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
#[cfg(feature = "arbitrary")]
use quickcheck::{Arbitrary, Gen};

#[cfg(feature = "rand-no-std")]
use rand::{
    distributions::{Distribution, Standard},
    Rng,
};

use num::One;
use num_complex::Complex;

use crate::base::dimension::{U1, U2};
use crate::base::storage::Storage;
use crate::base::{Matrix2, Scalar, Unit, Vector, Vector2};
use crate::geometry::{Rotation2, UnitComplex};
use simba::scalar::{RealField, SupersetOf};
use simba::simd::SimdRealField;

impl<T: SimdRealField> Default for UnitComplex<T>
where
    T::Element: SimdRealField,
{
    fn default() -> Self {
        Self::identity()
    }
}

/// # Identity
impl<T: SimdRealField> UnitComplex<T>
where
    T::Element: SimdRealField,
{
    /// The unit complex number multiplicative identity.
    ///
    /// # Example
    /// ```
    /// # use nalgebra::UnitComplex;
    /// let rot1 = UnitComplex::identity();
    /// let rot2 = UnitComplex::new(1.7);
    ///
    /// assert_eq!(rot1 * rot2, rot2);
    /// assert_eq!(rot2 * rot1, rot2);
    /// ```
    #[inline]
    pub fn identity() -> Self {
        Self::new_unchecked(Complex::new(T::one(), T::zero()))
    }
}

/// # Construction from a 2D rotation angle
impl<T: SimdRealField> UnitComplex<T>
where
    T::Element: SimdRealField,
{
    /// Builds the unit complex number corresponding to the rotation with the given angle.
    ///
    /// # Example
    ///
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use std::f32;
    /// # use nalgebra::{UnitComplex, Point2};
    /// let rot = UnitComplex::new(f32::consts::FRAC_PI_2);
    ///
    /// assert_relative_eq!(rot * Point2::new(3.0, 4.0), Point2::new(-4.0, 3.0));
    /// ```
    #[inline]
    pub fn new(angle: T) -> Self {
        let (sin, cos) = angle.simd_sin_cos();
        Self::from_cos_sin_unchecked(cos, sin)
    }

    /// Builds the unit complex number corresponding to the rotation with the angle.
    ///
    /// Same as `Self::new(angle)`.
    ///
    /// # Example
    ///
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use std::f32;
    /// # use nalgebra::{UnitComplex, Point2};
    /// let rot = UnitComplex::from_angle(f32::consts::FRAC_PI_2);
    ///
    /// assert_relative_eq!(rot * Point2::new(3.0, 4.0), Point2::new(-4.0, 3.0));
    /// ```
    // TODO: deprecate this.
    #[inline]
    pub fn from_angle(angle: T) -> Self {
        Self::new(angle)
    }

    /// Builds the unit complex number from the sinus and cosinus of the rotation angle.
    ///
    /// The input values are not checked to actually be cosines and sine of the same value.
    /// Is is generally preferable to use the `::new(angle)` constructor instead.
    ///
    /// # Example
    ///
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use std::f32;
    /// # use nalgebra::{UnitComplex, Vector2, Point2};
    /// let angle = f32::consts::FRAC_PI_2;
    /// let rot = UnitComplex::from_cos_sin_unchecked(angle.cos(), angle.sin());
    ///
    /// assert_relative_eq!(rot * Point2::new(3.0, 4.0), Point2::new(-4.0, 3.0));
    /// ```
    #[inline]
    pub fn from_cos_sin_unchecked(cos: T, sin: T) -> Self {
        Self::new_unchecked(Complex::new(cos, sin))
    }

    /// Builds a unit complex rotation from an angle in radian wrapped in a 1-dimensional vector.
    ///
    /// This is generally used in the context of generic programming. Using
    /// the `::new(angle)` method instead is more common.
    #[inline]
    pub fn from_scaled_axis<SB: Storage<T, U1>>(axisangle: Vector<T, U1, SB>) -> Self {
        Self::from_angle(axisangle[0].clone())
    }
}

/// # Construction from an existing 2D matrix or complex number
impl<T: SimdRealField> UnitComplex<T>
where
    T::Element: SimdRealField,
{
    /// Cast the components of `self` to another type.
    ///
    /// # Example
    /// ```
    /// #[macro_use] extern crate approx;
    /// # use nalgebra::UnitComplex;
    /// let c = UnitComplex::new(1.0f64);
    /// let c2 = c.cast::<f32>();
    /// assert_relative_eq!(c2, UnitComplex::new(1.0f32));
    /// ```
    pub fn cast<To: Scalar>(self) -> UnitComplex<To>
    where
        UnitComplex<To>: SupersetOf<Self>,
    {
        crate::convert(self)
    }

    /// The underlying complex number.
    ///
    /// Same as `self.as_ref()`.
    ///
    /// # Example
    /// ```
    /// # extern crate num_complex;
    /// # use num_complex::Complex;
    /// # use nalgebra::UnitComplex;
    /// let angle = 1.78f32;
    /// let rot = UnitComplex::new(angle);
    /// assert_eq!(*rot.complex(), Complex::new(angle.cos(), angle.sin()));
    /// ```
    #[inline]
    #[must_use]
    pub fn complex(&self) -> &Complex<T> {
        self.as_ref()
    }

    /// Creates a new unit complex number from a complex number.
    ///
    /// The input complex number will be normalized.
    #[inline]
    pub fn from_complex(q: Complex<T>) -> Self {
        Self::from_complex_and_get(q).0
    }

    /// Creates a new unit complex number from a complex number.
    ///
    /// The input complex number will be normalized. Returns the norm of the complex number as well.
    #[inline]
    pub fn from_complex_and_get(q: Complex<T>) -> (Self, T) {
        let norm = (q.im.clone() * q.im.clone() + q.re.clone() * q.re.clone()).simd_sqrt();
        (Self::new_unchecked(q / norm.clone()), norm)
    }

    /// Builds the unit complex number from the corresponding 2D rotation matrix.
    ///
    /// # Example
    /// ```
    /// # use nalgebra::{Rotation2, UnitComplex};
    /// let rot = Rotation2::new(1.7);
    /// let complex = UnitComplex::from_rotation_matrix(&rot);
    /// assert_eq!(complex, UnitComplex::new(1.7));
    /// ```
    // TODO: add UnitComplex::from(...) instead?
    #[inline]
    pub fn from_rotation_matrix(rotmat: &Rotation2<T>) -> Self {
        Self::new_unchecked(Complex::new(rotmat[(0, 0)].clone(), rotmat[(1, 0)].clone()))
    }

    /// Builds a rotation from a basis assumed to be orthonormal.
    ///
    /// In order to get a valid unit-quaternion, the input must be an
    /// orthonormal basis, i.e., all vectors are normalized, and the are
    /// all orthogonal to each other. These invariants are not checked
    /// by this method.
    pub fn from_basis_unchecked(basis: &[Vector2<T>; 2]) -> Self {
        let mat = Matrix2::from_columns(&basis[..]);
        let rot = Rotation2::from_matrix_unchecked(mat);
        Self::from_rotation_matrix(&rot)
    }

    /// Builds an unit complex by extracting the rotation part of the given transformation `m`.
    ///
    /// This is an iterative method. See `.from_matrix_eps` to provide mover
    /// convergence parameters and starting solution.
    /// This implements "A Robust Method to Extract the Rotational Part of Deformations" by Müller et al.
    pub fn from_matrix(m: &Matrix2<T>) -> Self
    where
        T: RealField,
    {
        Rotation2::from_matrix(m).into()
    }

    /// Builds an unit complex 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 `UnitQuaternion::identity()` if no other
    ///           guesses come to mind.
    pub fn from_matrix_eps(m: &Matrix2<T>, eps: T, max_iter: usize, guess: Self) -> Self
    where
        T: RealField,
    {
        let guess = Rotation2::from(guess);
        Rotation2::from_matrix_eps(m, eps, max_iter, guess).into()
    }

    /// The unit complex number 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::UnitComplex;
    /// let rot1 = UnitComplex::new(0.1);
    /// let rot2 = UnitComplex::new(1.7);
    /// let rot_to = rot1.rotation_to(&rot2);
    ///
    /// assert_relative_eq!(rot_to * rot1, rot2);
    /// assert_relative_eq!(rot_to.inverse() * rot2, rot1);
    /// ```
    #[inline]
    #[must_use]
    pub fn rotation_to(&self, other: &Self) -> Self {
        other / self
    }

    /// Raise this unit complex number to a given floating power.
    ///
    /// This returns the unit complex number that identifies a rotation angle equal to
    /// `self.angle() × n`.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::UnitComplex;
    /// let rot = UnitComplex::new(0.78);
    /// let pow = rot.powf(2.0);
    /// assert_relative_eq!(pow.angle(), 2.0 * 0.78);
    /// ```
    #[inline]
    #[must_use]
    pub fn powf(&self, n: T) -> Self {
        Self::from_angle(self.angle() * n)
    }
}

/// # Construction from two vectors
impl<T: SimdRealField> UnitComplex<T>
where
    T::Element: SimdRealField,
{
    /// The unit complex needed to make `a` and `b` be collinear and point toward the same
    /// direction.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::{Vector2, UnitComplex};
    /// let a = Vector2::new(1.0, 2.0);
    /// let b = Vector2::new(2.0, 1.0);
    /// let rot = UnitComplex::rotation_between(&a, &b);
    /// assert_relative_eq!(rot * a, b);
    /// assert_relative_eq!(rot.inverse() * b, a);
    /// ```
    #[inline]
    pub fn rotation_between<SB, SC>(a: &Vector<T, U2, SB>, b: &Vector<T, U2, SC>) -> Self
    where
        T: RealField,
        SB: Storage<T, U2>,
        SC: Storage<T, U2>,
    {
        Self::scaled_rotation_between(a, b, T::one())
    }

    /// The smallest rotation needed to make `a` and `b` collinear and point toward the same
    /// direction, raised to the power `s`.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::{Vector2, UnitComplex};
    /// let a = Vector2::new(1.0, 2.0);
    /// let b = Vector2::new(2.0, 1.0);
    /// let rot2 = UnitComplex::scaled_rotation_between(&a, &b, 0.2);
    /// let rot5 = UnitComplex::scaled_rotation_between(&a, &b, 0.5);
    /// assert_relative_eq!(rot2 * rot2 * rot2 * rot2 * rot2 * a, b, epsilon = 1.0e-6);
    /// assert_relative_eq!(rot5 * rot5 * a, b, epsilon = 1.0e-6);
    /// ```
    #[inline]
    pub fn scaled_rotation_between<SB, SC>(
        a: &Vector<T, U2, SB>,
        b: &Vector<T, U2, SC>,
        s: T,
    ) -> Self
    where
        T: RealField,
        SB: Storage<T, U2>,
        SC: Storage<T, U2>,
    {
        // TODO: code duplication with Rotation.
        if let (Some(na), Some(nb)) = (
            Unit::try_new(a.clone_owned(), T::zero()),
            Unit::try_new(b.clone_owned(), T::zero()),
        ) {
            Self::scaled_rotation_between_axis(&na, &nb, s)
        } else {
            Self::identity()
        }
    }

    /// The unit complex needed to make `a` and `b` be collinear and point toward the same
    /// direction.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::{Unit, Vector2, UnitComplex};
    /// let a = Unit::new_normalize(Vector2::new(1.0, 2.0));
    /// let b = Unit::new_normalize(Vector2::new(2.0, 1.0));
    /// let rot = UnitComplex::rotation_between_axis(&a, &b);
    /// assert_relative_eq!(rot * a, b);
    /// assert_relative_eq!(rot.inverse() * b, a);
    /// ```
    #[inline]
    pub fn rotation_between_axis<SB, SC>(
        a: &Unit<Vector<T, U2, SB>>,
        b: &Unit<Vector<T, U2, SC>>,
    ) -> Self
    where
        SB: Storage<T, U2>,
        SC: Storage<T, U2>,
    {
        Self::scaled_rotation_between_axis(a, b, T::one())
    }

    /// The smallest rotation needed to make `a` and `b` collinear and point toward the same
    /// direction, raised to the power `s`.
    ///
    /// # Example
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use nalgebra::{Unit, Vector2, UnitComplex};
    /// let a = Unit::new_normalize(Vector2::new(1.0, 2.0));
    /// let b = Unit::new_normalize(Vector2::new(2.0, 1.0));
    /// let rot2 = UnitComplex::scaled_rotation_between_axis(&a, &b, 0.2);
    /// let rot5 = UnitComplex::scaled_rotation_between_axis(&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_axis<SB, SC>(
        na: &Unit<Vector<T, U2, SB>>,
        nb: &Unit<Vector<T, U2, SC>>,
        s: T,
    ) -> Self
    where
        SB: Storage<T, U2>,
        SC: Storage<T, U2>,
    {
        let sang = na.perp(nb);
        let cang = na.dot(nb);

        Self::from_angle(sang.simd_atan2(cang) * s)
    }
}

impl<T: SimdRealField> One for UnitComplex<T>
where
    T::Element: SimdRealField,
{
    #[inline]
    fn one() -> Self {
        Self::identity()
    }
}

#[cfg(feature = "rand")]
impl<T: SimdRealField> Distribution<UnitComplex<T>> for Standard
where
    T::Element: SimdRealField,
    rand_distr::UnitCircle: Distribution<[T; 2]>,
{
    /// Generate a uniformly distributed random `UnitComplex`.
    #[inline]
    fn sample<'a, R: Rng + ?Sized>(&self, rng: &mut R) -> UnitComplex<T> {
        let x = rng.sample(rand_distr::UnitCircle);
        UnitComplex::new_unchecked(Complex::new(x[0].clone(), x[1].clone()))
    }
}

#[cfg(feature = "arbitrary")]
impl<T: SimdRealField + Arbitrary> Arbitrary for UnitComplex<T>
where
    T::Element: SimdRealField,
{
    #[inline]
    fn arbitrary(g: &mut Gen) -> Self {
        UnitComplex::from_angle(T::arbitrary(g))
    }
}