987 lines
33 KiB
Rust
987 lines
33 KiB
Rust
#[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, 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, -sia, 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])
|
||
}
|
||
}
|
||
|
||
/// # 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 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: &[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.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);
|
||
let _ = c.renormalize();
|
||
|
||
*self = Self::from_matrix_eps(self.matrix(), T::default_epsilon(), 0, c.into())
|
||
}
|
||
|
||
/// Raise the quaternion 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)].simd_atan2(self.matrix()[(0, 0)])
|
||
}
|
||
|
||
/// 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.simd_ne(T::zero()).if_else(
|
||
|| {
|
||
let ux = axis.as_ref()[0];
|
||
let uy = axis.as_ref()[1];
|
||
let uz = axis.as_ref()[2];
|
||
let sqx = ux * ux;
|
||
let sqy = uy * uy;
|
||
let sqz = uz * uz;
|
||
let (sin, cos) = angle.simd_sin_cos();
|
||
let one_m_cos = T::one() - cos;
|
||
|
||
Self::from_matrix_unchecked(SMatrix::<T, 3, 3>::new(
|
||
sqx + (T::one() - sqx) * cos,
|
||
ux * uy * one_m_cos - uz * sin,
|
||
ux * uz * one_m_cos + uy * sin,
|
||
ux * uy * one_m_cos + uz * sin,
|
||
sqy + (T::one() - sqy) * cos,
|
||
uy * uz * one_m_cos - ux * sin,
|
||
ux * uz * one_m_cos - uy * sin,
|
||
uy * uz * one_m_cos + ux * sin,
|
||
sqz + (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 * cp,
|
||
cy * sp * sr - sy * cr,
|
||
cy * sp * cr + sy * sr,
|
||
sy * cp,
|
||
sy * sp * sr + cy * cr,
|
||
sy * sp * cr - cy * sr,
|
||
-sp,
|
||
cp * 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>,
|
||
{
|
||
let zaxis = dir.normalize();
|
||
let xaxis = up.cross(&zaxis).normalize();
|
||
let yaxis = zaxis.cross(&xaxis).normalize();
|
||
|
||
Self::from_matrix_unchecked(SMatrix::<T, 3, 3>::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<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 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]
|
||
#[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 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<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_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() + T::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
|
||
T: RealField,
|
||
{
|
||
let mut c = UnitQuaternion::from(*self);
|
||
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)] + self.matrix()[(1, 1)] + self.matrix()[(2, 2)] - 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 axis = SVector::<T, 3>::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, 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 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]
|
||
#[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
|
||
if self[(2, 0)].abs() < T::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)] <= -T::one() {
|
||
(self[(0, 1)].atan2(self[(0, 2)]), T::frac_pi_2(), T::zero())
|
||
} else {
|
||
(
|
||
-self[(0, 1)].atan2(-self[(0, 2)]),
|
||
-T::frac_pi_2(),
|
||
T::zero(),
|
||
)
|
||
}
|
||
}
|
||
}
|
||
|
||
#[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,
|
||
ts,
|
||
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.simd_sqrt();
|
||
let v = Vector3::new(pc * sqrt_z, ps * sqrt_z, (T::one() - z).simd_sqrt());
|
||
let mut b = v * v.transpose();
|
||
b += b;
|
||
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))
|
||
}
|
||
}
|