nalgebra/geometry/rotation_specialization.rs
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#[cfg(feature = "arbitrary")]
use crate::base::storage::Owned;
#[cfg(feature = "arbitrary")]
use quickcheck::{Arbitrary, Gen};
use num::Zero;
#[cfg(feature = "rand-no-std")]
use rand::{
distributions::{uniform::SampleUniform, Distribution, OpenClosed01, Standard, Uniform},
Rng,
};
use simba::scalar::RealField;
use simba::simd::{SimdBool, SimdRealField};
use std::ops::Neg;
use crate::base::dimension::{U1, U2, U3};
use crate::base::storage::Storage;
use crate::base::{
Matrix2, Matrix3, SMatrix, SVector, Unit, UnitVector3, Vector, Vector1, Vector2, Vector3,
};
use crate::geometry::{Rotation2, Rotation3, UnitComplex, UnitQuaternion};
/*
*
* 2D Rotation matrix.
*
*/
/// # Construction from a 2D rotation angle
impl<T: SimdRealField> Rotation2<T> {
/// Builds a 2 dimensional rotation matrix from an angle in radian.
///
/// # Example
///
/// ```
/// # #[macro_use] extern crate approx;
/// # use std::f32;
/// # use nalgebra::{Rotation2, Point2};
/// let rot = Rotation2::new(f32::consts::FRAC_PI_2);
///
/// assert_relative_eq!(rot * Point2::new(3.0, 4.0), Point2::new(-4.0, 3.0));
/// ```
pub fn new(angle: T) -> Self {
let (sia, coa) = angle.simd_sin_cos();
Self::from_matrix_unchecked(Matrix2::new(coa.clone(), -sia.clone(), sia, coa))
}
/// Builds a 2 dimensional rotation matrix from an angle in radian wrapped in a 1-dimensional vector.
///
///
/// This is generally used in the context of generic programming. Using
/// the `::new(angle)` method instead is more common.
#[inline]
pub fn from_scaled_axis<SB: Storage<T, U1>>(axisangle: Vector<T, U1, SB>) -> Self {
Self::new(axisangle[0].clone())
}
}
/// # Construction from an existing 2D matrix or rotations
impl<T: SimdRealField> Rotation2<T> {
/// Builds a rotation from a basis assumed to be orthonormal.
///
/// In order to get a valid rotation matrix, 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: &[Vector2<T>; 2]) -> Self {
let mat = Matrix2::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: &Matrix2<T>) -> Self
where
T: RealField,
{
Self::from_matrix_eps(m, T::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`: an estimate of the solution. Convergence will be significantly faster if an initial solution close
/// to the actual solution is provided. Can be set to `Rotation2::identity()` if no other
/// guesses come to mind.
pub fn from_matrix_eps(m: &Matrix2<T>, eps: T, mut max_iter: usize, guess: Self) -> Self
where
T: 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).perp(&m.column(0)) + rot.column(1).perp(&m.column(1));
let denom = rot.column(0).dot(&m.column(0)) + rot.column(1).dot(&m.column(1));
let angle = axis / (denom.abs() + T::default_epsilon());
if angle.clone().abs() > eps {
rot = Self::new(angle) * rot;
} else {
break;
}
}
Self::from_matrix_unchecked(rot)
}
/// 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::{Vector2, Rotation2};
/// let a = Vector2::new(1.0, 2.0);
/// let b = Vector2::new(2.0, 1.0);
/// let rot = Rotation2::rotation_between(&a, &b);
/// assert_relative_eq!(rot * a, b);
/// assert_relative_eq!(rot.inverse() * b, a);
/// ```
#[inline]
pub fn rotation_between<SB, SC>(a: &Vector<T, U2, SB>, b: &Vector<T, U2, SC>) -> Self
where
T: RealField,
SB: Storage<T, U2>,
SC: Storage<T, U2>,
{
crate::convert(UnitComplex::rotation_between(a, b).to_rotation_matrix())
}
/// The smallest rotation needed to make `a` and `b` collinear and point toward the same
/// direction, raised to the power `s`.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::{Vector2, Rotation2};
/// let a = Vector2::new(1.0, 2.0);
/// let b = Vector2::new(2.0, 1.0);
/// let rot2 = Rotation2::scaled_rotation_between(&a, &b, 0.2);
/// let rot5 = Rotation2::scaled_rotation_between(&a, &b, 0.5);
/// 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<T, U2, SB>,
b: &Vector<T, U2, SC>,
s: T,
) -> Self
where
T: RealField,
SB: Storage<T, U2>,
SC: Storage<T, U2>,
{
crate::convert(UnitComplex::scaled_rotation_between(a, b, s).to_rotation_matrix())
}
/// 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::Rotation2;
/// let rot1 = Rotation2::new(0.1);
/// let rot2 = Rotation2::new(1.7);
/// let rot_to = rot1.rotation_to(&rot2);
///
/// assert_relative_eq!(rot_to * rot1, rot2);
/// assert_relative_eq!(rot_to.inverse() * rot2, rot1);
/// ```
#[inline]
#[must_use]
pub fn rotation_to(&self, other: &Self) -> Self {
other * self.inverse()
}
/// 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
T: RealField,
{
let mut c = UnitComplex::from(self.clone());
let _ = c.renormalize();
*self = Self::from_matrix_eps(self.matrix(), T::default_epsilon(), 0, c.into())
}
/// Raise the rotation to a given floating power, i.e., returns the rotation with the angle
/// of `self` multiplied by `n`.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::Rotation2;
/// let rot = Rotation2::new(0.78);
/// let pow = rot.powf(2.0);
/// assert_relative_eq!(pow.angle(), 2.0 * 0.78);
/// ```
#[inline]
#[must_use]
pub fn powf(&self, n: T) -> Self {
Self::new(self.angle() * n)
}
}
/// # 2D angle extraction
impl<T: SimdRealField> Rotation2<T> {
/// The rotation angle.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::Rotation2;
/// let rot = Rotation2::new(1.78);
/// assert_relative_eq!(rot.angle(), 1.78);
/// ```
#[inline]
#[must_use]
pub fn angle(&self) -> T {
self.matrix()[(1, 0)]
.clone()
.simd_atan2(self.matrix()[(0, 0)].clone())
}
/// The rotation angle needed to make `self` and `other` coincide.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use nalgebra::Rotation2;
/// let rot1 = Rotation2::new(0.1);
/// let rot2 = Rotation2::new(1.7);
/// assert_relative_eq!(rot1.angle_to(&rot2), 1.6);
/// ```
#[inline]
#[must_use]
pub fn angle_to(&self, other: &Self) -> T {
self.rotation_to(other).angle()
}
/// The rotation angle returned as a 1-dimensional vector.
///
/// This is generally used in the context of generic programming. Using
/// the `.angle()` method instead is more common.
#[inline]
#[must_use]
pub fn scaled_axis(&self) -> SVector<T, 1> {
Vector1::new(self.angle())
}
}
#[cfg(feature = "rand-no-std")]
impl<T: SimdRealField> Distribution<Rotation2<T>> for Standard
where
T::Element: SimdRealField,
T: SampleUniform,
{
/// Generate a uniformly distributed random rotation.
#[inline]
fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> Rotation2<T> {
let twopi = Uniform::new(T::zero(), T::simd_two_pi());
Rotation2::new(rng.sample(twopi))
}
}
#[cfg(feature = "arbitrary")]
impl<T: SimdRealField + Arbitrary> Arbitrary for Rotation2<T>
where
T::Element: SimdRealField,
Owned<T, U2, U2>: Send,
{
#[inline]
fn arbitrary(g: &mut Gen) -> Self {
Self::new(T::arbitrary(g))
}
}
/*
*
* 3D Rotation matrix.
*
*/
/// # Construction from a 3D axis and/or angles
impl<T: SimdRealField> Rotation3<T>
where
T::Element: SimdRealField,
{
/// Builds a 3 dimensional rotation matrix from an axis and an angle.
///
/// # Arguments
/// * `axisangle` - A vector representing the rotation. Its magnitude is the amount of rotation
/// in radian. Its direction is the axis of rotation.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use std::f32;
/// # use nalgebra::{Rotation3, Point3, Vector3};
/// let axisangle = Vector3::y() * f32::consts::FRAC_PI_2;
/// // Point and vector being transformed in the tests.
/// let pt = Point3::new(4.0, 5.0, 6.0);
/// let vec = Vector3::new(4.0, 5.0, 6.0);
/// let rot = Rotation3::new(axisangle);
///
/// assert_relative_eq!(rot * pt, Point3::new(6.0, 5.0, -4.0), epsilon = 1.0e-6);
/// assert_relative_eq!(rot * vec, Vector3::new(6.0, 5.0, -4.0), epsilon = 1.0e-6);
///
/// // A zero vector yields an identity.
/// assert_eq!(Rotation3::new(Vector3::<f32>::zeros()), Rotation3::identity());
/// ```
pub fn new<SB: Storage<T, U3>>(axisangle: Vector<T, U3, SB>) -> Self {
let axisangle = axisangle.into_owned();
let (axis, angle) = Unit::new_and_get(axisangle);
Self::from_axis_angle(&axis, angle)
}
/// Builds a 3D rotation matrix from an axis scaled by the rotation angle.
///
/// This is the same as `Self::new(axisangle)`.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use std::f32;
/// # use nalgebra::{Rotation3, Point3, Vector3};
/// let axisangle = Vector3::y() * f32::consts::FRAC_PI_2;
/// // Point and vector being transformed in the tests.
/// let pt = Point3::new(4.0, 5.0, 6.0);
/// let vec = Vector3::new(4.0, 5.0, 6.0);
/// let rot = Rotation3::new(axisangle);
///
/// assert_relative_eq!(rot * pt, Point3::new(6.0, 5.0, -4.0), epsilon = 1.0e-6);
/// assert_relative_eq!(rot * vec, Vector3::new(6.0, 5.0, -4.0), epsilon = 1.0e-6);
///
/// // A zero vector yields an identity.
/// assert_eq!(Rotation3::from_scaled_axis(Vector3::<f32>::zeros()), Rotation3::identity());
/// ```
pub fn from_scaled_axis<SB: Storage<T, U3>>(axisangle: Vector<T, U3, SB>) -> Self {
Self::new(axisangle)
}
/// Builds a 3D rotation matrix from an axis and a rotation angle.
///
/// # Example
/// ```
/// # #[macro_use] extern crate approx;
/// # use std::f32;
/// # use nalgebra::{Rotation3, Point3, Vector3};
/// let axis = Vector3::y_axis();
/// let angle = f32::consts::FRAC_PI_2;
/// // Point and vector being transformed in the tests.
/// let pt = Point3::new(4.0, 5.0, 6.0);
/// let vec = Vector3::new(4.0, 5.0, 6.0);
/// let rot = Rotation3::from_axis_angle(&axis, angle);
///
/// assert_eq!(rot.axis().unwrap(), axis);
/// assert_eq!(rot.angle(), angle);
/// assert_relative_eq!(rot * pt, Point3::new(6.0, 5.0, -4.0), epsilon = 1.0e-6);
/// assert_relative_eq!(rot * vec, Vector3::new(6.0, 5.0, -4.0), epsilon = 1.0e-6);
///
/// // A zero vector yields an identity.
/// assert_eq!(Rotation3::from_scaled_axis(Vector3::<f32>::zeros()), Rotation3::identity());
/// ```
pub fn from_axis_angle<SB>(axis: &Unit<Vector<T, U3, SB>>, angle: T) -> Self
where
SB: Storage<T, U3>,
{
angle.clone().simd_ne(T::zero()).if_else(
|| {
let ux = axis.as_ref()[0].clone();
let uy = axis.as_ref()[1].clone();
let uz = axis.as_ref()[2].clone();
let sqx = ux.clone() * ux.clone();
let sqy = uy.clone() * uy.clone();
let sqz = uz.clone() * uz.clone();
let (sin, cos) = angle.simd_sin_cos();
let one_m_cos = T::one() - cos.clone();
Self::from_matrix_unchecked(SMatrix::<T, 3, 3>::new(
sqx.clone() + (T::one() - sqx) * cos.clone(),
ux.clone() * uy.clone() * one_m_cos.clone() - uz.clone() * sin.clone(),
ux.clone() * uz.clone() * one_m_cos.clone() + uy.clone() * sin.clone(),
ux.clone() * uy.clone() * one_m_cos.clone() + uz.clone() * sin.clone(),
sqy.clone() + (T::one() - sqy) * cos.clone(),
uy.clone() * uz.clone() * one_m_cos.clone() - ux.clone() * sin.clone(),
ux.clone() * uz.clone() * one_m_cos.clone() - uy.clone() * sin.clone(),
uy * uz * one_m_cos + ux * sin,
sqz.clone() + (T::one() - sqz) * cos,
))
},
Self::identity,
)
}
/// Creates a new rotation from Euler angles.
///
/// The primitive rotations are applied in order: 1 roll − 2 pitch − 3 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 from_euler_angles(roll: T, pitch: T, yaw: T) -> Self {
let (sr, cr) = roll.simd_sin_cos();
let (sp, cp) = pitch.simd_sin_cos();
let (sy, cy) = yaw.simd_sin_cos();
Self::from_matrix_unchecked(SMatrix::<T, 3, 3>::new(
cy.clone() * cp.clone(),
cy.clone() * sp.clone() * sr.clone() - sy.clone() * cr.clone(),
cy.clone() * sp.clone() * cr.clone() + sy.clone() * sr.clone(),
sy.clone() * cp.clone(),
sy.clone() * sp.clone() * sr.clone() + cy.clone() * cr.clone(),
sy * sp.clone() * cr.clone() - cy * sr.clone(),
-sp,
cp.clone() * sr,
cp * cr,
))
}
}
/// # Construction from a 3D eye position and target point
impl<T: SimdRealField> Rotation3<T>
where
T::Element: SimdRealField,
{
/// Creates a rotation that corresponds to the local frame of an observer standing at the
/// origin and looking toward `dir`.
///
/// It maps the `z` axis to the direction `dir`.
///
/// # Arguments
/// * dir - The look direction, that is, direction the matrix `z` axis will be aligned with.
/// * up - The vertical direction. The only requirement of this parameter is to not be
/// collinear to `dir`. Non-collinearity is not checked.
///
/// # 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::face_towards(&dir, &up);
/// assert_relative_eq!(rot * Vector3::z(), dir.normalize());
/// ```
#[inline]
pub fn face_towards<SB, SC>(dir: &Vector<T, U3, SB>, up: &Vector<T, U3, SC>) -> Self
where
SB: Storage<T, U3>,
SC: Storage<T, U3>,
{
// Gram–Schmidt process
let zaxis = dir.normalize();
let xaxis = up.cross(&zaxis).normalize();
let yaxis = zaxis.cross(&xaxis);
Self::from_matrix_unchecked(SMatrix::<T, 3, 3>::new(
xaxis.x.clone(),
yaxis.x.clone(),
zaxis.x.clone(),
xaxis.y.clone(),
yaxis.y.clone(),
zaxis.y.clone(),
xaxis.z.clone(),
yaxis.z.clone(),
zaxis.z.clone(),
))
}
/// Deprecated: Use [`Rotation3::face_towards`] instead.
#[deprecated(note = "renamed to `face_towards`")]
pub fn new_observer_frames<SB, SC>(dir: &Vector<T, U3, SB>, up: &Vector<T, U3, SC>) -> Self
where
SB: Storage<T, U3>,
SC: Storage<T, 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<T, U3, SB>, up: &Vector<T, U3, SC>) -> Self
where
SB: Storage<T, U3>,
SC: Storage<T, 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<T, U3, SB>, up: &Vector<T, U3, SC>) -> Self
where
SB: Storage<T, U3>,
SC: Storage<T, U3>,
{
Self::face_towards(dir, up).inverse()
}
}
/// # Construction from an existing 3D matrix or rotations
impl<T: SimdRealField> Rotation3<T>
where
T::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<T, U3, SB>, b: &Vector<T, U3, SC>) -> Option<Self>
where
T: RealField,
SB: Storage<T, U3>,
SC: Storage<T, U3>,
{
Self::scaled_rotation_between(a, b, T::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<T, U3, SB>,
b: &Vector<T, U3, SC>,
n: T,
) -> Option<Self>
where
T: RealField,
SB: Storage<T, U3>,
SC: Storage<T, U3>,
{
// TODO: code duplication with Rotation.
if let (Some(na), Some(nb)) = (a.try_normalize(T::zero()), b.try_normalize(T::zero())) {
let c = na.cross(&nb);
if let Some(axis) = Unit::try_new(c, T::default_epsilon()) {
return Some(Self::from_axis_angle(&axis, na.dot(&nb).acos() * n));
}
// Zero or PI.
if na.dot(&nb) < T::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]
#[must_use]
pub fn rotation_to(&self, other: &Self) -> Self {
other * self.inverse()
}
/// Raise the rotation 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]
#[must_use]
pub fn powf(&self, n: T) -> Self
where
T: RealField,
{
if let Some(axis) = self.axis() {
Self::from_axis_angle(&axis, self.angle() * n)
} else if self.matrix()[(0, 0)] < T::zero() {
let minus_id = SMatrix::<T, 3, 3>::from_diagonal_element(-T::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 rotation matrix, 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<T>; 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<T>) -> Self
where
T: RealField,
{
Self::from_matrix_eps(m, T::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<T>, eps: T, mut max_iter: usize, guess: Self) -> Self
where
T: RealField,
{
if max_iter == 0 {
max_iter = usize::MAX;
}
// Using sqrt(eps) ensures we perturb with something larger than eps; clamp to eps to handle the case of eps > 1.0
let eps_disturbance = eps.clone().sqrt().max(eps.clone() * eps.clone());
let mut perturbation_axes = Vector3::x_axis();
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() + T::default_epsilon());
if let Some((axis, angle)) = Unit::try_new_and_get(axisangle, eps.clone()) {
rot = Rotation3::from_axis_angle(&axis, angle) * rot;
} else {
// Check if stuck in a maximum w.r.t. the norm (m - rot).norm()
let mut perturbed = rot.clone();
let norm_squared = (m - &rot).norm_squared();
let mut new_norm_squared: T;
// Perturb until the new norm is significantly different
loop {
perturbed *=
Rotation3::from_axis_angle(&perturbation_axes, eps_disturbance.clone());
new_norm_squared = (m - &perturbed).norm_squared();
if abs_diff_ne!(
norm_squared,
new_norm_squared,
epsilon = T::default_epsilon()
) {
break;
}
}
// If new norm is larger, it's a minimum
if norm_squared < new_norm_squared {
break;
}
// If not, continue from perturbed rotation, but use a different axes for the next perturbation
perturbation_axes = UnitVector3::new_unchecked(perturbation_axes.yzx());
rot = perturbed;
}
}
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
T: RealField,
{
let mut c = UnitQuaternion::from(self.clone());
let _ = c.renormalize();
*self = Self::from_matrix_eps(self.matrix(), T::default_epsilon(), 0, c.into())
}
}
/// # 3D axis and angle extraction
impl<T: SimdRealField> Rotation3<T> {
/// 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]
#[must_use]
pub fn angle(&self) -> T {
((self.matrix()[(0, 0)].clone()
+ self.matrix()[(1, 1)].clone()
+ self.matrix()[(2, 2)].clone()
- T::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]
#[must_use]
pub fn axis(&self) -> Option<Unit<Vector3<T>>>
where
T: RealField,
{
let rotmat = self.matrix();
let axis = SVector::<T, 3>::new(
rotmat[(2, 1)].clone() - rotmat[(1, 2)].clone(),
rotmat[(0, 2)].clone() - rotmat[(2, 0)].clone(),
rotmat[(1, 0)].clone() - rotmat[(0, 1)].clone(),
);
Unit::try_new(axis, T::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]
#[must_use]
pub fn scaled_axis(&self) -> Vector3<T>
where
T: 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 rotation matrix.
///
/// 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]
#[must_use]
pub fn axis_angle(&self) -> Option<(Unit<Vector3<T>>, T)>
where
T: RealField,
{
self.axis().map(|axis| (axis, self.angle()))
}
/// 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]
#[must_use]
pub fn angle_to(&self, other: &Self) -> T
where
T::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) -> (T, T, T)
where
T: 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);
/// ```
#[must_use]
pub fn euler_angles(&self) -> (T, T, T)
where
T: 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
// where roll, pitch, yaw angles are referred to as ψ, θ, ϕ,
if self[(2, 0)].clone().abs() < T::one() {
let pitch = -self[(2, 0)].clone().asin();
let theta_cos = pitch.clone().cos();
let roll = (self[(2, 1)].clone() / theta_cos.clone())
.atan2(self[(2, 2)].clone() / theta_cos.clone());
let yaw =
(self[(1, 0)].clone() / theta_cos.clone()).atan2(self[(0, 0)].clone() / theta_cos);
(roll, pitch, yaw)
} else if self[(2, 0)].clone() <= -T::one() {
(
self[(0, 1)].clone().atan2(self[(0, 2)].clone()),
T::frac_pi_2(),
T::zero(),
)
} else {
(
-self[(0, 1)].clone().atan2(-self[(0, 2)].clone()),
-T::frac_pi_2(),
T::zero(),
)
}
}
/// Represent this rotation as Euler angles.
///
/// Returns the angles produced in the order provided by seq parameter, along with the
/// observability flag. The Euler axes passed to seq must form an orthonormal basis. If the
/// rotation is gimbal locked, then the observability flag is false.
///
/// # Panics
///
/// Panics if the Euler axes in `seq` are not orthonormal.
///
/// # Example 1:
/// ```
/// use std::f64::consts::PI;
/// use approx::assert_relative_eq;
/// use nalgebra::{Matrix3, Rotation3, Unit, Vector3};
///
/// // 3-1-2
/// let n = [
/// Unit::new_unchecked(Vector3::new(0.0, 0.0, 1.0)),
/// Unit::new_unchecked(Vector3::new(1.0, 0.0, 0.0)),
/// Unit::new_unchecked(Vector3::new(0.0, 1.0, 0.0)),
/// ];
///
/// let r1 = Rotation3::from_axis_angle(&n[2], 20.0 * PI / 180.0);
/// let r2 = Rotation3::from_axis_angle(&n[1], 30.0 * PI / 180.0);
/// let r3 = Rotation3::from_axis_angle(&n[0], 45.0 * PI / 180.0);
///
/// let d = r3 * r2 * r1;
///
/// let (angles, observable) = d.euler_angles_ordered(n, false);
/// assert!(observable);
/// assert_relative_eq!(angles[0] * 180.0 / PI, 45.0, epsilon = 1e-12);
/// assert_relative_eq!(angles[1] * 180.0 / PI, 30.0, epsilon = 1e-12);
/// assert_relative_eq!(angles[2] * 180.0 / PI, 20.0, epsilon = 1e-12);
/// ```
///
/// # Example 2:
/// ```
/// use std::f64::consts::PI;
/// use approx::assert_relative_eq;
/// use nalgebra::{Matrix3, Rotation3, Unit, Vector3};
///
/// let sqrt_2 = 2.0_f64.sqrt();
/// let n = [
/// Unit::new_unchecked(Vector3::new(1.0 / sqrt_2, 1.0 / sqrt_2, 0.0)),
/// Unit::new_unchecked(Vector3::new(1.0 / sqrt_2, -1.0 / sqrt_2, 0.0)),
/// Unit::new_unchecked(Vector3::new(0.0, 0.0, 1.0)),
/// ];
///
/// let r1 = Rotation3::from_axis_angle(&n[2], 20.0 * PI / 180.0);
/// let r2 = Rotation3::from_axis_angle(&n[1], 30.0 * PI / 180.0);
/// let r3 = Rotation3::from_axis_angle(&n[0], 45.0 * PI / 180.0);
///
/// let d = r3 * r2 * r1;
///
/// let (angles, observable) = d.euler_angles_ordered(n, false);
/// assert!(observable);
/// assert_relative_eq!(angles[0] * 180.0 / PI, 45.0, epsilon = 1e-12);
/// assert_relative_eq!(angles[1] * 180.0 / PI, 30.0, epsilon = 1e-12);
/// assert_relative_eq!(angles[2] * 180.0 / PI, 20.0, epsilon = 1e-12);
/// ```
///
/// Algorithm based on:
/// Malcolm D. Shuster, F. Landis Markley, “General formula for extraction the Euler
/// angles”, Journal of guidance, control, and dynamics, vol. 29.1, pp. 215-221. 2006,
/// and modified to be able to produce extrinsic rotations.
#[must_use]
pub fn euler_angles_ordered(
&self,
mut seq: [Unit<Vector3<T>>; 3],
extrinsic: bool,
) -> ([T; 3], bool)
where
T: RealField + Copy,
{
let mut angles = [T::zero(); 3];
let eps = T::from_subset(&1e-7);
let two = T::from_subset(&2.0);
if extrinsic {
seq.reverse();
}
let [n1, n2, n3] = &seq;
assert_relative_eq!(n1.dot(n2), T::zero(), epsilon = eps);
assert_relative_eq!(n3.dot(n1), T::zero(), epsilon = eps);
let n1_c_n2 = n1.cross(n2);
let s1 = n1_c_n2.dot(n3);
let c1 = n1.dot(n3);
let lambda = s1.atan2(c1);
let mut c = Matrix3::zeros();
c.column_mut(0).copy_from(n2);
c.column_mut(1).copy_from(&n1_c_n2);
c.column_mut(2).copy_from(n1);
c.transpose_mut();
let r1l = Matrix3::new(
T::one(),
T::zero(),
T::zero(),
T::zero(),
c1,
s1,
T::zero(),
-s1,
c1,
);
let o_t = c * self.matrix() * (c.transpose() * r1l);
angles[1] = o_t.m33.acos();
let safe1 = angles[1].abs() >= eps;
let safe2 = (angles[1] - T::pi()).abs() >= eps;
let observable = safe1 && safe2;
angles[1] += lambda;
if observable {
angles[0] = o_t.m13.atan2(-o_t.m23);
angles[2] = o_t.m31.atan2(o_t.m32);
} else {
// gimbal lock detected
if extrinsic {
// angle1 is initialized to zero
if !safe1 {
angles[2] = (o_t.m12 - o_t.m21).atan2(o_t.m11 + o_t.m22);
} else {
angles[2] = -(o_t.m12 + o_t.m21).atan2(o_t.m11 - o_t.m22);
};
} else {
// angle3 is initialized to zero
if !safe1 {
angles[0] = (o_t.m12 - o_t.m21).atan2(o_t.m11 + o_t.m22);
} else {
angles[0] = (o_t.m12 + o_t.m21).atan2(o_t.m11 - o_t.m22);
};
};
};
let adjust = if seq[0] == seq[2] {
// lambda = 0, so ensure angle2 -> [0, pi]
angles[1] < T::zero() || angles[1] > T::pi()
} else {
// lamda = + or - pi/2, so ensure angle2 -> [-pi/2, pi/2]
angles[1] < -T::frac_pi_2() || angles[1] > T::frac_pi_2()
};
// dont adjust gimbal locked rotation
if adjust && observable {
angles[0] += T::pi();
angles[1] = two * lambda - angles[1];
angles[2] -= T::pi();
}
// ensure all angles are within [-pi, pi]
for angle in angles.as_mut_slice().iter_mut() {
if *angle < -T::pi() {
*angle += T::two_pi();
} else if *angle > T::pi() {
*angle -= T::two_pi();
}
}
if extrinsic {
angles.reverse();
}
(angles, observable)
}
}
#[cfg(feature = "rand-no-std")]
impl<T: SimdRealField> Distribution<Rotation3<T>> for Standard
where
T::Element: SimdRealField,
OpenClosed01: Distribution<T>,
T: SampleUniform,
{
/// Generate a uniformly distributed random rotation.
#[inline]
fn sample<'a, R: Rng + ?Sized>(&self, rng: &mut R) -> Rotation3<T> {
// 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(T::zero(), T::simd_two_pi());
let theta = rng.sample(&twopi);
let (ts, tc) = theta.simd_sin_cos();
let a = SMatrix::<T, 3, 3>::new(
tc.clone(),
ts.clone(),
T::zero(),
-ts,
tc,
T::zero(),
T::zero(),
T::zero(),
T::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.clone().simd_sqrt();
let v = Vector3::new(pc * sqrt_z.clone(), ps * sqrt_z, (T::one() - z).simd_sqrt());
let mut b = v.clone() * v.transpose();
b += b.clone();
b -= SMatrix::<T, 3, 3>::identity();
Rotation3::from_matrix_unchecked(b * a)
}
}
#[cfg(feature = "arbitrary")]
impl<T: SimdRealField + Arbitrary> Arbitrary for Rotation3<T>
where
T::Element: SimdRealField,
Owned<T, U3, U3>: Send,
Owned<T, U3>: Send,
{
#[inline]
fn arbitrary(g: &mut Gen) -> Self {
Self::new(SVector::arbitrary(g))
}
}