1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
#[cfg(feature = "arbitrary")]
use crate::base::storage::Owned;
#[cfg(feature = "arbitrary")]
use quickcheck::{Arbitrary, Gen};

use alga::general::RealField;
use num::Zero;
use rand::distributions::{Distribution, OpenClosed01, Standard};
use rand::Rng;
use std::ops::Neg;

use crate::base::dimension::{U1, U2, U3};
use crate::base::storage::Storage;
use crate::base::{Matrix2, Matrix3, MatrixN, Unit, Vector, Vector1, Vector3, VectorN};

use crate::geometry::{Rotation2, Rotation3, UnitComplex, UnitQuaternion};

/*
 *
 * 2D Rotation matrix.
 *
 */
impl<N: RealField> Rotation2<N> {
    /// 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: N) -> Self {
        let (sia, coa) = angle.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<N, U1>>(axisangle: Vector<N, U1, SB>) -> Self {
        Self::new(axisangle[0])
    }

    /// 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<N>) -> Self {
        Self::from_matrix_eps(m, N::default_epsilon(), 0, Self::identity())
    }

    /// Builds a rotation matrix by extracting the rotation part of the given transformation `m`.
    ///
    /// This implements "A Robust Method to Extract the Rotational Part of Deformations" by Müller et al.
    ///
    /// # Parameters
    ///
    /// * `m`: the matrix from which the rotational part is to be extracted.
    /// * `eps`: the angular errors tolerated between the current rotation and the optimal one.
    /// * `max_iter`: the maximum number of iterations. Loops indefinitely until convergence if set to `0`.
    /// * `guess`: 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<N>, eps: N, mut max_iter: usize, guess: Self) -> Self {
        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() + N::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<N, U2, SB>, b: &Vector<N, U2, SC>) -> Self
    where
        SB: Storage<N, U2>,
        SC: Storage<N, 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<N, U2, SB>,
        b: &Vector<N, U2, SC>,
        s: N,
    ) -> Self
    where
        SB: Storage<N, U2>,
        SC: Storage<N, U2>,
    {
        crate::convert(UnitComplex::scaled_rotation_between(a, b, s).to_rotation_matrix())
    }

    /// 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]
    pub fn angle(&self) -> N {
        self.matrix()[(1, 0)].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]
    pub fn angle_to(&self, other: &Self) -> N {
        self.rotation_to(other).angle()
    }

    /// 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]
    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) {
        let mut c = UnitComplex::from(*self);
        let _ = c.renormalize();

        *self = Self::from_matrix_eps(self.matrix(), N::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]
    pub fn powf(&self, n: N) -> Self {
        Self::new(self.angle() * n)
    }

    /// 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]
    pub fn scaled_axis(&self) -> VectorN<N, U1> {
        Vector1::new(self.angle())
    }
}

impl<N: RealField> Distribution<Rotation2<N>> for Standard
where OpenClosed01: Distribution<N>
{
    /// Generate a uniformly distributed random rotation.
    #[inline]
    fn sample<'a, R: Rng + ?Sized>(&self, rng: &'a mut R) -> Rotation2<N> {
        Rotation2::new(rng.sample(OpenClosed01) * N::two_pi())
    }
}

#[cfg(feature = "arbitrary")]
impl<N: RealField + Arbitrary> Arbitrary for Rotation2<N>
where Owned<N, U2, U2>: Send
{
    #[inline]
    fn arbitrary<G: Gen>(g: &mut G) -> Self {
        Self::new(N::arbitrary(g))
    }
}

/*
 *
 * 3D Rotation matrix.
 *
 */
impl<N: RealField> Rotation3<N> {
    /// 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<N, U3>>(axisangle: Vector<N, U3, SB>) -> Self {
        let axisangle = axisangle.into_owned();
        let (axis, angle) = Unit::new_and_get(axisangle);
        Self::from_axis_angle(&axis, angle)
    }

    /// Builds a rotation matrix by extracting the rotation part of the given transformation `m`.
    ///
    /// This is an iterative method. See `.from_matrix_eps` to provide mover
    /// convergence parameters and starting solution.
    /// This implements "A Robust Method to Extract the Rotational Part of Deformations" by Müller et al.
    pub fn from_matrix(m: &Matrix3<N>) -> Self {
        Self::from_matrix_eps(m, N::default_epsilon(), 0, Self::identity())
    }

    /// Builds a rotation matrix by extracting the rotation part of the given transformation `m`.
    ///
    /// This implements "A Robust Method to Extract the Rotational Part of Deformations" by Müller et al.
    ///
    /// # Parameters
    ///
    /// * `m`: the matrix from which the rotational part is to be extracted.
    /// * `eps`: the angular errors tolerated between the current rotation and the optimal one.
    /// * `max_iter`: the maximum number of iterations. Loops indefinitely until convergence if set to `0`.
    /// * `guess`: a guess of the solution. Convergence will be significantly faster if an initial solution close
    ///           to the actual solution is provided. Can be set to `Rotation3::identity()` if no other
    ///           guesses come to mind.
    pub fn from_matrix_eps(m: &Matrix3<N>, eps: N, mut max_iter: usize, guess: Self) -> Self {
        if max_iter == 0 {
            max_iter = usize::max_value();
        }

        let mut rot = guess.into_inner();

        for _ in 0..max_iter {
            let axis = rot.column(0).cross(&m.column(0)) +
                rot.column(1).cross(&m.column(1)) +
                rot.column(2).cross(&m.column(2));
            let denom = rot.column(0).dot(&m.column(0)) +
                rot.column(1).dot(&m.column(1)) +
                rot.column(2).dot(&m.column(2));

            let axisangle = axis / (denom.abs() + N::default_epsilon());

            if let Some((axis, angle)) = Unit::try_new_and_get(axisangle, eps) {
                rot = Rotation3::from_axis_angle(&axis, angle) * rot;
            } else {
                break;
            }
        }

        Self::from_matrix_unchecked(rot)
    }

    /// 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<N, U3>>(axisangle: Vector<N, 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<N, U3, SB>>, angle: N) -> Self
    where SB: Storage<N, U3> {
        if angle.is_zero() {
            Self::identity()
        } 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.sin_cos();
            let one_m_cos = N::one() - cos;

            Self::from_matrix_unchecked(MatrixN::<N, U3>::new(
                sqx + (N::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 + (N::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 + (N::one() - sqz) * cos,
            ))
        }
    }

    /// 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: N, pitch: N, yaw: N) -> Self {
        let (sr, cr) = roll.sin_cos();
        let (sp, cp) = pitch.sin_cos();
        let (sy, cy) = yaw.sin_cos();

        Self::from_matrix_unchecked(MatrixN::<N, U3>::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,
        ))
    }

    /// Creates Euler angles from a rotation.
    ///
    /// The angles are produced in the form (roll, pitch, yaw).
    #[deprecated(note = "This is renamed to use `.euler_angles()`.")]
    pub fn to_euler_angles(&self) -> (N, N, N) {
        self.euler_angles()
    }

    /// Euler angles corresponding to this rotation from a rotation.
    ///
    /// The angles are produced in the form (roll, pitch, yaw).
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::Rotation3;
    /// let rot = Rotation3::from_euler_angles(0.1, 0.2, 0.3);
    /// let euler = rot.euler_angles();
    /// assert_relative_eq!(euler.0, 0.1, epsilon = 1.0e-6);
    /// assert_relative_eq!(euler.1, 0.2, epsilon = 1.0e-6);
    /// assert_relative_eq!(euler.2, 0.3, epsilon = 1.0e-6);
    /// ```
    pub fn euler_angles(&self) -> (N, N, N) {
        // Implementation informed by "Computing Euler angles from a rotation matrix", by Gregory G. Slabaugh
        //  http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.371.6578
        if self[(2, 0)].abs() < N::one() {
            let yaw = -self[(2, 0)].asin();
            let roll = (self[(2, 1)] / yaw.cos()).atan2(self[(2, 2)] / yaw.cos());
            let pitch = (self[(1, 0)] / yaw.cos()).atan2(self[(0, 0)] / yaw.cos());
            (roll, yaw, pitch)
        } else if self[(2, 0)] <= -N::one() {
            (self[(0, 1)].atan2(self[(0, 2)]), N::frac_pi_2(), N::zero())
        } else {
            (
                -self[(0, 1)].atan2(-self[(0, 2)]),
                -N::frac_pi_2(),
                N::zero(),
            )
        }
    }

    /// 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) {
        let mut c = UnitQuaternion::from(*self);
        let _ = c.renormalize();

        *self = Self::from_matrix_eps(self.matrix(), N::default_epsilon(), 0, c.into())
    }

    /// 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<N, U3, SB>, up: &Vector<N, U3, SC>) -> Self
    where
        SB: Storage<N, U3>,
        SC: Storage<N, U3>,
    {
        let zaxis = dir.normalize();
        let xaxis = up.cross(&zaxis).normalize();
        let yaxis = zaxis.cross(&xaxis).normalize();

        Self::from_matrix_unchecked(MatrixN::<N, U3>::new(
            xaxis.x, yaxis.x, zaxis.x, xaxis.y, yaxis.y, zaxis.y, xaxis.z, yaxis.z, zaxis.z,
        ))
    }

    /// Deprecated: Use [Rotation3::face_towards] instead.
    #[deprecated(note="renamed to `face_towards`")]
    pub fn new_observer_frames<SB, SC>(dir: &Vector<N, U3, SB>, up: &Vector<N, U3, SC>) -> Self
    where
        SB: Storage<N, U3>,
        SC: Storage<N, U3>,
    {
        Self::face_towards(dir, up)
    }

    /// Builds a right-handed look-at view matrix without translation.
    ///
    /// It maps the view direction `dir` to the **negative** `z` axis.
    /// This conforms to the common notion of right handed look-at matrix from the computer
    /// graphics community.
    ///
    /// # Arguments
    ///   * dir - The direction toward which the camera looks.
    ///   * up - A vector approximately aligned with required the vertical axis. The only
    ///   requirement of this parameter is to not be collinear to `dir`.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use std::f32;
    /// # use nalgebra::{Rotation3, Vector3};
    /// let dir = Vector3::new(1.0, 2.0, 3.0);
    /// let up = Vector3::y();
    ///
    /// let rot = Rotation3::look_at_rh(&dir, &up);
    /// assert_relative_eq!(rot * dir.normalize(), -Vector3::z());
    /// ```
    #[inline]
    pub fn look_at_rh<SB, SC>(dir: &Vector<N, U3, SB>, up: &Vector<N, U3, SC>) -> Self
    where
        SB: Storage<N, U3>,
        SC: Storage<N, U3>,
    {
        Self::face_towards(&dir.neg(), up).inverse()
    }

    /// Builds a left-handed look-at view matrix without translation.
    ///
    /// It maps the view direction `dir` to the **positive** `z` axis.
    /// This conforms to the common notion of left handed look-at matrix from the computer
    /// graphics community.
    ///
    /// # Arguments
    ///   * dir - The direction toward which the camera looks.
    ///   * up - A vector approximately aligned with required the vertical axis. The only
    ///   requirement of this parameter is to not be collinear to `dir`.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use std::f32;
    /// # use nalgebra::{Rotation3, Vector3};
    /// let dir = Vector3::new(1.0, 2.0, 3.0);
    /// let up = Vector3::y();
    ///
    /// let rot = Rotation3::look_at_lh(&dir, &up);
    /// assert_relative_eq!(rot * dir.normalize(), Vector3::z());
    /// ```
    #[inline]
    pub fn look_at_lh<SB, SC>(dir: &Vector<N, U3, SB>, up: &Vector<N, U3, SC>) -> Self
    where
        SB: Storage<N, U3>,
        SC: Storage<N, U3>,
    {
        Self::face_towards(dir, up).inverse()
    }

    /// The rotation matrix required to align `a` and `b` but with its angle.
    ///
    /// This is the rotation `R` such that `(R * a).angle(b) == 0 && (R * a).dot(b).is_positive()`.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::{Vector3, Rotation3};
    /// let a = Vector3::new(1.0, 2.0, 3.0);
    /// let b = Vector3::new(3.0, 1.0, 2.0);
    /// let rot = Rotation3::rotation_between(&a, &b).unwrap();
    /// assert_relative_eq!(rot * a, b, epsilon = 1.0e-6);
    /// assert_relative_eq!(rot.inverse() * b, a, epsilon = 1.0e-6);
    /// ```
    #[inline]
    pub fn rotation_between<SB, SC>(a: &Vector<N, U3, SB>, b: &Vector<N, U3, SC>) -> Option<Self>
    where
        SB: Storage<N, U3>,
        SC: Storage<N, U3>,
    {
        Self::scaled_rotation_between(a, b, N::one())
    }

    /// The smallest rotation needed to make `a` and `b` collinear and point toward the same
    /// direction, raised to the power `s`.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::{Vector3, Rotation3};
    /// let a = Vector3::new(1.0, 2.0, 3.0);
    /// let b = Vector3::new(3.0, 1.0, 2.0);
    /// let rot2 = Rotation3::scaled_rotation_between(&a, &b, 0.2).unwrap();
    /// let rot5 = Rotation3::scaled_rotation_between(&a, &b, 0.5).unwrap();
    /// assert_relative_eq!(rot2 * rot2 * rot2 * rot2 * rot2 * a, b, epsilon = 1.0e-6);
    /// assert_relative_eq!(rot5 * rot5 * a, b, epsilon = 1.0e-6);
    /// ```
    #[inline]
    pub fn scaled_rotation_between<SB, SC>(
        a: &Vector<N, U3, SB>,
        b: &Vector<N, U3, SC>,
        n: N,
    ) -> Option<Self>
    where
        SB: Storage<N, U3>,
        SC: Storage<N, U3>,
    {
        // FIXME: code duplication with Rotation.
        if let (Some(na), Some(nb)) = (a.try_normalize(N::zero()), b.try_normalize(N::zero())) {
            let c = na.cross(&nb);

            if let Some(axis) = Unit::try_new(c, N::default_epsilon()) {
                return Some(Self::from_axis_angle(&axis, na.dot(&nb).acos() * n));
            }

            // Zero or PI.
            if na.dot(&nb) < N::zero() {
                // PI
                //
                // The rotation axis is undefined but the angle not zero. This is not a
                // simple rotation.
                return None;
            }
        }

        Some(Self::identity())
    }

    /// The rotation angle in [0; pi].
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::{Unit, Rotation3, Vector3};
    /// let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));
    /// let rot = Rotation3::from_axis_angle(&axis, 1.78);
    /// assert_relative_eq!(rot.angle(), 1.78);
    /// ```
    #[inline]
    pub fn angle(&self) -> N {
        ((self.matrix()[(0, 0)] + self.matrix()[(1, 1)] + self.matrix()[(2, 2)] - N::one())
            / crate::convert(2.0))
        .acos()
    }

    /// The rotation axis. Returns `None` if the rotation angle is zero or PI.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::{Rotation3, Vector3, Unit};
    /// let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));
    /// let angle = 1.2;
    /// let rot = Rotation3::from_axis_angle(&axis, angle);
    /// assert_relative_eq!(rot.axis().unwrap(), axis);
    ///
    /// // Case with a zero angle.
    /// let rot = Rotation3::from_axis_angle(&axis, 0.0);
    /// assert!(rot.axis().is_none());
    /// ```
    #[inline]
    pub fn axis(&self) -> Option<Unit<Vector3<N>>> {
        let axis = VectorN::<N, U3>::new(
            self.matrix()[(2, 1)] - self.matrix()[(1, 2)],
            self.matrix()[(0, 2)] - self.matrix()[(2, 0)],
            self.matrix()[(1, 0)] - self.matrix()[(0, 1)],
        );

        Unit::try_new(axis, N::default_epsilon())
    }

    /// The rotation axis multiplied by the rotation angle.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::{Rotation3, Vector3, Unit};
    /// let axisangle = Vector3::new(0.1, 0.2, 0.3);
    /// let rot = Rotation3::new(axisangle);
    /// assert_relative_eq!(rot.scaled_axis(), axisangle, epsilon = 1.0e-6);
    /// ```
    #[inline]
    pub fn scaled_axis(&self) -> Vector3<N> {
        if let Some(axis) = self.axis() {
            axis.into_inner() * self.angle()
        } else {
            Vector::zero()
        }
    }

    /// The rotation axis and angle in ]0, pi] of this unit quaternion.
    ///
    /// Returns `None` if the angle is zero.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::{Rotation3, Vector3, Unit};
    /// let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));
    /// let angle = 1.2;
    /// let rot = Rotation3::from_axis_angle(&axis, angle);
    /// let axis_angle = rot.axis_angle().unwrap();
    /// assert_relative_eq!(axis_angle.0, axis);
    /// assert_relative_eq!(axis_angle.1, angle);
    ///
    /// // Case with a zero angle.
    /// let rot = Rotation3::from_axis_angle(&axis, 0.0);
    /// assert!(rot.axis_angle().is_none());
    /// ```
    #[inline]
    pub fn axis_angle(&self) -> Option<(Unit<Vector3<N>>, N)> {
        if let Some(axis) = self.axis() {
            Some((axis, self.angle()))
        } else {
            None
        }
    }

    /// The rotation angle needed to make `self` and `other` coincide.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::{Rotation3, Vector3};
    /// let rot1 = Rotation3::from_axis_angle(&Vector3::y_axis(), 1.0);
    /// let rot2 = Rotation3::from_axis_angle(&Vector3::x_axis(), 0.1);
    /// assert_relative_eq!(rot1.angle_to(&rot2), 1.0045657, epsilon = 1.0e-6);
    /// ```
    #[inline]
    pub fn angle_to(&self, other: &Self) -> N {
        self.rotation_to(other).angle()
    }

    /// The rotation matrix needed to make `self` and `other` coincide.
    ///
    /// The result is such that: `self.rotation_to(other) * self == other`.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::{Rotation3, Vector3};
    /// let rot1 = Rotation3::from_axis_angle(&Vector3::y_axis(), 1.0);
    /// let rot2 = Rotation3::from_axis_angle(&Vector3::x_axis(), 0.1);
    /// let rot_to = rot1.rotation_to(&rot2);
    /// assert_relative_eq!(rot_to * rot1, rot2, epsilon = 1.0e-6);
    /// ```
    #[inline]
    pub fn rotation_to(&self, other: &Self) -> Self {
        other * self.inverse()
    }

    /// Raise the quaternion to a given floating power, i.e., returns the rotation with the same
    /// axis as `self` and an angle equal to `self.angle()` multiplied by `n`.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::{Rotation3, Vector3, Unit};
    /// let axis = Unit::new_normalize(Vector3::new(1.0, 2.0, 3.0));
    /// let angle = 1.2;
    /// let rot = Rotation3::from_axis_angle(&axis, angle);
    /// let pow = rot.powf(2.0);
    /// assert_relative_eq!(pow.axis().unwrap(), axis, epsilon = 1.0e-6);
    /// assert_eq!(pow.angle(), 2.4);
    /// ```
    #[inline]
    pub fn powf(&self, n: N) -> Self {
        if let Some(axis) = self.axis() {
            Self::from_axis_angle(&axis, self.angle() * n)
        } else if self.matrix()[(0, 0)] < N::zero() {
            let minus_id = MatrixN::<N, U3>::from_diagonal_element(-N::one());
            Self::from_matrix_unchecked(minus_id)
        } else {
            Self::identity()
        }
    }
}

impl<N: RealField> Distribution<Rotation3<N>> for Standard
where OpenClosed01: Distribution<N>
{
    /// Generate a uniformly distributed random rotation.
    #[inline]
    fn sample<'a, R: Rng + ?Sized>(&self, rng: &mut R) -> Rotation3<N> {
        // James Arvo.
        // Fast random rotation matrices.
        // In D. Kirk, editor, Graphics Gems III, pages 117-120. Academic, New York, 1992.

        // Compute a random rotation around Z
        let theta = N::two_pi() * rng.sample(OpenClosed01);
        let (ts, tc) = theta.sin_cos();
        let a = MatrixN::<N, U3>::new(
            tc,
            ts,
            N::zero(),
            -ts,
            tc,
            N::zero(),
            N::zero(),
            N::zero(),
            N::one(),
        );

        // Compute a random rotation *of* Z
        let phi = N::two_pi() * rng.sample(OpenClosed01);
        let z = rng.sample(OpenClosed01);
        let (ps, pc) = phi.sin_cos();
        let sqrt_z = z.sqrt();
        let v = Vector3::new(pc * sqrt_z, ps * sqrt_z, (N::one() - z).sqrt());
        let mut b = v * v.transpose();
        b += b;
        b -= MatrixN::<N, U3>::identity();

        Rotation3::from_matrix_unchecked(b * a)
    }
}

#[cfg(feature = "arbitrary")]
impl<N: RealField + Arbitrary> Arbitrary for Rotation3<N>
where
    Owned<N, U3, U3>: Send,
    Owned<N, U3>: Send,
{
    #[inline]
    fn arbitrary<G: Gen>(g: &mut G) -> Self {
        Self::new(VectorN::arbitrary(g))
    }
}