
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///
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/// // A zero vector yields an identity.
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/// assert_eq!(Rotation3::from_scaled_axis(Vector3::<f32>::zeros()), Rotation3::identity());
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/// ```
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pub fn from_axis_angle<SB>(axis: &Unit<Vector<N, U3, SB>>, angle: N) -> Self
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where
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SB: Storage<N, U3>,
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{
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angle.simd_ne(N::zero()).if_else(
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|| {
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let ux = axis.as_ref()[0];
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let uy = axis.as_ref()[1];
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let uz = axis.as_ref()[2];
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let sqx = ux * ux;
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let sqy = uy * uy;
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let sqz = uz * uz;
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let (sin, cos) = angle.simd_sin_cos();
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let one_m_cos = N::one() - cos;
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Self::from_matrix_unchecked(MatrixN::<N, U3>::new(
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sqx + (N::one() - sqx) * cos,
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ux * uy * one_m_cos - uz * sin,
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ux * uz * one_m_cos + uy * sin,
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ux * uy * one_m_cos + uz * sin,
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sqy + (N::one() - sqy) * cos,
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uy * uz * one_m_cos - ux * sin,
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ux * uz * one_m_cos - uy * sin,
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uy * uz * one_m_cos + ux * sin,
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sqz + (N::one() - sqz) * cos,
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))
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},
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Self::identity,
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)
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}
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/// Creates a new rotation from Euler angles.
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///
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/// The primitive rotations are applied in order: 1 roll − 2 pitch − 3 yaw.
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///
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/// # Example
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/// ```
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/// # #[macro_use] extern crate approx;
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/// # use nalgebra::Rotation3;
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/// let rot = Rotation3::from_euler_angles(0.1, 0.2, 0.3);
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/// let euler = rot.euler_angles();
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/// assert_relative_eq!(euler.0, 0.1, epsilon = 1.0e-6);
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/// assert_relative_eq!(euler.1, 0.2, epsilon = 1.0e-6);
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/// assert_relative_eq!(euler.2, 0.3, epsilon = 1.0e-6);
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/// ```
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pub fn from_euler_angles(roll: N, pitch: N, yaw: N) -> Self {
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let (sr, cr) = roll.simd_sin_cos();
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let (sp, cp) = pitch.simd_sin_cos();
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let (sy, cy) = yaw.simd_sin_cos();
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Self::from_matrix_unchecked(MatrixN::<N, U3>::new(
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cy * cp,
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cy * sp * sr - sy * cr,
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cy * sp * cr + sy * sr,
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sy * cp,
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sy * sp * sr + cy * cr,
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sy * sp * cr - cy * sr,
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-sp,
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cp * sr,
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cp * cr,
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))
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}
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}
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/// # Construction from a 3D eye position and target point
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impl<N: SimdRealField> Rotation3<N>
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where
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N::Element: SimdRealField,
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{
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/// Creates a rotation that corresponds to the local frame of an observer standing at the
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/// origin and looking toward `dir`.
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///
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/// It maps the `z` axis to the direction `dir`.
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///
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/// # Arguments
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/// * dir - The look direction, that is, direction the matrix `z` axis will be aligned with.
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/// * up - The vertical direction. The only requirement of this parameter is to not be
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/// collinear to `dir`. Non-collinearity is not checked.
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///
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/// # Example
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/// ```
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/// # #[macro_use] extern crate approx;
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/// # use std::f32;
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||
/// # use nalgebra::{Rotation3, Vector3};
|
||
/// let dir = Vector3::new(1.0, 2.0, 3.0);
|
||
/// let up = Vector3::y();
|
||
///
|
||
/// let rot = Rotation3::face_towards(&dir, &up);
|
||
/// assert_relative_eq!(rot * Vector3::z(), dir.normalize());
|
||
/// ```
|
||
#[inline]
|
||
pub fn face_towards<SB, SC>(dir: &Vector<N, U3, SB>, up: &Vector<N, U3, SC>) -> Self
|
||
where
|
||
SB: Storage<N, U3>,
|
||
SC: Storage<N, U3>,
|
||
{
|
||
let zaxis = dir.normalize();
|
||
let xaxis = up.cross(&zaxis).normalize();
|
||
let yaxis = zaxis.cross(&xaxis).normalize();
|
||
|
||
Self::from_matrix_unchecked(MatrixN::<N, U3>::new(
|
||
xaxis.x, yaxis.x, zaxis.x, xaxis.y, yaxis.y, zaxis.y, xaxis.z, yaxis.z, zaxis.z,
|
||
))
|
||
}
|
||
|
||
/// Deprecated: Use [Rotation3::face_towards] instead.
|
||
#[deprecated(note = "renamed to `face_towards`")]
|
||
pub fn new_observer_frames<SB, SC>(dir: &Vector<N, U3, SB>, up: &Vector<N, U3, SC>) -> Self
|
||
where
|
||
SB: Storage<N, U3>,
|
||
SC: Storage<N, U3>,
|
||
{
|
||
Self::face_towards(dir, up)
|
||
}
|
||
|
||
/// Builds a right-handed look-at view matrix without translation.
|
||
///
|
||
/// It maps the view direction `dir` to the **negative** `z` axis.
|
||
/// This conforms to the common notion of right handed look-at matrix from the computer
|
||
/// graphics community.
|
||
///
|
||
/// # Arguments
|
||
/// * dir - The direction toward which the camera looks.
|
||
/// * up - A vector approximately aligned with required the vertical axis. The only
|
||
/// requirement of this parameter is to not be collinear to `dir`.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use std::f32;
|
||
/// # use nalgebra::{Rotation3, Vector3};
|
||
/// let dir = Vector3::new(1.0, 2.0, 3.0);
|
||
/// let up = Vector3::y();
|
||
///
|
||
/// let rot = Rotation3::look_at_rh(&dir, &up);
|
||
/// assert_relative_eq!(rot * dir.normalize(), -Vector3::z());
|
||
/// ```
|
||
#[inline]
|
||
pub fn look_at_rh<SB, SC>(dir: &Vector<N, U3, SB>, up: &Vector<N, U3, SC>) -> Self
|
||
where
|
||
SB: Storage<N, U3>,
|
||
SC: Storage<N, U3>,
|
||
{
|
||
Self::face_towards(&dir.neg(), up).inverse()
|
||
}
|
||
|
||
/// Builds a left-handed look-at view matrix without translation.
|
||
///
|
||
/// It maps the view direction `dir` to the **positive** `z` axis.
|
||
/// This conforms to the common notion of left handed look-at matrix from the computer
|
||
/// graphics community.
|
||
///
|
||
/// # Arguments
|
||
/// * dir - The direction toward which the camera looks.
|
||
/// * up - A vector approximately aligned with required the vertical axis. The only
|
||
/// requirement of this parameter is to not be collinear to `dir`.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use std::f32;
|
||
/// # use nalgebra::{Rotation3, Vector3};
|
||
/// let dir = Vector3::new(1.0, 2.0, 3.0);
|
||
/// let up = Vector3::y();
|
||
///
|
||
/// let rot = Rotation3::look_at_lh(&dir, &up);
|
||
/// assert_relative_eq!(rot * dir.normalize(), Vector3::z());
|
||
/// ```
|
||
#[inline]
|
||
pub fn look_at_lh<SB, SC>(dir: &Vector<N, U3, SB>, up: &Vector<N, U3, SC>) -> Self
|
||
where
|
||
SB: Storage<N, U3>,
|
||
SC: Storage<N, U3>,
|
||
{
|
||
Self::face_towards(dir, up).inverse()
|
||
}
|
||
}
|
||
|
||
/// # Construction from an existing 3D matrix or rotations
|
||
impl<N: SimdRealField> Rotation3<N>
|
||
where
|
||
N::Element: SimdRealField,
|
||
{
|
||
/// The rotation matrix required to align `a` and `b` but with its angle.
|
||
///
|
||
/// This is the rotation `R` such that `(R * a).angle(b) == 0 && (R * a).dot(b).is_positive()`.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::{Vector3, Rotation3};
|
||
/// let a = Vector3::new(1.0, 2.0, 3.0);
|
||
/// let b = Vector3::new(3.0, 1.0, 2.0);
|
||
/// let rot = Rotation3::rotation_between(&a, &b).unwrap();
|
||
/// assert_relative_eq!(rot * a, b, epsilon = 1.0e-6);
|
||
/// assert_relative_eq!(rot.inverse() * b, a, epsilon = 1.0e-6);
|
||
/// ```
|
||
#[inline]
|
||
pub fn rotation_between<SB, SC>(a: &Vector<N, U3, SB>, b: &Vector<N, U3, SC>) -> Option<Self>
|
||
where
|
||
N: RealField,
|
||
SB: Storage<N, U3>,
|
||
SC: Storage<N, U3>,
|
||
{
|
||
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`.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::{Vector3, Rotation3};
|
||
/// let a = Vector3::new(1.0, 2.0, 3.0);
|
||
/// let b = Vector3::new(3.0, 1.0, 2.0);
|
||
/// let rot2 = Rotation3::scaled_rotation_between(&a, &b, 0.2).unwrap();
|
||
/// let rot5 = Rotation3::scaled_rotation_between(&a, &b, 0.5).unwrap();
|
||
/// assert_relative_eq!(rot2 * rot2 * rot2 * rot2 * rot2 * a, b, epsilon = 1.0e-6);
|
||
/// assert_relative_eq!(rot5 * rot5 * a, b, epsilon = 1.0e-6);
|
||
/// ```
|
||
#[inline]
|
||
pub fn scaled_rotation_between<SB, SC>(
|
||
a: &Vector<N, U3, SB>,
|
||
b: &Vector<N, U3, SC>,
|
||
n: N,
|
||
) -> Option<Self>
|
||
where
|
||
N: RealField,
|
||
SB: Storage<N, U3>,
|
||
SC: Storage<N, U3>,
|
||
{
|
||
// TODO: code duplication with Rotation.
|
||
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())
|
||
}
|
||
|
||
/// The rotation matrix needed to make `self` and `other` coincide.
|
||
///
|
||
/// The result is such that: `self.rotation_to(other) * self == other`.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::{Rotation3, Vector3};
|
||
/// let rot1 = Rotation3::from_axis_angle(&Vector3::y_axis(), 1.0);
|
||
/// let rot2 = Rotation3::from_axis_angle(&Vector3::x_axis(), 0.1);
|
||
/// let rot_to = rot1.rotation_to(&rot2);
|
||
/// assert_relative_eq!(rot_to * rot1, rot2, epsilon = 1.0e-6);
|
||
/// ```
|
||
#[inline]
|
||
pub fn rotation_to(&self, other: &Self) -> Self {
|
||
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`.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::{Rotation3, Vector3, Unit};
|
||
/// let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));
|
||
/// let angle = 1.2;
|
||
/// let rot = Rotation3::from_axis_angle(&axis, angle);
|
||
/// let pow = rot.powf(2.0);
|
||
/// assert_relative_eq!(pow.axis().unwrap(), axis, epsilon = 1.0e-6);
|
||
/// assert_eq!(pow.angle(), 2.4);
|
||
/// ```
|
||
#[inline]
|
||
pub fn powf(&self, n: N) -> Self
|
||
where
|
||
N: RealField,
|
||
{
|
||
if let Some(axis) = self.axis() {
|
||
Self::from_axis_angle(&axis, self.angle() * n)
|
||
} else if self.matrix()[(0, 0)] < N::zero() {
|
||
let minus_id = MatrixN::<N, U3>::from_diagonal_element(-N::one());
|
||
Self::from_matrix_unchecked(minus_id)
|
||
} else {
|
||
Self::identity()
|
||
}
|
||
}
|
||
|
||
/// Builds a rotation from a basis assumed to be orthonormal.
|
||
///
|
||
/// In order to get a valid unit-quaternion, the input must be an
|
||
/// orthonormal basis, i.e., all vectors are normalized, and the are
|
||
/// all orthogonal to each other. These invariants are not checked
|
||
/// by this method.
|
||
pub fn from_basis_unchecked(basis: &[Vector3<N>; 3]) -> Self {
|
||
let mat = Matrix3::from_columns(&basis[..]);
|
||
Self::from_matrix_unchecked(mat)
|
||
}
|
||
|
||
/// Builds a rotation matrix by extracting the rotation part of the given transformation `m`.
|
||
///
|
||
/// This is an iterative method. See `.from_matrix_eps` to provide mover
|
||
/// convergence parameters and starting solution.
|
||
/// This implements "A Robust Method to Extract the Rotational Part of Deformations" by Müller et al.
|
||
pub fn from_matrix(m: &Matrix3<N>) -> Self
|
||
where
|
||
N: RealField,
|
||
{
|
||
Self::from_matrix_eps(m, N::default_epsilon(), 0, Self::identity())
|
||
}
|
||
|
||
/// Builds a rotation matrix by extracting the rotation part of the given transformation `m`.
|
||
///
|
||
/// This implements "A Robust Method to Extract the Rotational Part of Deformations" by Müller et al.
|
||
///
|
||
/// # Parameters
|
||
///
|
||
/// * `m`: the matrix from which the rotational part is to be extracted.
|
||
/// * `eps`: the angular errors tolerated between the current rotation and the optimal one.
|
||
/// * `max_iter`: the maximum number of iterations. Loops indefinitely until convergence if set to `0`.
|
||
/// * `guess`: a guess of the solution. Convergence will be significantly faster if an initial solution close
|
||
/// to the actual solution is provided. Can be set to `Rotation3::identity()` if no other
|
||
/// guesses come to mind.
|
||
pub fn from_matrix_eps(m: &Matrix3<N>, eps: N, mut max_iter: usize, guess: Self) -> Self
|
||
where
|
||
N: RealField,
|
||
{
|
||
if max_iter == 0 {
|
||
max_iter = usize::max_value();
|
||
}
|
||
|
||
let mut rot = guess.into_inner();
|
||
|
||
for _ in 0..max_iter {
|
||
let axis = rot.column(0).cross(&m.column(0))
|
||
+ rot.column(1).cross(&m.column(1))
|
||
+ rot.column(2).cross(&m.column(2));
|
||
let denom = rot.column(0).dot(&m.column(0))
|
||
+ rot.column(1).dot(&m.column(1))
|
||
+ rot.column(2).dot(&m.column(2));
|
||
|
||
let axisangle = axis / (denom.abs() + N::default_epsilon());
|
||
|
||
if let Some((axis, angle)) = Unit::try_new_and_get(axisangle, eps) {
|
||
rot = Rotation3::from_axis_angle(&axis, angle) * rot;
|
||
} else {
|
||
break;
|
||
}
|
||
}
|
||
|
||
Self::from_matrix_unchecked(rot)
|
||
}
|
||
|
||
/// Ensure this rotation is an orthonormal rotation matrix. This is useful when repeated
|
||
/// computations might cause the matrix from progressively not being orthonormal anymore.
|
||
#[inline]
|
||
pub fn renormalize(&mut self)
|
||
where
|
||
N: RealField,
|
||
{
|
||
let mut c = UnitQuaternion::from(*self);
|
||
let _ = c.renormalize();
|
||
|
||
*self = Self::from_matrix_eps(self.matrix(), N::default_epsilon(), 0, c.into())
|
||
}
|
||
}
|
||
|
||
/// # 3D axis and angle extraction
|
||
impl<N: SimdRealField> Rotation3<N> {
|
||
/// The rotation angle in [0; pi].
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::{Unit, Rotation3, Vector3};
|
||
/// let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));
|
||
/// let rot = Rotation3::from_axis_angle(&axis, 1.78);
|
||
/// assert_relative_eq!(rot.angle(), 1.78);
|
||
/// ```
|
||
#[inline]
|
||
pub fn angle(&self) -> N {
|
||
((self.matrix()[(0, 0)] + self.matrix()[(1, 1)] + self.matrix()[(2, 2)] - N::one())
|
||
/ crate::convert(2.0))
|
||
.simd_acos()
|
||
}
|
||
|
||
/// The rotation axis. Returns `None` if the rotation angle is zero or PI.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::{Rotation3, Vector3, Unit};
|
||
/// let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));
|
||
/// let angle = 1.2;
|
||
/// let rot = Rotation3::from_axis_angle(&axis, angle);
|
||
/// assert_relative_eq!(rot.axis().unwrap(), axis);
|
||
///
|
||
/// // Case with a zero angle.
|
||
/// let rot = Rotation3::from_axis_angle(&axis, 0.0);
|
||
/// assert!(rot.axis().is_none());
|
||
/// ```
|
||
#[inline]
|
||
pub fn axis(&self) -> Option<Unit<Vector3<N>>>
|
||
where
|
||
N: RealField,
|
||
{
|
||
let axis = VectorN::<N, U3>::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.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::{Rotation3, Vector3, Unit};
|
||
/// let axisangle = Vector3::new(0.1, 0.2, 0.3);
|
||
/// let rot = Rotation3::new(axisangle);
|
||
/// assert_relative_eq!(rot.scaled_axis(), axisangle, epsilon = 1.0e-6);
|
||
/// ```
|
||
#[inline]
|
||
pub fn scaled_axis(&self) -> Vector3<N>
|
||
where
|
||
N: RealField,
|
||
{
|
||
if let Some(axis) = self.axis() {
|
||
axis.into_inner() * self.angle()
|
||
} else {
|
||
Vector::zero()
|
||
}
|
||
}
|
||
|
||
/// The rotation axis and angle in ]0, pi] of this unit quaternion.
|
||
///
|
||
/// Returns `None` if the angle is zero.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::{Rotation3, Vector3, Unit};
|
||
/// let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));
|
||
/// let angle = 1.2;
|
||
/// let rot = Rotation3::from_axis_angle(&axis, angle);
|
||
/// let axis_angle = rot.axis_angle().unwrap();
|
||
/// assert_relative_eq!(axis_angle.0, axis);
|
||
/// assert_relative_eq!(axis_angle.1, angle);
|
||
///
|
||
/// // Case with a zero angle.
|
||
/// let rot = Rotation3::from_axis_angle(&axis, 0.0);
|
||
/// assert!(rot.axis_angle().is_none());
|
||
/// ```
|
||
#[inline]
|
||
pub fn axis_angle(&self) -> Option<(Unit<Vector3<N>>, N)>
|
||
where
|
||
N: RealField,
|
||
{
|
||
if let Some(axis) = self.axis() {
|
||
Some((axis, self.angle()))
|
||
} else {
|
||
None
|
||
}
|
||
}
|
||
|
||
/// The rotation angle needed to make `self` and `other` coincide.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::{Rotation3, Vector3};
|
||
/// let rot1 = Rotation3::from_axis_angle(&Vector3::y_axis(), 1.0);
|
||
/// let rot2 = Rotation3::from_axis_angle(&Vector3::x_axis(), 0.1);
|
||
/// assert_relative_eq!(rot1.angle_to(&rot2), 1.0045657, epsilon = 1.0e-6);
|
||
/// ```
|
||
#[inline]
|
||
pub fn angle_to(&self, other: &Self) -> N
|
||
where
|
||
N::Element: SimdRealField,
|
||
{
|
||
self.rotation_to(other).angle()
|
||
}
|
||
|
||
/// Creates Euler angles from a rotation.
|
||
///
|
||
/// The angles are produced in the form (roll, pitch, yaw).
|
||
#[deprecated(note = "This is renamed to use `.euler_angles()`.")]
|
||
pub fn to_euler_angles(&self) -> (N, N, N)
|
||
where
|
||
N: RealField,
|
||
{
|
||
self.euler_angles()
|
||
}
|
||
|
||
/// Euler angles corresponding to this rotation from a rotation.
|
||
///
|
||
/// The angles are produced in the form (roll, pitch, yaw).
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::Rotation3;
|
||
/// let rot = Rotation3::from_euler_angles(0.1, 0.2, 0.3);
|
||
/// let euler = rot.euler_angles();
|
||
/// assert_relative_eq!(euler.0, 0.1, epsilon = 1.0e-6);
|
||
/// assert_relative_eq!(euler.1, 0.2, epsilon = 1.0e-6);
|
||
/// assert_relative_eq!(euler.2, 0.3, epsilon = 1.0e-6);
|
||
/// ```
|
||
pub fn euler_angles(&self) -> (N, N, N)
|
||
where
|
||
N: RealField,
|
||
{
|
||
// Implementation informed by "Computing Euler angles from a rotation matrix", by Gregory G. Slabaugh
|
||
// https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.371.6578
|
||
if self[(2, 0)].abs() < N::one() {
|
||
let yaw = -self[(2, 0)].asin();
|
||
let roll = (self[(2, 1)] / yaw.cos()).atan2(self[(2, 2)] / yaw.cos());
|
||
let pitch = (self[(1, 0)] / yaw.cos()).atan2(self[(0, 0)] / yaw.cos());
|
||
(roll, yaw, pitch)
|
||
} else if self[(2, 0)] <= -N::one() {
|
||
(self[(0, 1)].atan2(self[(0, 2)]), N::frac_pi_2(), N::zero())
|
||
} else {
|
||
(
|
||
-self[(0, 1)].atan2(-self[(0, 2)]),
|
||
-N::frac_pi_2(),
|
||
N::zero(),
|
||
)
|
||
}
|
||
}
|
||
}
|
||
|
||
#[cfg(feature = "rand-no-std")]
|
||
impl<N: SimdRealField> Distribution<Rotation3<N>> for Standard
|
||
where
|
||
N::Element: SimdRealField,
|
||
OpenClosed01: Distribution<N>,
|
||
N: SampleUniform,
|
||
{
|
||
/// Generate a uniformly distributed random rotation.
|
||
#[inline]
|
||
fn sample<'a, R: Rng + ?Sized>(&self, rng: &mut R) -> Rotation3<N> {
|
||
// James Arvo.
|
||
// Fast random rotation matrices.
|
||
// In D. Kirk, editor, Graphics Gems III, pages 117-120. Academic, New York, 1992.
|
||
|
||
// Compute a random rotation around Z
|
||
let twopi = Uniform::new(N::zero(), N::simd_two_pi());
|
||
let theta = rng.sample(&twopi);
|
||
let (ts, tc) = theta.simd_sin_cos();
|
||
let a = MatrixN::<N, U3>::new(
|
||
tc,
|
||
ts,
|
||
N::zero(),
|
||
-ts,
|
||
tc,
|
||
N::zero(),
|
||
N::zero(),
|
||
N::zero(),
|
||
N::one(),
|
||
);
|
||
|
||
// Compute a random rotation *of* Z
|
||
let phi = rng.sample(&twopi);
|
||
let z = rng.sample(OpenClosed01);
|
||
let (ps, pc) = phi.simd_sin_cos();
|
||
let sqrt_z = z.simd_sqrt();
|
||
let v = Vector3::new(pc * sqrt_z, ps * sqrt_z, (N::one() - z).simd_sqrt());
|
||
let mut b = v * v.transpose();
|
||
b += b;
|
||
b -= MatrixN::<N, U3>::identity();
|
||
|
||
Rotation3::from_matrix_unchecked(b * a)
|
||
}
|
||
}
|
||
|
||
#[cfg(feature = "arbitrary")]
|
||
impl<N: SimdRealField + Arbitrary> Arbitrary for Rotation3<N>
|
||
where
|
||
N::Element: SimdRealField,
|
||
Owned<N, U3, U3>: Send,
|
||
Owned<N, U3>: Send,
|
||
{
|
||
#[inline]
|
||
fn arbitrary(g: &mut Gen) -> Self {
|
||
Self::new(VectorN::arbitrary(g))
|
||
}
|
||
}
|