1640 lines
51 KiB
Rust
Executable File
1640 lines
51 KiB
Rust
Executable File
use approx::{AbsDiffEq, RelativeEq, UlpsEq};
|
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use num::Zero;
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use std::fmt;
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#[cfg(feature = "abomonation-serialize")]
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use std::io::{Result as IOResult, Write};
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#[cfg(feature = "serde-serialize")]
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use crate::base::storage::Owned;
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#[cfg(feature = "serde-serialize")]
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use serde::{Deserialize, Deserializer, Serialize, Serializer};
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#[cfg(feature = "abomonation-serialize")]
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use abomonation::Abomonation;
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use simba::scalar::{ClosedNeg, RealField};
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use simba::simd::{SimdBool, SimdOption, SimdRealField};
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use crate::base::dimension::{U1, U3, U4};
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use crate::base::storage::{CStride, RStride};
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use crate::base::{
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Matrix3, Matrix4, MatrixSlice, MatrixSliceMut, Normed, Scalar, Unit, Vector3, Vector4,
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};
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use crate::geometry::{Point3, Rotation};
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/// A quaternion. See the type alias `UnitQuaternion = Unit<Quaternion>` for a quaternion
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/// that may be used as a rotation.
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#[repr(C)]
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#[derive(Debug, Copy, Clone, Hash, PartialEq, Eq)]
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pub struct Quaternion<N: Scalar> {
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/// This quaternion as a 4D vector of coordinates in the `[ x, y, z, w ]` storage order.
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pub coords: Vector4<N>,
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}
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impl<N: Scalar + Zero> Default for Quaternion<N> {
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fn default() -> Self {
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Quaternion {
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coords: Vector4::zeros(),
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}
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}
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}
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#[cfg(feature = "bytemuck")]
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unsafe impl<N: Scalar> bytemuck::Zeroable for Quaternion<N> where Vector4<N>: bytemuck::Zeroable {}
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#[cfg(feature = "bytemuck")]
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unsafe impl<N: Scalar> bytemuck::Pod for Quaternion<N>
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where
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Vector4<N>: bytemuck::Pod,
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N: Copy,
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{
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}
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#[cfg(feature = "abomonation-serialize")]
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impl<N: Scalar> Abomonation for Quaternion<N>
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where
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Vector4<N>: Abomonation,
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{
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unsafe fn entomb<W: Write>(&self, writer: &mut W) -> IOResult<()> {
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self.coords.entomb(writer)
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}
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fn extent(&self) -> usize {
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self.coords.extent()
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}
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unsafe fn exhume<'a, 'b>(&'a mut self, bytes: &'b mut [u8]) -> Option<&'b mut [u8]> {
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self.coords.exhume(bytes)
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}
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}
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#[cfg(feature = "serde-serialize")]
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impl<N: Scalar> Serialize for Quaternion<N>
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where
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Owned<N, U4>: Serialize,
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{
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fn serialize<S>(&self, serializer: S) -> Result<S::Ok, S::Error>
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where
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S: Serializer,
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{
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self.coords.serialize(serializer)
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}
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}
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#[cfg(feature = "serde-serialize")]
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impl<'a, N: Scalar> Deserialize<'a> for Quaternion<N>
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where
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Owned<N, U4>: Deserialize<'a>,
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{
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fn deserialize<Des>(deserializer: Des) -> Result<Self, Des::Error>
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where
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Des: Deserializer<'a>,
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{
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let coords = Vector4::<N>::deserialize(deserializer)?;
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Ok(Self::from(coords))
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}
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}
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impl<N: SimdRealField> Quaternion<N>
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where
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N::Element: SimdRealField,
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{
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/// Moves this unit quaternion into one that owns its data.
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#[inline]
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#[deprecated(note = "This method is a no-op and will be removed in a future release.")]
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pub fn into_owned(self) -> Self {
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self
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}
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/// Clones this unit quaternion into one that owns its data.
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#[inline]
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#[deprecated(note = "This method is a no-op and will be removed in a future release.")]
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pub fn clone_owned(&self) -> Self {
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Self::from(self.coords.clone_owned())
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}
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/// Normalizes this quaternion.
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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::Quaternion;
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/// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
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/// let q_normalized = q.normalize();
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/// relative_eq!(q_normalized.norm(), 1.0);
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/// ```
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#[inline]
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#[must_use = "Did you mean to use normalize_mut()?"]
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pub fn normalize(&self) -> Self {
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Self::from(self.coords.normalize())
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}
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/// The imaginary part of this quaternion.
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#[inline]
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pub fn imag(&self) -> Vector3<N> {
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self.coords.xyz()
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}
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/// The conjugate of this quaternion.
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///
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/// # Example
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/// ```
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/// # use nalgebra::Quaternion;
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/// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
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/// let conj = q.conjugate();
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/// assert!(conj.i == -2.0 && conj.j == -3.0 && conj.k == -4.0 && conj.w == 1.0);
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/// ```
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#[inline]
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#[must_use = "Did you mean to use conjugate_mut()?"]
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pub fn conjugate(&self) -> Self {
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Self::from_parts(self.w, -self.imag())
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}
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/// Linear interpolation between two quaternion.
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///
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/// Computes `self * (1 - t) + other * t`.
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///
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/// # Example
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/// ```
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/// # use nalgebra::Quaternion;
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/// let q1 = Quaternion::new(1.0, 2.0, 3.0, 4.0);
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/// let q2 = Quaternion::new(10.0, 20.0, 30.0, 40.0);
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///
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/// assert_eq!(q1.lerp(&q2, 0.1), Quaternion::new(1.9, 3.8, 5.7, 7.6));
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/// ```
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#[inline]
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pub fn lerp(&self, other: &Self, t: N) -> Self {
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self * (N::one() - t) + other * t
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}
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/// The vector part `(i, j, k)` of this quaternion.
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///
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/// # Example
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/// ```
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/// # use nalgebra::Quaternion;
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/// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
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/// assert_eq!(q.vector()[0], 2.0);
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/// assert_eq!(q.vector()[1], 3.0);
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/// assert_eq!(q.vector()[2], 4.0);
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/// ```
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#[inline]
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pub fn vector(&self) -> MatrixSlice<N, U3, U1, RStride<N, U4, U1>, CStride<N, U4, U1>> {
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self.coords.fixed_rows::<U3>(0)
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}
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/// The scalar part `w` of this quaternion.
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///
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/// # Example
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/// ```
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/// # use nalgebra::Quaternion;
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/// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
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/// assert_eq!(q.scalar(), 1.0);
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/// ```
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#[inline]
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pub fn scalar(&self) -> N {
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self.coords[3]
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}
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/// Reinterprets this quaternion as a 4D vector.
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///
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/// # Example
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/// ```
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/// # use nalgebra::{Vector4, Quaternion};
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/// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
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/// // Recall that the quaternion is stored internally as (i, j, k, w)
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/// // while the crate::new constructor takes the arguments as (w, i, j, k).
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/// assert_eq!(*q.as_vector(), Vector4::new(2.0, 3.0, 4.0, 1.0));
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/// ```
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#[inline]
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pub fn as_vector(&self) -> &Vector4<N> {
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&self.coords
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}
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/// The norm of this quaternion.
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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::Quaternion;
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/// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
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/// assert_relative_eq!(q.norm(), 5.47722557, epsilon = 1.0e-6);
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/// ```
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#[inline]
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pub fn norm(&self) -> N {
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self.coords.norm()
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}
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/// A synonym for the norm of this quaternion.
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///
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/// Aka the length.
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/// This is the same as `.norm()`
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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::Quaternion;
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/// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
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/// assert_relative_eq!(q.magnitude(), 5.47722557, epsilon = 1.0e-6);
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/// ```
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#[inline]
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pub fn magnitude(&self) -> N {
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self.norm()
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}
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/// The squared norm of this quaternion.
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///
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/// # Example
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/// ```
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/// # use nalgebra::Quaternion;
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/// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
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/// assert_eq!(q.magnitude_squared(), 30.0);
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/// ```
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#[inline]
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pub fn norm_squared(&self) -> N {
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self.coords.norm_squared()
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}
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/// A synonym for the squared norm of this quaternion.
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///
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/// Aka the squared length.
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/// This is the same as `.norm_squared()`
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///
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/// # Example
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/// ```
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/// # use nalgebra::Quaternion;
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/// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
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/// assert_eq!(q.magnitude_squared(), 30.0);
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/// ```
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#[inline]
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pub fn magnitude_squared(&self) -> N {
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self.norm_squared()
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}
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/// The dot product of two quaternions.
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///
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/// # Example
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/// ```
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/// # use nalgebra::Quaternion;
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/// let q1 = Quaternion::new(1.0, 2.0, 3.0, 4.0);
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/// let q2 = Quaternion::new(5.0, 6.0, 7.0, 8.0);
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/// assert_eq!(q1.dot(&q2), 70.0);
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/// ```
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#[inline]
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pub fn dot(&self, rhs: &Self) -> N {
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self.coords.dot(&rhs.coords)
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}
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}
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impl<N: SimdRealField> Quaternion<N>
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where
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N::Element: SimdRealField,
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{
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/// Inverts this quaternion if it is not zero.
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///
|
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/// This method also does not works with SIMD components (see `simd_try_inverse` instead).
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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::Quaternion;
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/// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
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/// let inv_q = q.try_inverse();
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///
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/// assert!(inv_q.is_some());
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/// assert_relative_eq!(inv_q.unwrap() * q, Quaternion::identity());
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///
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/// //Non-invertible case
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/// let q = Quaternion::new(0.0, 0.0, 0.0, 0.0);
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/// let inv_q = q.try_inverse();
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///
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/// assert!(inv_q.is_none());
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/// ```
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#[inline]
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#[must_use = "Did you mean to use try_inverse_mut()?"]
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pub fn try_inverse(&self) -> Option<Self>
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where
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N: RealField,
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{
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let mut res = *self;
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||
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if res.try_inverse_mut() {
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Some(res)
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} else {
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None
|
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}
|
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}
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|
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/// Attempt to inverse this quaternion.
|
||
///
|
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/// This method also works with SIMD components.
|
||
#[inline]
|
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#[must_use = "Did you mean to use try_inverse_mut()?"]
|
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pub fn simd_try_inverse(&self) -> SimdOption<Self> {
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let norm_squared = self.norm_squared();
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let ge = norm_squared.simd_ge(N::simd_default_epsilon());
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SimdOption::new(self.conjugate() / norm_squared, ge)
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}
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/// Calculates the inner product (also known as the dot product).
|
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/// See "Foundations of Game Engine Development, Volume 1: Mathematics" by Lengyel
|
||
/// Formula 4.89.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::Quaternion;
|
||
/// let a = Quaternion::new(0.0, 2.0, 3.0, 4.0);
|
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/// let b = Quaternion::new(0.0, 5.0, 2.0, 1.0);
|
||
/// let expected = Quaternion::new(-20.0, 0.0, 0.0, 0.0);
|
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/// let result = a.inner(&b);
|
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/// assert_relative_eq!(expected, result, epsilon = 1.0e-5);
|
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#[inline]
|
||
pub fn inner(&self, other: &Self) -> Self {
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(self * other + other * self).half()
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||
}
|
||
|
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/// Calculates the outer product (also known as the wedge product).
|
||
/// See "Foundations of Game Engine Development, Volume 1: Mathematics" by Lengyel
|
||
/// Formula 4.89.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::Quaternion;
|
||
/// let a = Quaternion::new(0.0, 2.0, 3.0, 4.0);
|
||
/// let b = Quaternion::new(0.0, 5.0, 2.0, 1.0);
|
||
/// let expected = Quaternion::new(0.0, -5.0, 18.0, -11.0);
|
||
/// let result = a.outer(&b);
|
||
/// assert_relative_eq!(expected, result, epsilon = 1.0e-5);
|
||
/// ```
|
||
#[inline]
|
||
pub fn outer(&self, other: &Self) -> Self {
|
||
#[allow(clippy::eq_op)]
|
||
(self * other - other * self).half()
|
||
}
|
||
|
||
/// Calculates the projection of `self` onto `other` (also known as the parallel).
|
||
/// See "Foundations of Game Engine Development, Volume 1: Mathematics" by Lengyel
|
||
/// Formula 4.94.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::Quaternion;
|
||
/// let a = Quaternion::new(0.0, 2.0, 3.0, 4.0);
|
||
/// let b = Quaternion::new(0.0, 5.0, 2.0, 1.0);
|
||
/// let expected = Quaternion::new(0.0, 3.333333333333333, 1.3333333333333333, 0.6666666666666666);
|
||
/// let result = a.project(&b).unwrap();
|
||
/// assert_relative_eq!(expected, result, epsilon = 1.0e-5);
|
||
/// ```
|
||
#[inline]
|
||
pub fn project(&self, other: &Self) -> Option<Self>
|
||
where
|
||
N: RealField,
|
||
{
|
||
self.inner(other).right_div(other)
|
||
}
|
||
|
||
/// Calculates the rejection of `self` from `other` (also known as the perpendicular).
|
||
/// See "Foundations of Game Engine Development, Volume 1: Mathematics" by Lengyel
|
||
/// Formula 4.94.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::Quaternion;
|
||
/// let a = Quaternion::new(0.0, 2.0, 3.0, 4.0);
|
||
/// let b = Quaternion::new(0.0, 5.0, 2.0, 1.0);
|
||
/// let expected = Quaternion::new(0.0, -1.3333333333333333, 1.6666666666666665, 3.3333333333333335);
|
||
/// let result = a.reject(&b).unwrap();
|
||
/// assert_relative_eq!(expected, result, epsilon = 1.0e-5);
|
||
/// ```
|
||
#[inline]
|
||
pub fn reject(&self, other: &Self) -> Option<Self>
|
||
where
|
||
N: RealField,
|
||
{
|
||
self.outer(other).right_div(other)
|
||
}
|
||
|
||
/// The polar decomposition of this quaternion.
|
||
///
|
||
/// Returns, from left to right: the quaternion norm, the half rotation angle, the rotation
|
||
/// axis. If the rotation angle is zero, the rotation axis is set to `None`.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # use std::f32;
|
||
/// # use nalgebra::{Vector3, Quaternion};
|
||
/// let q = Quaternion::new(0.0, 5.0, 0.0, 0.0);
|
||
/// let (norm, half_ang, axis) = q.polar_decomposition();
|
||
/// assert_eq!(norm, 5.0);
|
||
/// assert_eq!(half_ang, f32::consts::FRAC_PI_2);
|
||
/// assert_eq!(axis, Some(Vector3::x_axis()));
|
||
/// ```
|
||
pub fn polar_decomposition(&self) -> (N, N, Option<Unit<Vector3<N>>>)
|
||
where
|
||
N: RealField,
|
||
{
|
||
if let Some((q, n)) = Unit::try_new_and_get(*self, N::zero()) {
|
||
if let Some(axis) = Unit::try_new(self.vector().clone_owned(), N::zero()) {
|
||
let angle = q.angle() / crate::convert(2.0f64);
|
||
|
||
(n, angle, Some(axis))
|
||
} else {
|
||
(n, N::zero(), None)
|
||
}
|
||
} else {
|
||
(N::zero(), N::zero(), None)
|
||
}
|
||
}
|
||
|
||
/// Compute the natural logarithm of a quaternion.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::Quaternion;
|
||
/// let q = Quaternion::new(2.0, 5.0, 0.0, 0.0);
|
||
/// assert_relative_eq!(q.ln(), Quaternion::new(1.683647, 1.190289, 0.0, 0.0), epsilon = 1.0e-6)
|
||
/// ```
|
||
#[inline]
|
||
pub fn ln(&self) -> Self {
|
||
let n = self.norm();
|
||
let v = self.vector();
|
||
let s = self.scalar();
|
||
|
||
Self::from_parts(n.simd_ln(), v.normalize() * (s / n).simd_acos())
|
||
}
|
||
|
||
/// Compute the exponential of a quaternion.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::Quaternion;
|
||
/// let q = Quaternion::new(1.683647, 1.190289, 0.0, 0.0);
|
||
/// assert_relative_eq!(q.exp(), Quaternion::new(2.0, 5.0, 0.0, 0.0), epsilon = 1.0e-5)
|
||
/// ```
|
||
#[inline]
|
||
pub fn exp(&self) -> Self {
|
||
self.exp_eps(N::simd_default_epsilon())
|
||
}
|
||
|
||
/// Compute the exponential of a quaternion. Returns the identity if the vector part of this quaternion
|
||
/// has a norm smaller than `eps`.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::Quaternion;
|
||
/// let q = Quaternion::new(1.683647, 1.190289, 0.0, 0.0);
|
||
/// assert_relative_eq!(q.exp_eps(1.0e-6), Quaternion::new(2.0, 5.0, 0.0, 0.0), epsilon = 1.0e-5);
|
||
///
|
||
/// // Singular case.
|
||
/// let q = Quaternion::new(0.0000001, 0.0, 0.0, 0.0);
|
||
/// assert_eq!(q.exp_eps(1.0e-6), Quaternion::identity());
|
||
/// ```
|
||
#[inline]
|
||
pub fn exp_eps(&self, eps: N) -> Self {
|
||
let v = self.vector();
|
||
let nn = v.norm_squared();
|
||
let le = nn.simd_le(eps * eps);
|
||
le.if_else(Self::identity, || {
|
||
let w_exp = self.scalar().simd_exp();
|
||
let n = nn.simd_sqrt();
|
||
let nv = v * (w_exp * n.simd_sin() / n);
|
||
|
||
Self::from_parts(w_exp * n.simd_cos(), nv)
|
||
})
|
||
}
|
||
|
||
/// Raise the quaternion to a given floating power.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::Quaternion;
|
||
/// let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
|
||
/// assert_relative_eq!(q.powf(1.5), Quaternion::new( -6.2576659, 4.1549037, 6.2323556, 8.3098075), epsilon = 1.0e-6);
|
||
/// ```
|
||
#[inline]
|
||
pub fn powf(&self, n: N) -> Self {
|
||
(self.ln() * n).exp()
|
||
}
|
||
|
||
/// Transforms this quaternion into its 4D vector form (Vector part, Scalar part).
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # use nalgebra::{Quaternion, Vector4};
|
||
/// let mut q = Quaternion::identity();
|
||
/// *q.as_vector_mut() = Vector4::new(1.0, 2.0, 3.0, 4.0);
|
||
/// assert!(q.i == 1.0 && q.j == 2.0 && q.k == 3.0 && q.w == 4.0);
|
||
/// ```
|
||
#[inline]
|
||
pub fn as_vector_mut(&mut self) -> &mut Vector4<N> {
|
||
&mut self.coords
|
||
}
|
||
|
||
/// The mutable vector part `(i, j, k)` of this quaternion.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # use nalgebra::{Quaternion, Vector4};
|
||
/// let mut q = Quaternion::identity();
|
||
/// {
|
||
/// let mut v = q.vector_mut();
|
||
/// v[0] = 2.0;
|
||
/// v[1] = 3.0;
|
||
/// v[2] = 4.0;
|
||
/// }
|
||
/// assert!(q.i == 2.0 && q.j == 3.0 && q.k == 4.0 && q.w == 1.0);
|
||
/// ```
|
||
#[inline]
|
||
pub fn vector_mut(
|
||
&mut self,
|
||
) -> MatrixSliceMut<N, U3, U1, RStride<N, U4, U1>, CStride<N, U4, U1>> {
|
||
self.coords.fixed_rows_mut::<U3>(0)
|
||
}
|
||
|
||
/// Replaces this quaternion by its conjugate.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # use nalgebra::Quaternion;
|
||
/// let mut q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
|
||
/// q.conjugate_mut();
|
||
/// assert!(q.i == -2.0 && q.j == -3.0 && q.k == -4.0 && q.w == 1.0);
|
||
/// ```
|
||
#[inline]
|
||
pub fn conjugate_mut(&mut self) {
|
||
self.coords[0] = -self.coords[0];
|
||
self.coords[1] = -self.coords[1];
|
||
self.coords[2] = -self.coords[2];
|
||
}
|
||
|
||
/// Inverts this quaternion in-place if it is not zero.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::Quaternion;
|
||
/// let mut q = Quaternion::new(1.0f32, 2.0, 3.0, 4.0);
|
||
///
|
||
/// assert!(q.try_inverse_mut());
|
||
/// assert_relative_eq!(q * Quaternion::new(1.0, 2.0, 3.0, 4.0), Quaternion::identity());
|
||
///
|
||
/// //Non-invertible case
|
||
/// let mut q = Quaternion::new(0.0f32, 0.0, 0.0, 0.0);
|
||
/// assert!(!q.try_inverse_mut());
|
||
/// ```
|
||
#[inline]
|
||
pub fn try_inverse_mut(&mut self) -> N::SimdBool {
|
||
let norm_squared = self.norm_squared();
|
||
let ge = norm_squared.simd_ge(N::simd_default_epsilon());
|
||
*self = ge.if_else(|| self.conjugate() / norm_squared, || *self);
|
||
ge
|
||
}
|
||
|
||
/// Normalizes this quaternion.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::Quaternion;
|
||
/// let mut q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
|
||
/// q.normalize_mut();
|
||
/// assert_relative_eq!(q.norm(), 1.0);
|
||
/// ```
|
||
#[inline]
|
||
pub fn normalize_mut(&mut self) -> N {
|
||
self.coords.normalize_mut()
|
||
}
|
||
|
||
/// Calculates square of a quaternion.
|
||
#[inline]
|
||
pub fn squared(&self) -> Self {
|
||
self * self
|
||
}
|
||
|
||
/// Divides quaternion into two.
|
||
#[inline]
|
||
pub fn half(&self) -> Self {
|
||
self / crate::convert(2.0f64)
|
||
}
|
||
|
||
/// Calculates square root.
|
||
#[inline]
|
||
pub fn sqrt(&self) -> Self {
|
||
self.powf(crate::convert(0.5))
|
||
}
|
||
|
||
/// Check if the quaternion is pure.
|
||
///
|
||
/// A quaternion is pure if it has no real part (`self.w == 0.0`).
|
||
#[inline]
|
||
pub fn is_pure(&self) -> bool {
|
||
self.w.is_zero()
|
||
}
|
||
|
||
/// Convert quaternion to pure quaternion.
|
||
#[inline]
|
||
pub fn pure(&self) -> Self {
|
||
Self::from_imag(self.imag())
|
||
}
|
||
|
||
/// Left quaternionic division.
|
||
///
|
||
/// Calculates B<sup>-1</sup> * A where A = self, B = other.
|
||
#[inline]
|
||
pub fn left_div(&self, other: &Self) -> Option<Self>
|
||
where
|
||
N: RealField,
|
||
{
|
||
other.try_inverse().map(|inv| inv * self)
|
||
}
|
||
|
||
/// Right quaternionic division.
|
||
///
|
||
/// Calculates A * B<sup>-1</sup> where A = self, B = other.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::Quaternion;
|
||
/// let a = Quaternion::new(0.0, 1.0, 2.0, 3.0);
|
||
/// let b = Quaternion::new(0.0, 5.0, 2.0, 1.0);
|
||
/// let result = a.right_div(&b).unwrap();
|
||
/// let expected = Quaternion::new(0.4, 0.13333333333333336, -0.4666666666666667, 0.26666666666666666);
|
||
/// assert_relative_eq!(expected, result, epsilon = 1.0e-7);
|
||
/// ```
|
||
#[inline]
|
||
pub fn right_div(&self, other: &Self) -> Option<Self>
|
||
where
|
||
N: RealField,
|
||
{
|
||
other.try_inverse().map(|inv| self * inv)
|
||
}
|
||
|
||
/// Calculates the quaternionic cosinus.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::Quaternion;
|
||
/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
|
||
/// let expected = Quaternion::new(58.93364616794395, -34.086183690465596, -51.1292755356984, -68.17236738093119);
|
||
/// let result = input.cos();
|
||
/// assert_relative_eq!(expected, result, epsilon = 1.0e-7);
|
||
/// ```
|
||
#[inline]
|
||
pub fn cos(&self) -> Self {
|
||
let z = self.imag().magnitude();
|
||
let w = -self.w.simd_sin() * z.simd_sinhc();
|
||
Self::from_parts(self.w.simd_cos() * z.simd_cosh(), self.imag() * w)
|
||
}
|
||
|
||
/// Calculates the quaternionic arccosinus.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::Quaternion;
|
||
/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
|
||
/// let result = input.cos().acos();
|
||
/// assert_relative_eq!(input, result, epsilon = 1.0e-7);
|
||
/// ```
|
||
#[inline]
|
||
pub fn acos(&self) -> Self {
|
||
let u = Self::from_imag(self.imag().normalize());
|
||
let identity = Self::identity();
|
||
|
||
let z = (self + (self.squared() - identity).sqrt()).ln();
|
||
|
||
-(u * z)
|
||
}
|
||
|
||
/// Calculates the quaternionic sinus.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::Quaternion;
|
||
/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
|
||
/// let expected = Quaternion::new(91.78371578403467, 21.886486853029176, 32.82973027954377, 43.77297370605835);
|
||
/// let result = input.sin();
|
||
/// assert_relative_eq!(expected, result, epsilon = 1.0e-7);
|
||
/// ```
|
||
#[inline]
|
||
pub fn sin(&self) -> Self {
|
||
let z = self.imag().magnitude();
|
||
let w = self.w.simd_cos() * z.simd_sinhc();
|
||
Self::from_parts(self.w.simd_sin() * z.simd_cosh(), self.imag() * w)
|
||
}
|
||
|
||
/// Calculates the quaternionic arcsinus.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::Quaternion;
|
||
/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
|
||
/// let result = input.sin().asin();
|
||
/// assert_relative_eq!(input, result, epsilon = 1.0e-7);
|
||
/// ```
|
||
#[inline]
|
||
pub fn asin(&self) -> Self {
|
||
let u = Self::from_imag(self.imag().normalize());
|
||
let identity = Self::identity();
|
||
|
||
let z = ((u * self) + (identity - self.squared()).sqrt()).ln();
|
||
|
||
-(u * z)
|
||
}
|
||
|
||
/// Calculates the quaternionic tangent.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::Quaternion;
|
||
/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
|
||
/// let expected = Quaternion::new(0.00003821631725009489, 0.3713971716439371, 0.5570957574659058, 0.7427943432878743);
|
||
/// let result = input.tan();
|
||
/// assert_relative_eq!(expected, result, epsilon = 1.0e-7);
|
||
/// ```
|
||
#[inline]
|
||
pub fn tan(&self) -> Self
|
||
where
|
||
N: RealField,
|
||
{
|
||
self.sin().right_div(&self.cos()).unwrap()
|
||
}
|
||
|
||
/// Calculates the quaternionic arctangent.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::Quaternion;
|
||
/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
|
||
/// let result = input.tan().atan();
|
||
/// assert_relative_eq!(input, result, epsilon = 1.0e-7);
|
||
/// ```
|
||
#[inline]
|
||
pub fn atan(&self) -> Self
|
||
where
|
||
N: RealField,
|
||
{
|
||
let u = Self::from_imag(self.imag().normalize());
|
||
let num = u + self;
|
||
let den = u - self;
|
||
let fr = num.right_div(&den).unwrap();
|
||
let ln = fr.ln();
|
||
(u.half()) * ln
|
||
}
|
||
|
||
/// Calculates the hyperbolic quaternionic sinus.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::Quaternion;
|
||
/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
|
||
/// let expected = Quaternion::new(0.7323376060463428, -0.4482074499805421, -0.6723111749708133, -0.8964148999610843);
|
||
/// let result = input.sinh();
|
||
/// assert_relative_eq!(expected, result, epsilon = 1.0e-7);
|
||
/// ```
|
||
#[inline]
|
||
pub fn sinh(&self) -> Self {
|
||
(self.exp() - (-self).exp()).half()
|
||
}
|
||
|
||
/// Calculates the hyperbolic quaternionic arcsinus.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::Quaternion;
|
||
/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
|
||
/// let expected = Quaternion::new(2.385889902585242, 0.514052600662788, 0.7710789009941821, 1.028105201325576);
|
||
/// let result = input.asinh();
|
||
/// assert_relative_eq!(expected, result, epsilon = 1.0e-7);
|
||
/// ```
|
||
#[inline]
|
||
pub fn asinh(&self) -> Self {
|
||
let identity = Self::identity();
|
||
(self + (identity + self.squared()).sqrt()).ln()
|
||
}
|
||
|
||
/// Calculates the hyperbolic quaternionic cosinus.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::Quaternion;
|
||
/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
|
||
/// let expected = Quaternion::new(0.9615851176369566, -0.3413521745610167, -0.5120282618415251, -0.6827043491220334);
|
||
/// let result = input.cosh();
|
||
/// assert_relative_eq!(expected, result, epsilon = 1.0e-7);
|
||
/// ```
|
||
#[inline]
|
||
pub fn cosh(&self) -> Self {
|
||
(self.exp() + (-self).exp()).half()
|
||
}
|
||
|
||
/// Calculates the hyperbolic quaternionic arccosinus.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::Quaternion;
|
||
/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
|
||
/// let expected = Quaternion::new(2.4014472020074007, 0.5162761016176176, 0.7744141524264264, 1.0325522032352352);
|
||
/// let result = input.acosh();
|
||
/// assert_relative_eq!(expected, result, epsilon = 1.0e-7);
|
||
/// ```
|
||
#[inline]
|
||
pub fn acosh(&self) -> Self {
|
||
let identity = Self::identity();
|
||
(self + (self + identity).sqrt() * (self - identity).sqrt()).ln()
|
||
}
|
||
|
||
/// Calculates the hyperbolic quaternionic tangent.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::Quaternion;
|
||
/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
|
||
/// let expected = Quaternion::new(1.0248695360556623, -0.10229568178876419, -0.1534435226831464, -0.20459136357752844);
|
||
/// let result = input.tanh();
|
||
/// assert_relative_eq!(expected, result, epsilon = 1.0e-7);
|
||
/// ```
|
||
#[inline]
|
||
pub fn tanh(&self) -> Self
|
||
where
|
||
N: RealField,
|
||
{
|
||
self.sinh().right_div(&self.cosh()).unwrap()
|
||
}
|
||
|
||
/// Calculates the hyperbolic quaternionic arctangent.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::Quaternion;
|
||
/// let input = Quaternion::new(1.0, 2.0, 3.0, 4.0);
|
||
/// let expected = Quaternion::new(0.03230293287000163, 0.5173453683196951, 0.7760180524795426, 1.0346907366393903);
|
||
/// let result = input.atanh();
|
||
/// assert_relative_eq!(expected, result, epsilon = 1.0e-7);
|
||
/// ```
|
||
#[inline]
|
||
pub fn atanh(&self) -> Self {
|
||
let identity = Self::identity();
|
||
((identity + self).ln() - (identity - self).ln()).half()
|
||
}
|
||
}
|
||
|
||
impl<N: RealField + AbsDiffEq<Epsilon = N>> AbsDiffEq for Quaternion<N> {
|
||
type Epsilon = N;
|
||
|
||
#[inline]
|
||
fn default_epsilon() -> Self::Epsilon {
|
||
N::default_epsilon()
|
||
}
|
||
|
||
#[inline]
|
||
fn abs_diff_eq(&self, other: &Self, epsilon: Self::Epsilon) -> bool {
|
||
self.as_vector().abs_diff_eq(other.as_vector(), epsilon) ||
|
||
// Account for the double-covering of S², i.e. q = -q
|
||
self.as_vector().iter().zip(other.as_vector().iter()).all(|(a, b)| a.abs_diff_eq(&-*b, epsilon))
|
||
}
|
||
}
|
||
|
||
impl<N: RealField + RelativeEq<Epsilon = N>> RelativeEq for Quaternion<N> {
|
||
#[inline]
|
||
fn default_max_relative() -> Self::Epsilon {
|
||
N::default_max_relative()
|
||
}
|
||
|
||
#[inline]
|
||
fn relative_eq(
|
||
&self,
|
||
other: &Self,
|
||
epsilon: Self::Epsilon,
|
||
max_relative: Self::Epsilon,
|
||
) -> bool {
|
||
self.as_vector().relative_eq(other.as_vector(), epsilon, max_relative) ||
|
||
// Account for the double-covering of S², i.e. q = -q
|
||
self.as_vector().iter().zip(other.as_vector().iter()).all(|(a, b)| a.relative_eq(&-*b, epsilon, max_relative))
|
||
}
|
||
}
|
||
|
||
impl<N: RealField + UlpsEq<Epsilon = N>> UlpsEq for Quaternion<N> {
|
||
#[inline]
|
||
fn default_max_ulps() -> u32 {
|
||
N::default_max_ulps()
|
||
}
|
||
|
||
#[inline]
|
||
fn ulps_eq(&self, other: &Self, epsilon: Self::Epsilon, max_ulps: u32) -> bool {
|
||
self.as_vector().ulps_eq(other.as_vector(), epsilon, max_ulps) ||
|
||
// Account for the double-covering of S², i.e. q = -q.
|
||
self.as_vector().iter().zip(other.as_vector().iter()).all(|(a, b)| a.ulps_eq(&-*b, epsilon, max_ulps))
|
||
}
|
||
}
|
||
|
||
impl<N: RealField + fmt::Display> fmt::Display for Quaternion<N> {
|
||
fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
|
||
write!(
|
||
f,
|
||
"Quaternion {} − ({}, {}, {})",
|
||
self[3], self[0], self[1], self[2]
|
||
)
|
||
}
|
||
}
|
||
|
||
/// A unit quaternions. May be used to represent a rotation.
|
||
pub type UnitQuaternion<N> = Unit<Quaternion<N>>;
|
||
|
||
impl<N: Scalar + ClosedNeg + PartialEq> PartialEq for UnitQuaternion<N> {
|
||
#[inline]
|
||
fn eq(&self, rhs: &Self) -> bool {
|
||
self.coords == rhs.coords ||
|
||
// Account for the double-covering of S², i.e. q = -q
|
||
self.coords.iter().zip(rhs.coords.iter()).all(|(a, b)| *a == -b.inlined_clone())
|
||
}
|
||
}
|
||
|
||
impl<N: Scalar + ClosedNeg + Eq> Eq for UnitQuaternion<N> {}
|
||
|
||
impl<N: SimdRealField> Normed for Quaternion<N> {
|
||
type Norm = N::SimdRealField;
|
||
|
||
#[inline]
|
||
fn norm(&self) -> N::SimdRealField {
|
||
self.coords.norm()
|
||
}
|
||
|
||
#[inline]
|
||
fn norm_squared(&self) -> N::SimdRealField {
|
||
self.coords.norm_squared()
|
||
}
|
||
|
||
#[inline]
|
||
fn scale_mut(&mut self, n: Self::Norm) {
|
||
self.coords.scale_mut(n)
|
||
}
|
||
|
||
#[inline]
|
||
fn unscale_mut(&mut self, n: Self::Norm) {
|
||
self.coords.unscale_mut(n)
|
||
}
|
||
}
|
||
|
||
impl<N: SimdRealField> UnitQuaternion<N>
|
||
where
|
||
N::Element: SimdRealField,
|
||
{
|
||
/// The rotation angle in [0; pi] of this unit quaternion.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # use nalgebra::{Unit, UnitQuaternion, Vector3};
|
||
/// let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));
|
||
/// let rot = UnitQuaternion::from_axis_angle(&axis, 1.78);
|
||
/// assert_eq!(rot.angle(), 1.78);
|
||
/// ```
|
||
#[inline]
|
||
pub fn angle(&self) -> N {
|
||
let w = self.quaternion().scalar().simd_abs();
|
||
self.quaternion().imag().norm().simd_atan2(w) * crate::convert(2.0f64)
|
||
}
|
||
|
||
/// The underlying quaternion.
|
||
///
|
||
/// Same as `self.as_ref()`.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # use nalgebra::{UnitQuaternion, Quaternion};
|
||
/// let axis = UnitQuaternion::identity();
|
||
/// assert_eq!(*axis.quaternion(), Quaternion::new(1.0, 0.0, 0.0, 0.0));
|
||
/// ```
|
||
#[inline]
|
||
pub fn quaternion(&self) -> &Quaternion<N> {
|
||
self.as_ref()
|
||
}
|
||
|
||
/// Compute the conjugate of this unit quaternion.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # use nalgebra::{Unit, UnitQuaternion, Vector3};
|
||
/// let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));
|
||
/// let rot = UnitQuaternion::from_axis_angle(&axis, 1.78);
|
||
/// let conj = rot.conjugate();
|
||
/// assert_eq!(conj, UnitQuaternion::from_axis_angle(&-axis, 1.78));
|
||
/// ```
|
||
#[inline]
|
||
#[must_use = "Did you mean to use conjugate_mut()?"]
|
||
pub fn conjugate(&self) -> Self {
|
||
Self::new_unchecked(self.as_ref().conjugate())
|
||
}
|
||
|
||
/// Inverts this quaternion if it is not zero.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # use nalgebra::{Unit, UnitQuaternion, Vector3};
|
||
/// let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));
|
||
/// let rot = UnitQuaternion::from_axis_angle(&axis, 1.78);
|
||
/// let inv = rot.inverse();
|
||
/// assert_eq!(rot * inv, UnitQuaternion::identity());
|
||
/// assert_eq!(inv * rot, UnitQuaternion::identity());
|
||
/// ```
|
||
#[inline]
|
||
#[must_use = "Did you mean to use inverse_mut()?"]
|
||
pub fn inverse(&self) -> Self {
|
||
self.conjugate()
|
||
}
|
||
|
||
/// The rotation angle needed to make `self` and `other` coincide.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::{UnitQuaternion, Vector3};
|
||
/// let rot1 = UnitQuaternion::from_axis_angle(&Vector3::y_axis(), 1.0);
|
||
/// let rot2 = UnitQuaternion::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 {
|
||
let delta = self.rotation_to(other);
|
||
delta.angle()
|
||
}
|
||
|
||
/// The unit quaternion 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::{UnitQuaternion, Vector3};
|
||
/// let rot1 = UnitQuaternion::from_axis_angle(&Vector3::y_axis(), 1.0);
|
||
/// let rot2 = UnitQuaternion::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
|
||
}
|
||
|
||
/// Linear interpolation between two unit quaternions.
|
||
///
|
||
/// The result is not normalized.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # use nalgebra::{UnitQuaternion, Quaternion};
|
||
/// let q1 = UnitQuaternion::new_normalize(Quaternion::new(1.0, 0.0, 0.0, 0.0));
|
||
/// let q2 = UnitQuaternion::new_normalize(Quaternion::new(0.0, 1.0, 0.0, 0.0));
|
||
/// assert_eq!(q1.lerp(&q2, 0.1), Quaternion::new(0.9, 0.1, 0.0, 0.0));
|
||
/// ```
|
||
#[inline]
|
||
pub fn lerp(&self, other: &Self, t: N) -> Quaternion<N> {
|
||
self.as_ref().lerp(other.as_ref(), t)
|
||
}
|
||
|
||
/// Normalized linear interpolation between two unit quaternions.
|
||
///
|
||
/// This is the same as `self.lerp` except that the result is normalized.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # use nalgebra::{UnitQuaternion, Quaternion};
|
||
/// let q1 = UnitQuaternion::new_normalize(Quaternion::new(1.0, 0.0, 0.0, 0.0));
|
||
/// let q2 = UnitQuaternion::new_normalize(Quaternion::new(0.0, 1.0, 0.0, 0.0));
|
||
/// assert_eq!(q1.nlerp(&q2, 0.1), UnitQuaternion::new_normalize(Quaternion::new(0.9, 0.1, 0.0, 0.0)));
|
||
/// ```
|
||
#[inline]
|
||
pub fn nlerp(&self, other: &Self, t: N) -> Self {
|
||
let mut res = self.lerp(other, t);
|
||
let _ = res.normalize_mut();
|
||
|
||
Self::new_unchecked(res)
|
||
}
|
||
|
||
/// Spherical linear interpolation between two unit quaternions.
|
||
///
|
||
/// Panics if the angle between both quaternion is 180 degrees (in which case the interpolation
|
||
/// is not well-defined). Use `.try_slerp` instead to avoid the panic.
|
||
///
|
||
/// # Examples:
|
||
///
|
||
/// ```
|
||
/// # use nalgebra::geometry::UnitQuaternion;
|
||
///
|
||
/// let q1 = UnitQuaternion::from_euler_angles(std::f32::consts::FRAC_PI_4, 0.0, 0.0);
|
||
/// let q2 = UnitQuaternion::from_euler_angles(-std::f32::consts::PI, 0.0, 0.0);
|
||
///
|
||
/// let q = q1.slerp(&q2, 1.0 / 3.0);
|
||
///
|
||
/// assert_eq!(q.euler_angles(), (std::f32::consts::FRAC_PI_2, 0.0, 0.0));
|
||
/// ```
|
||
#[inline]
|
||
pub fn slerp(&self, other: &Self, t: N) -> Self
|
||
where
|
||
N: RealField,
|
||
{
|
||
self.try_slerp(other, t, N::default_epsilon())
|
||
.expect("Quaternion slerp: ambiguous configuration.")
|
||
}
|
||
|
||
/// Computes the spherical linear interpolation between two unit quaternions or returns `None`
|
||
/// if both quaternions are approximately 180 degrees apart (in which case the interpolation is
|
||
/// not well-defined).
|
||
///
|
||
/// # Arguments
|
||
/// * `self`: the first quaternion to interpolate from.
|
||
/// * `other`: the second quaternion to interpolate toward.
|
||
/// * `t`: the interpolation parameter. Should be between 0 and 1.
|
||
/// * `epsilon`: the value below which the sinus of the angle separating both quaternion
|
||
/// must be to return `None`.
|
||
#[inline]
|
||
pub fn try_slerp(&self, other: &Self, t: N, epsilon: N) -> Option<Self>
|
||
where
|
||
N: RealField,
|
||
{
|
||
let coords = if self.coords.dot(&other.coords) < N::zero() {
|
||
Unit::new_unchecked(self.coords).try_slerp(
|
||
&Unit::new_unchecked(-other.coords),
|
||
t,
|
||
epsilon,
|
||
)
|
||
} else {
|
||
Unit::new_unchecked(self.coords).try_slerp(
|
||
&Unit::new_unchecked(other.coords),
|
||
t,
|
||
epsilon,
|
||
)
|
||
};
|
||
|
||
coords.map(|q| Unit::new_unchecked(Quaternion::from(q.into_inner())))
|
||
}
|
||
|
||
/// Compute the conjugate of this unit quaternion in-place.
|
||
#[inline]
|
||
pub fn conjugate_mut(&mut self) {
|
||
self.as_mut_unchecked().conjugate_mut()
|
||
}
|
||
|
||
/// Inverts this quaternion if it is not zero.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::{UnitQuaternion, Vector3, Unit};
|
||
/// let axisangle = Vector3::new(0.1, 0.2, 0.3);
|
||
/// let mut rot = UnitQuaternion::new(axisangle);
|
||
/// rot.inverse_mut();
|
||
/// assert_relative_eq!(rot * UnitQuaternion::new(axisangle), UnitQuaternion::identity());
|
||
/// assert_relative_eq!(UnitQuaternion::new(axisangle) * rot, UnitQuaternion::identity());
|
||
/// ```
|
||
#[inline]
|
||
pub fn inverse_mut(&mut self) {
|
||
self.as_mut_unchecked().conjugate_mut()
|
||
}
|
||
|
||
/// The rotation axis of this unit quaternion or `None` if the rotation is zero.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # use nalgebra::{UnitQuaternion, Vector3, Unit};
|
||
/// let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));
|
||
/// let angle = 1.2;
|
||
/// let rot = UnitQuaternion::from_axis_angle(&axis, angle);
|
||
/// assert_eq!(rot.axis(), Some(axis));
|
||
///
|
||
/// // Case with a zero angle.
|
||
/// let rot = UnitQuaternion::from_axis_angle(&axis, 0.0);
|
||
/// assert!(rot.axis().is_none());
|
||
/// ```
|
||
#[inline]
|
||
pub fn axis(&self) -> Option<Unit<Vector3<N>>>
|
||
where
|
||
N: RealField,
|
||
{
|
||
let v = if self.quaternion().scalar() >= N::zero() {
|
||
self.as_ref().vector().clone_owned()
|
||
} else {
|
||
-self.as_ref().vector()
|
||
};
|
||
|
||
Unit::try_new(v, N::zero())
|
||
}
|
||
|
||
/// The rotation axis of this unit quaternion multiplied by the rotation angle.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::{UnitQuaternion, Vector3, Unit};
|
||
/// let axisangle = Vector3::new(0.1, 0.2, 0.3);
|
||
/// let rot = UnitQuaternion::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 {
|
||
Vector3::zero()
|
||
}
|
||
}
|
||
|
||
/// The rotation axis and angle in ]0, pi] of this unit quaternion.
|
||
///
|
||
/// Returns `None` if the angle is zero.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # use nalgebra::{UnitQuaternion, Vector3, Unit};
|
||
/// let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));
|
||
/// let angle = 1.2;
|
||
/// let rot = UnitQuaternion::from_axis_angle(&axis, angle);
|
||
/// assert_eq!(rot.axis_angle(), Some((axis, angle)));
|
||
///
|
||
/// // Case with a zero angle.
|
||
/// let rot = UnitQuaternion::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,
|
||
{
|
||
self.axis().map(|axis| (axis, self.angle()))
|
||
}
|
||
|
||
/// Compute the exponential of a quaternion.
|
||
///
|
||
/// Note that this function yields a `Quaternion<N>` because it loses the unit property.
|
||
#[inline]
|
||
pub fn exp(&self) -> Quaternion<N> {
|
||
self.as_ref().exp()
|
||
}
|
||
|
||
/// Compute the natural logarithm of a quaternion.
|
||
///
|
||
/// Note that this function yields a `Quaternion<N>` because it loses the unit property.
|
||
/// The vector part of the return value corresponds to the axis-angle representation (divided
|
||
/// by 2.0) of this unit quaternion.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::{Vector3, UnitQuaternion};
|
||
/// let axisangle = Vector3::new(0.1, 0.2, 0.3);
|
||
/// let q = UnitQuaternion::new(axisangle);
|
||
/// assert_relative_eq!(q.ln().vector().into_owned(), axisangle, epsilon = 1.0e-6);
|
||
/// ```
|
||
#[inline]
|
||
pub fn ln(&self) -> Quaternion<N>
|
||
where
|
||
N: RealField,
|
||
{
|
||
if let Some(v) = self.axis() {
|
||
Quaternion::from_imag(v.into_inner() * self.angle())
|
||
} else {
|
||
Quaternion::zero()
|
||
}
|
||
}
|
||
|
||
/// Raise the quaternion to a given floating power.
|
||
///
|
||
/// This returns the unit quaternion that identifies a rotation with axis `self.axis()` and
|
||
/// angle `self.angle() × n`.
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::{UnitQuaternion, Vector3, Unit};
|
||
/// let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));
|
||
/// let angle = 1.2;
|
||
/// let rot = UnitQuaternion::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(v) = self.axis() {
|
||
Self::from_axis_angle(&v, self.angle() * n)
|
||
} else {
|
||
Self::identity()
|
||
}
|
||
}
|
||
|
||
/// Builds a rotation matrix from this unit quaternion.
|
||
///
|
||
/// # Example
|
||
///
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use std::f32;
|
||
/// # use nalgebra::{UnitQuaternion, Vector3, Matrix3};
|
||
/// let q = UnitQuaternion::from_axis_angle(&Vector3::z_axis(), f32::consts::FRAC_PI_6);
|
||
/// let rot = q.to_rotation_matrix();
|
||
/// let expected = Matrix3::new(0.8660254, -0.5, 0.0,
|
||
/// 0.5, 0.8660254, 0.0,
|
||
/// 0.0, 0.0, 1.0);
|
||
///
|
||
/// assert_relative_eq!(*rot.matrix(), expected, epsilon = 1.0e-6);
|
||
/// ```
|
||
#[inline]
|
||
pub fn to_rotation_matrix(&self) -> Rotation<N, U3> {
|
||
let i = self.as_ref()[0];
|
||
let j = self.as_ref()[1];
|
||
let k = self.as_ref()[2];
|
||
let w = self.as_ref()[3];
|
||
|
||
let ww = w * w;
|
||
let ii = i * i;
|
||
let jj = j * j;
|
||
let kk = k * k;
|
||
let ij = i * j * crate::convert(2.0f64);
|
||
let wk = w * k * crate::convert(2.0f64);
|
||
let wj = w * j * crate::convert(2.0f64);
|
||
let ik = i * k * crate::convert(2.0f64);
|
||
let jk = j * k * crate::convert(2.0f64);
|
||
let wi = w * i * crate::convert(2.0f64);
|
||
|
||
Rotation::from_matrix_unchecked(Matrix3::new(
|
||
ww + ii - jj - kk,
|
||
ij - wk,
|
||
wj + ik,
|
||
wk + ij,
|
||
ww - ii + jj - kk,
|
||
jk - wi,
|
||
ik - wj,
|
||
wi + jk,
|
||
ww - ii - jj + kk,
|
||
))
|
||
}
|
||
|
||
/// Converts this unit quaternion into its equivalent Euler angles.
|
||
///
|
||
/// The angles are produced in the form (roll, pitch, yaw).
|
||
#[inline]
|
||
#[deprecated(note = "This is renamed to use `.euler_angles()`.")]
|
||
pub fn to_euler_angles(&self) -> (N, N, N)
|
||
where
|
||
N: RealField,
|
||
{
|
||
self.euler_angles()
|
||
}
|
||
|
||
/// Retrieves the euler angles corresponding to this unit quaternion.
|
||
///
|
||
/// The angles are produced in the form (roll, pitch, yaw).
|
||
///
|
||
/// # Example
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use nalgebra::UnitQuaternion;
|
||
/// let rot = UnitQuaternion::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);
|
||
/// ```
|
||
#[inline]
|
||
pub fn euler_angles(&self) -> (N, N, N)
|
||
where
|
||
N: RealField,
|
||
{
|
||
self.to_rotation_matrix().euler_angles()
|
||
}
|
||
|
||
/// Converts this unit quaternion into its equivalent homogeneous transformation matrix.
|
||
///
|
||
/// # Example
|
||
///
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use std::f32;
|
||
/// # use nalgebra::{UnitQuaternion, Vector3, Matrix4};
|
||
/// let rot = UnitQuaternion::from_axis_angle(&Vector3::z_axis(), f32::consts::FRAC_PI_6);
|
||
/// let expected = Matrix4::new(0.8660254, -0.5, 0.0, 0.0,
|
||
/// 0.5, 0.8660254, 0.0, 0.0,
|
||
/// 0.0, 0.0, 1.0, 0.0,
|
||
/// 0.0, 0.0, 0.0, 1.0);
|
||
///
|
||
/// assert_relative_eq!(rot.to_homogeneous(), expected, epsilon = 1.0e-6);
|
||
/// ```
|
||
#[inline]
|
||
pub fn to_homogeneous(&self) -> Matrix4<N> {
|
||
self.to_rotation_matrix().to_homogeneous()
|
||
}
|
||
|
||
/// Rotate a point by this unit quaternion.
|
||
///
|
||
/// This is the same as the multiplication `self * pt`.
|
||
///
|
||
/// # Example
|
||
///
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use std::f32;
|
||
/// # use nalgebra::{UnitQuaternion, Vector3, Point3};
|
||
/// let rot = UnitQuaternion::from_axis_angle(&Vector3::y_axis(), f32::consts::FRAC_PI_2);
|
||
/// let transformed_point = rot.transform_point(&Point3::new(1.0, 2.0, 3.0));
|
||
///
|
||
/// assert_relative_eq!(transformed_point, Point3::new(3.0, 2.0, -1.0), epsilon = 1.0e-6);
|
||
/// ```
|
||
#[inline]
|
||
pub fn transform_point(&self, pt: &Point3<N>) -> Point3<N> {
|
||
self * pt
|
||
}
|
||
|
||
/// Rotate a vector by this unit quaternion.
|
||
///
|
||
/// This is the same as the multiplication `self * v`.
|
||
///
|
||
/// # Example
|
||
///
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use std::f32;
|
||
/// # use nalgebra::{UnitQuaternion, Vector3};
|
||
/// let rot = UnitQuaternion::from_axis_angle(&Vector3::y_axis(), f32::consts::FRAC_PI_2);
|
||
/// let transformed_vector = rot.transform_vector(&Vector3::new(1.0, 2.0, 3.0));
|
||
///
|
||
/// assert_relative_eq!(transformed_vector, Vector3::new(3.0, 2.0, -1.0), epsilon = 1.0e-6);
|
||
/// ```
|
||
#[inline]
|
||
pub fn transform_vector(&self, v: &Vector3<N>) -> Vector3<N> {
|
||
self * v
|
||
}
|
||
|
||
/// Rotate a point by the inverse of this unit quaternion. This may be
|
||
/// cheaper than inverting the unit quaternion and transforming the
|
||
/// point.
|
||
///
|
||
/// # Example
|
||
///
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use std::f32;
|
||
/// # use nalgebra::{UnitQuaternion, Vector3, Point3};
|
||
/// let rot = UnitQuaternion::from_axis_angle(&Vector3::y_axis(), f32::consts::FRAC_PI_2);
|
||
/// let transformed_point = rot.inverse_transform_point(&Point3::new(1.0, 2.0, 3.0));
|
||
///
|
||
/// assert_relative_eq!(transformed_point, Point3::new(-3.0, 2.0, 1.0), epsilon = 1.0e-6);
|
||
/// ```
|
||
#[inline]
|
||
pub fn inverse_transform_point(&self, pt: &Point3<N>) -> Point3<N> {
|
||
// TODO: would it be useful performancewise not to call inverse explicitly (i-e. implement
|
||
// the inverse transformation explicitly here) ?
|
||
self.inverse() * pt
|
||
}
|
||
|
||
/// Rotate a vector by the inverse of this unit quaternion. This may be
|
||
/// cheaper than inverting the unit quaternion and transforming the
|
||
/// vector.
|
||
///
|
||
/// # Example
|
||
///
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use std::f32;
|
||
/// # use nalgebra::{UnitQuaternion, Vector3};
|
||
/// let rot = UnitQuaternion::from_axis_angle(&Vector3::y_axis(), f32::consts::FRAC_PI_2);
|
||
/// let transformed_vector = rot.inverse_transform_vector(&Vector3::new(1.0, 2.0, 3.0));
|
||
///
|
||
/// assert_relative_eq!(transformed_vector, Vector3::new(-3.0, 2.0, 1.0), epsilon = 1.0e-6);
|
||
/// ```
|
||
#[inline]
|
||
pub fn inverse_transform_vector(&self, v: &Vector3<N>) -> Vector3<N> {
|
||
self.inverse() * v
|
||
}
|
||
|
||
/// Rotate a vector by the inverse of this unit quaternion. This may be
|
||
/// cheaper than inverting the unit quaternion and transforming the
|
||
/// vector.
|
||
///
|
||
/// # Example
|
||
///
|
||
/// ```
|
||
/// # #[macro_use] extern crate approx;
|
||
/// # use std::f32;
|
||
/// # use nalgebra::{UnitQuaternion, Vector3};
|
||
/// let rot = UnitQuaternion::from_axis_angle(&Vector3::z_axis(), f32::consts::FRAC_PI_2);
|
||
/// let transformed_vector = rot.inverse_transform_unit_vector(&Vector3::x_axis());
|
||
///
|
||
/// assert_relative_eq!(transformed_vector, -Vector3::y_axis(), epsilon = 1.0e-6);
|
||
/// ```
|
||
#[inline]
|
||
pub fn inverse_transform_unit_vector(&self, v: &Unit<Vector3<N>>) -> Unit<Vector3<N>> {
|
||
self.inverse() * v
|
||
}
|
||
|
||
/// Appends to `self` a rotation given in the axis-angle form, using a linearized formulation.
|
||
///
|
||
/// This is faster, but approximate, way to compute `UnitQuaternion::new(axisangle) * self`.
|
||
#[inline]
|
||
pub fn append_axisangle_linearized(&self, axisangle: &Vector3<N>) -> Self {
|
||
let half: N = crate::convert(0.5);
|
||
let q1 = self.into_inner();
|
||
let q2 = Quaternion::from_imag(axisangle * half);
|
||
Unit::new_normalize(q1 + q2 * q1)
|
||
}
|
||
}
|
||
|
||
impl<N: RealField> Default for UnitQuaternion<N> {
|
||
fn default() -> Self {
|
||
Self::identity()
|
||
}
|
||
}
|
||
|
||
impl<N: RealField + fmt::Display> fmt::Display for UnitQuaternion<N> {
|
||
fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
|
||
if let Some(axis) = self.axis() {
|
||
let axis = axis.into_inner();
|
||
write!(
|
||
f,
|
||
"UnitQuaternion angle: {} − axis: ({}, {}, {})",
|
||
self.angle(),
|
||
axis[0],
|
||
axis[1],
|
||
axis[2]
|
||
)
|
||
} else {
|
||
write!(
|
||
f,
|
||
"UnitQuaternion angle: {} − axis: (undefined)",
|
||
self.angle()
|
||
)
|
||
}
|
||
}
|
||
}
|
||
|
||
impl<N: RealField + AbsDiffEq<Epsilon = N>> AbsDiffEq for UnitQuaternion<N> {
|
||
type Epsilon = N;
|
||
|
||
#[inline]
|
||
fn default_epsilon() -> Self::Epsilon {
|
||
N::default_epsilon()
|
||
}
|
||
|
||
#[inline]
|
||
fn abs_diff_eq(&self, other: &Self, epsilon: Self::Epsilon) -> bool {
|
||
self.as_ref().abs_diff_eq(other.as_ref(), epsilon)
|
||
}
|
||
}
|
||
|
||
impl<N: RealField + RelativeEq<Epsilon = N>> RelativeEq for UnitQuaternion<N> {
|
||
#[inline]
|
||
fn default_max_relative() -> Self::Epsilon {
|
||
N::default_max_relative()
|
||
}
|
||
|
||
#[inline]
|
||
fn relative_eq(
|
||
&self,
|
||
other: &Self,
|
||
epsilon: Self::Epsilon,
|
||
max_relative: Self::Epsilon,
|
||
) -> bool {
|
||
self.as_ref()
|
||
.relative_eq(other.as_ref(), epsilon, max_relative)
|
||
}
|
||
}
|
||
|
||
impl<N: RealField + UlpsEq<Epsilon = N>> UlpsEq for UnitQuaternion<N> {
|
||
#[inline]
|
||
fn default_max_ulps() -> u32 {
|
||
N::default_max_ulps()
|
||
}
|
||
|
||
#[inline]
|
||
fn ulps_eq(&self, other: &Self, epsilon: Self::Epsilon, max_ulps: u32) -> bool {
|
||
self.as_ref().ulps_eq(other.as_ref(), epsilon, max_ulps)
|
||
}
|
||
}
|