nalgebra/geometry/
rotation.rs

1// Needed otherwise the rkyv macros generate code incompatible with rust-2024
2#![cfg_attr(feature = "rkyv-serialize", allow(unsafe_op_in_unsafe_fn))]
3
4use approx::{AbsDiffEq, RelativeEq, UlpsEq};
5use num::{One, Zero};
6use std::fmt;
7use std::hash;
8
9#[cfg(feature = "serde-serialize-no-std")]
10use serde::{Deserialize, Deserializer, Serialize, Serializer};
11
12#[cfg(feature = "serde-serialize-no-std")]
13use crate::base::storage::Owned;
14
15use simba::scalar::RealField;
16use simba::simd::SimdRealField;
17
18use crate::base::allocator::Allocator;
19use crate::base::dimension::{DimNameAdd, DimNameSum, U1};
20use crate::base::{Const, DefaultAllocator, OMatrix, SMatrix, SVector, Scalar, Unit};
21use crate::geometry::Point;
22
23#[cfg(feature = "rkyv-serialize")]
24use rkyv::bytecheck;
25
26/// A rotation matrix.
27///
28/// This is also known as an element of a Special Orthogonal (SO) group.
29/// The `Rotation` type can either represent a 2D or 3D rotation, represented as a matrix.
30/// For a rotation based on quaternions, see [`UnitQuaternion`](crate::UnitQuaternion) instead.
31///
32/// Note that instead of using the [`Rotation`](crate::Rotation) type in your code directly, you should use one
33/// of its aliases: [`Rotation2`](crate::Rotation2), or [`Rotation3`](crate::Rotation3). Though
34/// keep in mind that all the documentation of all the methods of these aliases will also appears on
35/// this page.
36///
37/// # Construction
38/// * [Identity <span style="float:right;">`identity`</span>](#identity)
39/// * [From a 2D rotation angle <span style="float:right;">`new`…</span>](#construction-from-a-2d-rotation-angle)
40/// * [From an existing 2D matrix or rotations <span style="float:right;">`from_matrix`, `rotation_between`, `powf`…</span>](#construction-from-an-existing-2d-matrix-or-rotations)
41/// * [From a 3D axis and/or angles <span style="float:right;">`new`, `from_euler_angles`, `from_axis_angle`…</span>](#construction-from-a-3d-axis-andor-angles)
42/// * [From a 3D eye position and target point <span style="float:right;">`look_at`, `look_at_lh`, `rotation_between`…</span>](#construction-from-a-3d-eye-position-and-target-point)
43/// * [From an existing 3D matrix or rotations <span style="float:right;">`from_matrix`, `rotation_between`, `powf`…</span>](#construction-from-an-existing-3d-matrix-or-rotations)
44///
45/// # Transformation and composition
46/// Note that transforming vectors and points can be done by multiplication, e.g., `rotation * point`.
47/// Composing an rotation with another transformation can also be done by multiplication or division.
48/// * [3D axis and angle extraction <span style="float:right;">`angle`, `euler_angles`, `scaled_axis`, `angle_to`…</span>](#3d-axis-and-angle-extraction)
49/// * [2D angle extraction <span style="float:right;">`angle`, `angle_to`…</span>](#2d-angle-extraction)
50/// * [Transformation of a vector or a point <span style="float:right;">`transform_vector`, `inverse_transform_point`…</span>](#transformation-of-a-vector-or-a-point)
51/// * [Transposition and inversion <span style="float:right;">`transpose`, `inverse`…</span>](#transposition-and-inversion)
52/// * [Interpolation <span style="float:right;">`slerp`…</span>](#interpolation)
53///
54/// # Conversion
55/// * [Conversion to a matrix <span style="float:right;">`matrix`, `to_homogeneous`…</span>](#conversion-to-a-matrix)
56///
57#[repr(C)]
58#[cfg_attr(
59    feature = "rkyv-serialize-no-std",
60    derive(rkyv::Archive, rkyv::Serialize, rkyv::Deserialize),
61    archive(
62        as = "Rotation<T::Archived, D>",
63        bound(archive = "
64        T: rkyv::Archive,
65        SMatrix<T, D, D>: rkyv::Archive<Archived = SMatrix<T::Archived, D, D>>
66    ")
67    )
68)]
69#[cfg_attr(feature = "rkyv-serialize", derive(bytecheck::CheckBytes))]
70#[cfg_attr(feature = "defmt", derive(defmt::Format))]
71#[derive(Copy, Clone)]
72pub struct Rotation<T, const D: usize> {
73    matrix: SMatrix<T, D, D>,
74}
75
76impl<T: fmt::Debug, const D: usize> fmt::Debug for Rotation<T, D> {
77    fn fmt(&self, formatter: &mut fmt::Formatter<'_>) -> Result<(), fmt::Error> {
78        self.matrix.fmt(formatter)
79    }
80}
81
82impl<T: Scalar + hash::Hash, const D: usize> hash::Hash for Rotation<T, D>
83where
84    <DefaultAllocator as Allocator<Const<D>, Const<D>>>::Buffer<T>: hash::Hash,
85{
86    fn hash<H: hash::Hasher>(&self, state: &mut H) {
87        self.matrix.hash(state)
88    }
89}
90
91#[cfg(feature = "bytemuck")]
92unsafe impl<T, const D: usize> bytemuck::Zeroable for Rotation<T, D>
93where
94    T: Scalar + bytemuck::Zeroable,
95    SMatrix<T, D, D>: bytemuck::Zeroable,
96{
97}
98
99#[cfg(feature = "bytemuck")]
100unsafe impl<T, const D: usize> bytemuck::Pod for Rotation<T, D>
101where
102    T: Scalar + bytemuck::Pod,
103    SMatrix<T, D, D>: bytemuck::Pod,
104{
105}
106
107#[cfg(feature = "serde-serialize-no-std")]
108impl<T: Scalar, const D: usize> Serialize for Rotation<T, D>
109where
110    Owned<T, Const<D>, Const<D>>: Serialize,
111{
112    fn serialize<S>(&self, serializer: S) -> Result<S::Ok, S::Error>
113    where
114        S: Serializer,
115    {
116        self.matrix.serialize(serializer)
117    }
118}
119
120#[cfg(feature = "serde-serialize-no-std")]
121impl<'a, T: Scalar, const D: usize> Deserialize<'a> for Rotation<T, D>
122where
123    Owned<T, Const<D>, Const<D>>: Deserialize<'a>,
124{
125    fn deserialize<Des>(deserializer: Des) -> Result<Self, Des::Error>
126    where
127        Des: Deserializer<'a>,
128    {
129        let matrix = SMatrix::<T, D, D>::deserialize(deserializer)?;
130
131        Ok(Self::from_matrix_unchecked(matrix))
132    }
133}
134
135impl<T, const D: usize> Rotation<T, D> {
136    /// Creates a new rotation from the given square matrix.
137    ///
138    /// The matrix orthonormality is not checked.
139    ///
140    /// # Example
141    /// ```
142    /// # use nalgebra::{Rotation2, Rotation3, Matrix2, Matrix3};
143    /// # use std::f32;
144    /// let mat = Matrix3::new(0.8660254, -0.5,      0.0,
145    ///                        0.5,       0.8660254, 0.0,
146    ///                        0.0,       0.0,       1.0);
147    /// let rot = Rotation3::from_matrix_unchecked(mat);
148    ///
149    /// assert_eq!(*rot.matrix(), mat);
150    ///
151    ///
152    /// let mat = Matrix2::new(0.8660254, -0.5,
153    ///                        0.5,       0.8660254);
154    /// let rot = Rotation2::from_matrix_unchecked(mat);
155    ///
156    /// assert_eq!(*rot.matrix(), mat);
157    /// ```
158    #[inline]
159    pub const fn from_matrix_unchecked(matrix: SMatrix<T, D, D>) -> Self {
160        Self { matrix }
161    }
162}
163
164/// # Conversion to a matrix
165impl<T: Scalar, const D: usize> Rotation<T, D> {
166    /// A reference to the underlying matrix representation of this rotation.
167    ///
168    /// # Example
169    /// ```
170    /// # use nalgebra::{Rotation2, Rotation3, Vector3, Matrix2, Matrix3};
171    /// # use std::f32;
172    /// let rot = Rotation3::from_axis_angle(&Vector3::z_axis(), f32::consts::FRAC_PI_6);
173    /// let expected = Matrix3::new(0.8660254, -0.5,      0.0,
174    ///                             0.5,       0.8660254, 0.0,
175    ///                             0.0,       0.0,       1.0);
176    /// assert_eq!(*rot.matrix(), expected);
177    ///
178    ///
179    /// let rot = Rotation2::new(f32::consts::FRAC_PI_6);
180    /// let expected = Matrix2::new(0.8660254, -0.5,
181    ///                             0.5,       0.8660254);
182    /// assert_eq!(*rot.matrix(), expected);
183    /// ```
184    #[inline]
185    #[must_use]
186    pub const fn matrix(&self) -> &SMatrix<T, D, D> {
187        &self.matrix
188    }
189
190    /// A mutable reference to the underlying matrix representation of this rotation.
191    ///
192    /// # Safety
193    ///
194    /// Invariants of the rotation matrix should not be violated.
195    #[inline]
196    #[deprecated(note = "Use `.matrix_mut_unchecked()` instead.")]
197    pub const unsafe fn matrix_mut(&mut self) -> &mut SMatrix<T, D, D> {
198        &mut self.matrix
199    }
200
201    /// A mutable reference to the underlying matrix representation of this rotation.
202    ///
203    /// This is suffixed by "_unchecked" because this allows the user to replace the
204    /// matrix by another one that is non-inversible or non-orthonormal. If one of
205    /// those properties is broken, subsequent method calls may return bogus results.
206    #[inline]
207    pub const fn matrix_mut_unchecked(&mut self) -> &mut SMatrix<T, D, D> {
208        &mut self.matrix
209    }
210
211    /// Unwraps the underlying matrix.
212    ///
213    /// # Example
214    /// ```
215    /// # use nalgebra::{Rotation2, Rotation3, Vector3, Matrix2, Matrix3};
216    /// # use std::f32;
217    /// let rot = Rotation3::from_axis_angle(&Vector3::z_axis(), f32::consts::FRAC_PI_6);
218    /// let mat = rot.into_inner();
219    /// let expected = Matrix3::new(0.8660254, -0.5,      0.0,
220    ///                             0.5,       0.8660254, 0.0,
221    ///                             0.0,       0.0,       1.0);
222    /// assert_eq!(mat, expected);
223    ///
224    ///
225    /// let rot = Rotation2::new(f32::consts::FRAC_PI_6);
226    /// let mat = rot.into_inner();
227    /// let expected = Matrix2::new(0.8660254, -0.5,
228    ///                             0.5,       0.8660254);
229    /// assert_eq!(mat, expected);
230    /// ```
231    #[inline]
232    pub fn into_inner(self) -> SMatrix<T, D, D> {
233        self.matrix
234    }
235
236    /// Unwraps the underlying matrix.
237    /// Deprecated: Use [`Rotation::into_inner`] instead.
238    #[deprecated(note = "use `.into_inner()` instead")]
239    #[inline]
240    pub fn unwrap(self) -> SMatrix<T, D, D> {
241        self.matrix
242    }
243
244    /// Converts this rotation into its equivalent homogeneous transformation matrix.
245    ///
246    /// This is the same as `self.into()`.
247    ///
248    /// # Example
249    /// ```
250    /// # use nalgebra::{Rotation2, Rotation3, Vector3, Matrix3, Matrix4};
251    /// # use std::f32;
252    /// let rot = Rotation3::from_axis_angle(&Vector3::z_axis(), f32::consts::FRAC_PI_6);
253    /// let expected = Matrix4::new(0.8660254, -0.5,      0.0, 0.0,
254    ///                             0.5,       0.8660254, 0.0, 0.0,
255    ///                             0.0,       0.0,       1.0, 0.0,
256    ///                             0.0,       0.0,       0.0, 1.0);
257    /// assert_eq!(rot.to_homogeneous(), expected);
258    ///
259    ///
260    /// let rot = Rotation2::new(f32::consts::FRAC_PI_6);
261    /// let expected = Matrix3::new(0.8660254, -0.5,      0.0,
262    ///                             0.5,       0.8660254, 0.0,
263    ///                             0.0,       0.0,       1.0);
264    /// assert_eq!(rot.to_homogeneous(), expected);
265    /// ```
266    #[inline]
267    #[must_use]
268    pub fn to_homogeneous(&self) -> OMatrix<T, DimNameSum<Const<D>, U1>, DimNameSum<Const<D>, U1>>
269    where
270        T: Zero + One,
271        Const<D>: DimNameAdd<U1>,
272        DefaultAllocator: Allocator<DimNameSum<Const<D>, U1>, DimNameSum<Const<D>, U1>>,
273    {
274        // We could use `SMatrix::to_homogeneous()` here, but that would imply
275        // adding the additional traits `DimAdd` and `IsNotStaticOne`. Maybe
276        // these things will get nicer once specialization lands in Rust.
277        let mut res = OMatrix::<T, DimNameSum<Const<D>, U1>, DimNameSum<Const<D>, U1>>::identity();
278        res.fixed_view_mut::<D, D>(0, 0).copy_from(&self.matrix);
279
280        res
281    }
282}
283
284/// # Transposition and inversion
285impl<T: Scalar, const D: usize> Rotation<T, D> {
286    /// Transposes `self`.
287    ///
288    /// Same as `.inverse()` because the inverse of a rotation matrix is its transpose.
289    ///
290    /// # Example
291    /// ```
292    /// # #[macro_use] extern crate approx;
293    /// # use nalgebra::{Rotation2, Rotation3, Vector3};
294    /// let rot = Rotation3::new(Vector3::new(1.0, 2.0, 3.0));
295    /// let tr_rot = rot.transpose();
296    /// assert_relative_eq!(rot * tr_rot, Rotation3::identity(), epsilon = 1.0e-6);
297    /// assert_relative_eq!(tr_rot * rot, Rotation3::identity(), epsilon = 1.0e-6);
298    ///
299    /// let rot = Rotation2::new(1.2);
300    /// let tr_rot = rot.transpose();
301    /// assert_relative_eq!(rot * tr_rot, Rotation2::identity(), epsilon = 1.0e-6);
302    /// assert_relative_eq!(tr_rot * rot, Rotation2::identity(), epsilon = 1.0e-6);
303    /// ```
304    #[inline]
305    #[must_use = "Did you mean to use transpose_mut()?"]
306    pub fn transpose(&self) -> Self {
307        Self::from_matrix_unchecked(self.matrix.transpose())
308    }
309
310    /// Inverts `self`.
311    ///
312    /// Same as `.transpose()` because the inverse of a rotation matrix is its transpose.
313    ///
314    /// # Example
315    /// ```
316    /// # #[macro_use] extern crate approx;
317    /// # use nalgebra::{Rotation2, Rotation3, Vector3};
318    /// let rot = Rotation3::new(Vector3::new(1.0, 2.0, 3.0));
319    /// let inv = rot.inverse();
320    /// assert_relative_eq!(rot * inv, Rotation3::identity(), epsilon = 1.0e-6);
321    /// assert_relative_eq!(inv * rot, Rotation3::identity(), epsilon = 1.0e-6);
322    ///
323    /// let rot = Rotation2::new(1.2);
324    /// let inv = rot.inverse();
325    /// assert_relative_eq!(rot * inv, Rotation2::identity(), epsilon = 1.0e-6);
326    /// assert_relative_eq!(inv * rot, Rotation2::identity(), epsilon = 1.0e-6);
327    /// ```
328    #[inline]
329    #[must_use = "Did you mean to use inverse_mut()?"]
330    pub fn inverse(&self) -> Self {
331        self.transpose()
332    }
333
334    /// Transposes `self` in-place.
335    ///
336    /// Same as `.inverse_mut()` because the inverse of a rotation matrix is its transpose.
337    ///
338    /// # Example
339    /// ```
340    /// # #[macro_use] extern crate approx;
341    /// # use nalgebra::{Rotation2, Rotation3, Vector3};
342    /// let rot = Rotation3::new(Vector3::new(1.0, 2.0, 3.0));
343    /// let mut tr_rot = Rotation3::new(Vector3::new(1.0, 2.0, 3.0));
344    /// tr_rot.transpose_mut();
345    ///
346    /// assert_relative_eq!(rot * tr_rot, Rotation3::identity(), epsilon = 1.0e-6);
347    /// assert_relative_eq!(tr_rot * rot, Rotation3::identity(), epsilon = 1.0e-6);
348    ///
349    /// let rot = Rotation2::new(1.2);
350    /// let mut tr_rot = Rotation2::new(1.2);
351    /// tr_rot.transpose_mut();
352    ///
353    /// assert_relative_eq!(rot * tr_rot, Rotation2::identity(), epsilon = 1.0e-6);
354    /// assert_relative_eq!(tr_rot * rot, Rotation2::identity(), epsilon = 1.0e-6);
355    /// ```
356    #[inline]
357    pub fn transpose_mut(&mut self) {
358        self.matrix.transpose_mut()
359    }
360
361    /// Inverts `self` in-place.
362    ///
363    /// Same as `.transpose_mut()` because the inverse of a rotation matrix is its transpose.
364    ///
365    /// # Example
366    /// ```
367    /// # #[macro_use] extern crate approx;
368    /// # use nalgebra::{Rotation2, Rotation3, Vector3};
369    /// let rot = Rotation3::new(Vector3::new(1.0, 2.0, 3.0));
370    /// let mut inv = Rotation3::new(Vector3::new(1.0, 2.0, 3.0));
371    /// inv.inverse_mut();
372    ///
373    /// assert_relative_eq!(rot * inv, Rotation3::identity(), epsilon = 1.0e-6);
374    /// assert_relative_eq!(inv * rot, Rotation3::identity(), epsilon = 1.0e-6);
375    ///
376    /// let rot = Rotation2::new(1.2);
377    /// let mut inv = Rotation2::new(1.2);
378    /// inv.inverse_mut();
379    ///
380    /// assert_relative_eq!(rot * inv, Rotation2::identity(), epsilon = 1.0e-6);
381    /// assert_relative_eq!(inv * rot, Rotation2::identity(), epsilon = 1.0e-6);
382    /// ```
383    #[inline]
384    pub fn inverse_mut(&mut self) {
385        self.transpose_mut()
386    }
387}
388
389/// # Transformation of a vector or a point
390impl<T: SimdRealField, const D: usize> Rotation<T, D>
391where
392    T::Element: SimdRealField,
393{
394    /// Rotate the given point.
395    ///
396    /// This is the same as the multiplication `self * pt`.
397    ///
398    /// # Example
399    /// ```
400    /// # #[macro_use] extern crate approx;
401    /// # use std::f32;
402    /// # use nalgebra::{Point3, Rotation2, Rotation3, UnitQuaternion, Vector3};
403    /// let rot = Rotation3::new(Vector3::y() * f32::consts::FRAC_PI_2);
404    /// let transformed_point = rot.transform_point(&Point3::new(1.0, 2.0, 3.0));
405    ///
406    /// assert_relative_eq!(transformed_point, Point3::new(3.0, 2.0, -1.0), epsilon = 1.0e-6);
407    /// ```
408    #[inline]
409    #[must_use]
410    pub fn transform_point(&self, pt: &Point<T, D>) -> Point<T, D> {
411        self * pt
412    }
413
414    /// Rotate the given vector.
415    ///
416    /// This is the same as the multiplication `self * v`.
417    ///
418    /// # Example
419    /// ```
420    /// # #[macro_use] extern crate approx;
421    /// # use std::f32;
422    /// # use nalgebra::{Rotation2, Rotation3, UnitQuaternion, Vector3};
423    /// let rot = Rotation3::new(Vector3::y() * f32::consts::FRAC_PI_2);
424    /// let transformed_vector = rot.transform_vector(&Vector3::new(1.0, 2.0, 3.0));
425    ///
426    /// assert_relative_eq!(transformed_vector, Vector3::new(3.0, 2.0, -1.0), epsilon = 1.0e-6);
427    /// ```
428    #[inline]
429    #[must_use]
430    pub fn transform_vector(&self, v: &SVector<T, D>) -> SVector<T, D> {
431        self * v
432    }
433
434    /// Rotate the given point by the inverse of this rotation. This may be
435    /// cheaper than inverting the rotation and then transforming the given
436    /// point.
437    ///
438    /// # Example
439    /// ```
440    /// # #[macro_use] extern crate approx;
441    /// # use std::f32;
442    /// # use nalgebra::{Point3, Rotation2, Rotation3, UnitQuaternion, Vector3};
443    /// let rot = Rotation3::new(Vector3::y() * f32::consts::FRAC_PI_2);
444    /// let transformed_point = rot.inverse_transform_point(&Point3::new(1.0, 2.0, 3.0));
445    ///
446    /// assert_relative_eq!(transformed_point, Point3::new(-3.0, 2.0, 1.0), epsilon = 1.0e-6);
447    /// ```
448    #[inline]
449    #[must_use]
450    pub fn inverse_transform_point(&self, pt: &Point<T, D>) -> Point<T, D> {
451        Point::from(self.inverse_transform_vector(&pt.coords))
452    }
453
454    /// Rotate the given vector by the inverse of this rotation. This may be
455    /// cheaper than inverting the rotation and then transforming the given
456    /// vector.
457    ///
458    /// # Example
459    /// ```
460    /// # #[macro_use] extern crate approx;
461    /// # use std::f32;
462    /// # use nalgebra::{Rotation2, Rotation3, UnitQuaternion, Vector3};
463    /// let rot = Rotation3::new(Vector3::y() * f32::consts::FRAC_PI_2);
464    /// let transformed_vector = rot.inverse_transform_vector(&Vector3::new(1.0, 2.0, 3.0));
465    ///
466    /// assert_relative_eq!(transformed_vector, Vector3::new(-3.0, 2.0, 1.0), epsilon = 1.0e-6);
467    /// ```
468    #[inline]
469    #[must_use]
470    pub fn inverse_transform_vector(&self, v: &SVector<T, D>) -> SVector<T, D> {
471        self.matrix().tr_mul(v)
472    }
473
474    /// Rotate the given vector by the inverse of this rotation. This may be
475    /// cheaper than inverting the rotation and then transforming the given
476    /// vector.
477    ///
478    /// # Example
479    /// ```
480    /// # #[macro_use] extern crate approx;
481    /// # use std::f32;
482    /// # use nalgebra::{Rotation2, Rotation3, UnitQuaternion, Vector3};
483    /// let rot = Rotation3::new(Vector3::z() * f32::consts::FRAC_PI_2);
484    /// let transformed_vector = rot.inverse_transform_unit_vector(&Vector3::x_axis());
485    ///
486    /// assert_relative_eq!(transformed_vector, -Vector3::y_axis(), epsilon = 1.0e-6);
487    /// ```
488    #[inline]
489    #[must_use]
490    pub fn inverse_transform_unit_vector(&self, v: &Unit<SVector<T, D>>) -> Unit<SVector<T, D>> {
491        Unit::new_unchecked(self.inverse_transform_vector(&**v))
492    }
493}
494
495impl<T: Scalar + Eq, const D: usize> Eq for Rotation<T, D> {}
496
497impl<T: Scalar + PartialEq, const D: usize> PartialEq for Rotation<T, D> {
498    #[inline]
499    fn eq(&self, right: &Self) -> bool {
500        self.matrix == right.matrix
501    }
502}
503
504impl<T, const D: usize> AbsDiffEq for Rotation<T, D>
505where
506    T: Scalar + AbsDiffEq,
507    T::Epsilon: Clone,
508{
509    type Epsilon = T::Epsilon;
510
511    #[inline]
512    fn default_epsilon() -> Self::Epsilon {
513        T::default_epsilon()
514    }
515
516    #[inline]
517    fn abs_diff_eq(&self, other: &Self, epsilon: Self::Epsilon) -> bool {
518        self.matrix.abs_diff_eq(&other.matrix, epsilon)
519    }
520}
521
522impl<T, const D: usize> RelativeEq for Rotation<T, D>
523where
524    T: Scalar + RelativeEq,
525    T::Epsilon: Clone,
526{
527    #[inline]
528    fn default_max_relative() -> Self::Epsilon {
529        T::default_max_relative()
530    }
531
532    #[inline]
533    fn relative_eq(
534        &self,
535        other: &Self,
536        epsilon: Self::Epsilon,
537        max_relative: Self::Epsilon,
538    ) -> bool {
539        self.matrix
540            .relative_eq(&other.matrix, epsilon, max_relative)
541    }
542}
543
544impl<T, const D: usize> UlpsEq for Rotation<T, D>
545where
546    T: Scalar + UlpsEq,
547    T::Epsilon: Clone,
548{
549    #[inline]
550    fn default_max_ulps() -> u32 {
551        T::default_max_ulps()
552    }
553
554    #[inline]
555    fn ulps_eq(&self, other: &Self, epsilon: Self::Epsilon, max_ulps: u32) -> bool {
556        self.matrix.ulps_eq(&other.matrix, epsilon, max_ulps)
557    }
558}
559
560/*
561 *
562 * Display
563 *
564 */
565impl<T, const D: usize> fmt::Display for Rotation<T, D>
566where
567    T: RealField + fmt::Display,
568{
569    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
570        let precision = f.precision().unwrap_or(3);
571
572        writeln!(f, "Rotation matrix {{")?;
573        write!(f, "{:.*}", precision, self.matrix)?;
574        writeln!(f, "}}")
575    }
576}
577
578//          //         /*
579//          //          *
580//          //          * Absolute
581//          //          *
582//          //          */
583//          //         impl<T: Absolute> Absolute for $t<T> {
584//          //             type AbsoluteValue = $submatrix<T::AbsoluteValue>;
585//          //
586//          //             #[inline]
587//          //             fn abs(m: &$t<T>) -> $submatrix<T::AbsoluteValue> {
588//          //                 Absolute::abs(&m.submatrix)
589//          //             }
590//          //         }