Files
addr2line
adler
adler32
ahash
aho_corasick
angle
approx
backtrace
bitflags
blender
bytemuck
byteorder
case
cast_trait
cfg_if
chrono
color
color_quant
const_fn
crc32fast
crossbeam
crossbeam_channel
crossbeam_deque
crossbeam_epoch
crossbeam_queue
crossbeam_skiplist
crossbeam_utils
darling
darling_core
darling_macro
dds
deflate
densevec
derive_builder
derive_builder_core
dot
downcast_rs
dual_quat
either
erased_serde
failure
failure_derive
fixedbitset
float_cmp
fnv
freeimage
freeimage_sys
freetype
freetype_gl_sys
freetype_sys
freetypegl
futures
futures_channel
futures_core
futures_executor
futures_io
futures_macro
futures_sink
futures_task
futures_util
async_await
future
io
lock
sink
stream
task
fxhash
generational_arena
generic_array
getrandom
gif
gimli
glfw
glfw_sys
glin
glin_derive
glsl
half
harfbuzz
harfbuzz_ft_sys
harfbuzz_sys
hashbrown
human_sort
ident_case
image
indexmap
instant
itertools
itoa
jpeg_decoder
lazy_static
libc
libm
lock_api
log
lut_parser
matrixmultiply
memchr
memoffset
meshopt
miniz_oxide
monotonic_clock
mopa
mutiny_derive
na
nalgebra
base
geometry
linalg
ncollide3d
bounding_volume
interpolation
partitioning
pipeline
procedural
query
algorithms
closest_points
contact
distance
nonlinear_time_of_impact
point
proximity
ray
time_of_impact
visitors
shape
transformation
utils
nom
num_complex
num_cpus
num_integer
num_iter
num_rational
num_traits
numext_constructor
numext_fixed_uint
numext_fixed_uint_core
numext_fixed_uint_hack
object
once_cell
parking_lot
parking_lot_core
pathfinding
pennereq
petgraph
pin_project_lite
pin_utils
png
polygon2
ppv_lite86
proc_macro2
proc_macro_crate
proc_macro_hack
proc_macro_nested
quote
rand
rand_chacha
rand_core
rand_distr
raw_window_handle
rawpointer
rayon
rayon_core
rect_packer
regex
regex_syntax
retain_mut
rin
rin_app
rin_blender
rin_core
rin_gl
rin_graphics
rin_gui
rin_material
rin_math
rin_postpo
rin_scene
rin_util
rin_window
rinblender
rinecs
rinecs_derive
rinecs_derive_utils
ringui_derive
rustc_demangle
rusty_pool
ryu
scopeguard
seitan
seitan_derive
semver
semver_parser
serde
serde_derive
serde_json
shaderdata_derive
simba
slab
slice_of_array
slotmap
smallvec
std140_data
streaming_iterator
strsim
syn
synstructure
thiserror
thiserror_impl
thread_local
tiff
time
toml
typenum
unchecked_unwrap
unicode_xid
vec2
vec3
weezl
x11
zlib_sys
  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
use approx::{AbsDiffEq, RelativeEq, UlpsEq};
use std::fmt;
use std::hash;
#[cfg(feature = "abomonation-serialize")]
use std::io::{Result as IOResult, Write};

#[cfg(feature = "serde-serialize")]
use serde::{Deserialize, Serialize};

#[cfg(feature = "abomonation-serialize")]
use abomonation::Abomonation;

use simba::scalar::{RealField, SubsetOf};
use simba::simd::SimdRealField;

use crate::base::allocator::Allocator;
use crate::base::dimension::{DimName, DimNameAdd, DimNameSum, U1};
use crate::base::storage::Owned;
use crate::base::{DefaultAllocator, MatrixN, Scalar, Unit, VectorN};
use crate::geometry::{AbstractRotation, Point, Translation};

/// A direct isometry, i.e., a rotation followed by a translation (aka. a rigid-body motion).
///
/// This is also known as an element of a Special Euclidean (SE) group.
/// The `Isometry` type can either represent a 2D or 3D isometry.
/// A 2D isometry is composed of:
/// - A translation part of type [`Translation2`](crate::Translation2)
/// - A rotation part which can either be a [`UnitComplex`](crate::UnitComplex) or a [`Rotation2`](crate::Rotation2).
/// A 3D isometry is composed of:
/// - A translation part of type [`Translation3`](crate::Translation3)
/// - A rotation part which can either be a [`UnitQuaternion`](crate::UnitQuaternion) or a [`Rotation3`](crate::Rotation3).
///
/// Note that instead of using the [`Isometry`](crate::Isometry) type in your code directly, you should use one
/// of its aliases: [`Isometry2`](crate::Isometry2), [`Isometry3`](crate::Isometry3),
/// [`IsometryMatrix2`](crate::IsometryMatrix2), [`IsometryMatrix3`](crate::IsometryMatrix3). Though
/// keep in mind that all the documentation of all the methods of these aliases will also appears on
/// this page.
///
/// # Construction
/// * [From a 2D vector and/or an angle <span style="float:right;">`new`, `translation`, `rotation`…</span>](#construction-from-a-2d-vector-andor-a-rotation-angle)
/// * [From a 3D vector and/or an axis-angle <span style="float:right;">`new`, `translation`, `rotation`…</span>](#construction-from-a-3d-vector-andor-an-axis-angle)
/// * [From a 3D eye position and target point <span style="float:right;">`look_at`, `look_at_lh`, `face_towards`…</span>](#construction-from-a-3d-eye-position-and-target-point)
/// * [From the translation and rotation parts <span style="float:right;">`from_parts`…</span>](#from-the-translation-and-rotation-parts)
///
/// # Transformation and composition
/// Note that transforming vectors and points can be done by multiplication, e.g., `isometry * point`.
/// Composing an isometry with another transformation can also be done by multiplication or division.
///
/// * [Transformation of a vector or a point <span style="float:right;">`transform_vector`, `inverse_transform_point`…</span>](#transformation-of-a-vector-or-a-point)
/// * [Inversion and in-place composition <span style="float:right;">`inverse`, `append_rotation_wrt_point_mut`…</span>](#inversion-and-in-place-composition)
/// * [Interpolation <span style="float:right;">`lerp_slerp`…</span>](#interpolation)
///
/// # Conversion to a matrix
/// * [Conversion to a matrix <span style="float:right;">`to_matrix`…</span>](#conversion-to-a-matrix)
///
#[repr(C)]
#[derive(Debug)]
#[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
#[cfg_attr(
    feature = "serde-serialize",
    serde(bound(serialize = "R: Serialize,
                     DefaultAllocator: Allocator<N, D>,
                     Owned<N, D>: Serialize"))
)]
#[cfg_attr(
    feature = "serde-serialize",
    serde(bound(deserialize = "R: Deserialize<'de>,
                       DefaultAllocator: Allocator<N, D>,
                       Owned<N, D>: Deserialize<'de>"))
)]
pub struct Isometry<N: Scalar, D: DimName, R>
where
    DefaultAllocator: Allocator<N, D>,
{
    /// The pure rotational part of this isometry.
    pub rotation: R,
    /// The pure translational part of this isometry.
    pub translation: Translation<N, D>,
}

#[cfg(feature = "abomonation-serialize")]
impl<N, D, R> Abomonation for Isometry<N, D, R>
where
    N: SimdRealField,
    D: DimName,
    R: Abomonation,
    Translation<N, D>: Abomonation,
    DefaultAllocator: Allocator<N, D>,
{
    unsafe fn entomb<W: Write>(&self, writer: &mut W) -> IOResult<()> {
        self.rotation.entomb(writer)?;
        self.translation.entomb(writer)
    }

    fn extent(&self) -> usize {
        self.rotation.extent() + self.translation.extent()
    }

    unsafe fn exhume<'a, 'b>(&'a mut self, bytes: &'b mut [u8]) -> Option<&'b mut [u8]> {
        self.rotation
            .exhume(bytes)
            .and_then(|bytes| self.translation.exhume(bytes))
    }
}

impl<N: Scalar + hash::Hash, D: DimName + hash::Hash, R: hash::Hash> hash::Hash
    for Isometry<N, D, R>
where
    DefaultAllocator: Allocator<N, D>,
    Owned<N, D>: hash::Hash,
{
    fn hash<H: hash::Hasher>(&self, state: &mut H) {
        self.translation.hash(state);
        self.rotation.hash(state);
    }
}

impl<N: Scalar + Copy, D: DimName + Copy, R: Copy> Copy for Isometry<N, D, R>
where
    DefaultAllocator: Allocator<N, D>,
    Owned<N, D>: Copy,
{
}

impl<N: Scalar, D: DimName, R: Clone> Clone for Isometry<N, D, R>
where
    DefaultAllocator: Allocator<N, D>,
{
    #[inline]
    fn clone(&self) -> Self {
        Self {
            rotation: self.rotation.clone(),
            translation: self.translation.clone(),
        }
    }
}
/// # From the translation and rotation parts
impl<N: Scalar, D: DimName, R: AbstractRotation<N, D>> Isometry<N, D, R>
where
    DefaultAllocator: Allocator<N, D>,
{
    /// Creates a new isometry from its rotational and translational parts.
    ///
    /// # Example
    ///
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use std::f32;
    /// # use nalgebra::{Isometry3, Translation3, UnitQuaternion, Vector3, Point3};
    /// let tra = Translation3::new(0.0, 0.0, 3.0);
    /// let rot = UnitQuaternion::from_scaled_axis(Vector3::y() * f32::consts::PI);
    /// let iso = Isometry3::from_parts(tra, rot);
    ///
    /// assert_relative_eq!(iso * Point3::new(1.0, 2.0, 3.0), Point3::new(-1.0, 2.0, 0.0), epsilon = 1.0e-6);
    /// ```
    #[inline]
    pub fn from_parts(translation: Translation<N, D>, rotation: R) -> Self {
        Self {
            rotation,
            translation,
        }
    }
}

/// # Inversion and in-place composition
impl<N: SimdRealField, D: DimName, R: AbstractRotation<N, D>> Isometry<N, D, R>
where
    N::Element: SimdRealField,
    DefaultAllocator: Allocator<N, D>,
{
    /// Inverts `self`.
    ///
    /// # Example
    ///
    /// ```
    /// # use std::f32;
    /// # use nalgebra::{Isometry2, Point2, Vector2};
    /// let iso = Isometry2::new(Vector2::new(1.0, 2.0), f32::consts::FRAC_PI_2);
    /// let inv = iso.inverse();
    /// let pt = Point2::new(1.0, 2.0);
    ///
    /// assert_eq!(inv * (iso * pt), pt);
    /// ```
    #[inline]
    #[must_use = "Did you mean to use inverse_mut()?"]
    pub fn inverse(&self) -> Self {
        let mut res = self.clone();
        res.inverse_mut();
        res
    }

    /// Inverts `self` in-place.
    ///
    /// # Example
    ///
    /// ```
    /// # use std::f32;
    /// # use nalgebra::{Isometry2, Point2, Vector2};
    /// let mut iso = Isometry2::new(Vector2::new(1.0, 2.0), f32::consts::FRAC_PI_2);
    /// let pt = Point2::new(1.0, 2.0);
    /// let transformed_pt = iso * pt;
    /// iso.inverse_mut();
    ///
    /// assert_eq!(iso * transformed_pt, pt);
    /// ```
    #[inline]
    pub fn inverse_mut(&mut self) {
        self.rotation.inverse_mut();
        self.translation.inverse_mut();
        self.translation.vector = self.rotation.transform_vector(&self.translation.vector);
    }

    /// Computes `self.inverse() * rhs` in a more efficient way.
    ///
    /// # Example
    ///
    /// ```
    /// # use std::f32;
    /// # use nalgebra::{Isometry2, Point2, Vector2};
    /// let mut iso1 = Isometry2::new(Vector2::new(1.0, 2.0), f32::consts::FRAC_PI_2);
    /// let mut iso2 = Isometry2::new(Vector2::new(10.0, 20.0), f32::consts::FRAC_PI_4);
    ///
    /// assert_eq!(iso1.inverse() * iso2, iso1.inv_mul(&iso2));
    /// ```
    #[inline]
    pub fn inv_mul(&self, rhs: &Isometry<N, D, R>) -> Self {
        let inv_rot1 = self.rotation.inverse();
        let tr_12 = rhs.translation.vector.clone() - self.translation.vector.clone();
        Isometry::from_parts(
            inv_rot1.transform_vector(&tr_12).into(),
            inv_rot1 * rhs.rotation.clone(),
        )
    }

    /// Appends to `self` the given translation in-place.
    ///
    /// # Example
    ///
    /// ```
    /// # use std::f32;
    /// # use nalgebra::{Isometry2, Translation2, Vector2};
    /// let mut iso = Isometry2::new(Vector2::new(1.0, 2.0), f32::consts::FRAC_PI_2);
    /// let tra = Translation2::new(3.0, 4.0);
    /// // Same as `iso = tra * iso`.
    /// iso.append_translation_mut(&tra);
    ///
    /// assert_eq!(iso.translation, Translation2::new(4.0, 6.0));
    /// ```
    #[inline]
    pub fn append_translation_mut(&mut self, t: &Translation<N, D>) {
        self.translation.vector += &t.vector
    }

    /// Appends to `self` the given rotation in-place.
    ///
    /// # Example
    ///
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use std::f32;
    /// # use nalgebra::{Isometry2, Translation2, UnitComplex, Vector2};
    /// let mut iso = Isometry2::new(Vector2::new(1.0, 2.0), f32::consts::PI / 6.0);
    /// let rot = UnitComplex::new(f32::consts::PI / 2.0);
    /// // Same as `iso = rot * iso`.
    /// iso.append_rotation_mut(&rot);
    ///
    /// assert_relative_eq!(iso, Isometry2::new(Vector2::new(-2.0, 1.0), f32::consts::PI * 2.0 / 3.0), epsilon = 1.0e-6);
    /// ```
    #[inline]
    pub fn append_rotation_mut(&mut self, r: &R) {
        self.rotation = r.clone() * self.rotation.clone();
        self.translation.vector = r.transform_vector(&self.translation.vector);
    }

    /// Appends in-place to `self` a rotation centered at the point `p`, i.e., the rotation that
    /// lets `p` invariant.
    ///
    /// # Example
    ///
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use std::f32;
    /// # use nalgebra::{Isometry2, Translation2, UnitComplex, Vector2, Point2};
    /// let mut iso = Isometry2::new(Vector2::new(1.0, 2.0), f32::consts::FRAC_PI_2);
    /// let rot = UnitComplex::new(f32::consts::FRAC_PI_2);
    /// let pt = Point2::new(1.0, 0.0);
    /// iso.append_rotation_wrt_point_mut(&rot, &pt);
    ///
    /// assert_relative_eq!(iso * pt, Point2::new(-2.0, 0.0), epsilon = 1.0e-6);
    /// ```
    #[inline]
    pub fn append_rotation_wrt_point_mut(&mut self, r: &R, p: &Point<N, D>) {
        self.translation.vector -= &p.coords;
        self.append_rotation_mut(r);
        self.translation.vector += &p.coords;
    }

    /// Appends in-place to `self` a rotation centered at the point with coordinates
    /// `self.translation`.
    ///
    /// # Example
    ///
    /// ```
    /// # use std::f32;
    /// # use nalgebra::{Isometry2, Translation2, UnitComplex, Vector2, Point2};
    /// let mut iso = Isometry2::new(Vector2::new(1.0, 2.0), f32::consts::FRAC_PI_2);
    /// let rot = UnitComplex::new(f32::consts::FRAC_PI_2);
    /// iso.append_rotation_wrt_center_mut(&rot);
    ///
    /// // The translation part should not have changed.
    /// assert_eq!(iso.translation.vector, Vector2::new(1.0, 2.0));
    /// assert_eq!(iso.rotation, UnitComplex::new(f32::consts::PI));
    /// ```
    #[inline]
    pub fn append_rotation_wrt_center_mut(&mut self, r: &R) {
        self.rotation = r.clone() * self.rotation.clone();
    }
}

/// # Transformation of a vector or a point
impl<N: SimdRealField, D: DimName, R: AbstractRotation<N, D>> Isometry<N, D, R>
where
    N::Element: SimdRealField,
    DefaultAllocator: Allocator<N, D>,
{
    /// Transform the given point by this isometry.
    ///
    /// This is the same as the multiplication `self * pt`.
    ///
    /// # Example
    ///
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use std::f32;
    /// # use nalgebra::{Isometry3, Translation3, UnitQuaternion, Vector3, Point3};
    /// let tra = Translation3::new(0.0, 0.0, 3.0);
    /// let rot = UnitQuaternion::from_scaled_axis(Vector3::y() * f32::consts::FRAC_PI_2);
    /// let iso = Isometry3::from_parts(tra, rot);
    ///
    /// let transformed_point = iso.transform_point(&Point3::new(1.0, 2.0, 3.0));
    /// assert_relative_eq!(transformed_point, Point3::new(3.0, 2.0, 2.0), epsilon = 1.0e-6);
    /// ```
    #[inline]
    pub fn transform_point(&self, pt: &Point<N, D>) -> Point<N, D> {
        self * pt
    }

    /// Transform the given vector by this isometry, ignoring the translation
    /// component of the isometry.
    ///
    /// This is the same as the multiplication `self * v`.
    ///
    /// # Example
    ///
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use std::f32;
    /// # use nalgebra::{Isometry3, Translation3, UnitQuaternion, Vector3};
    /// let tra = Translation3::new(0.0, 0.0, 3.0);
    /// let rot = UnitQuaternion::from_scaled_axis(Vector3::y() * f32::consts::FRAC_PI_2);
    /// let iso = Isometry3::from_parts(tra, rot);
    ///
    /// let transformed_point = iso.transform_vector(&Vector3::new(1.0, 2.0, 3.0));
    /// assert_relative_eq!(transformed_point, Vector3::new(3.0, 2.0, -1.0), epsilon = 1.0e-6);
    /// ```
    #[inline]
    pub fn transform_vector(&self, v: &VectorN<N, D>) -> VectorN<N, D> {
        self * v
    }

    /// Transform the given point by the inverse of this isometry. This may be
    /// less expensive than computing the entire isometry inverse and then
    /// transforming the point.
    ///
    /// # Example
    ///
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use std::f32;
    /// # use nalgebra::{Isometry3, Translation3, UnitQuaternion, Vector3, Point3};
    /// let tra = Translation3::new(0.0, 0.0, 3.0);
    /// let rot = UnitQuaternion::from_scaled_axis(Vector3::y() * f32::consts::FRAC_PI_2);
    /// let iso = Isometry3::from_parts(tra, rot);
    ///
    /// let transformed_point = iso.inverse_transform_point(&Point3::new(1.0, 2.0, 3.0));
    /// assert_relative_eq!(transformed_point, Point3::new(0.0, 2.0, 1.0), epsilon = 1.0e-6);
    /// ```
    #[inline]
    pub fn inverse_transform_point(&self, pt: &Point<N, D>) -> Point<N, D> {
        self.rotation
            .inverse_transform_point(&(pt - &self.translation.vector))
    }

    /// Transform the given vector by the inverse of this isometry, ignoring the
    /// translation component of the isometry. This may be
    /// less expensive than computing the entire isometry inverse and then
    /// transforming the point.
    ///
    /// # Example
    ///
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use std::f32;
    /// # use nalgebra::{Isometry3, Translation3, UnitQuaternion, Vector3};
    /// let tra = Translation3::new(0.0, 0.0, 3.0);
    /// let rot = UnitQuaternion::from_scaled_axis(Vector3::y() * f32::consts::FRAC_PI_2);
    /// let iso = Isometry3::from_parts(tra, rot);
    ///
    /// let transformed_point = iso.inverse_transform_vector(&Vector3::new(1.0, 2.0, 3.0));
    /// assert_relative_eq!(transformed_point, Vector3::new(-3.0, 2.0, 1.0), epsilon = 1.0e-6);
    /// ```
    #[inline]
    pub fn inverse_transform_vector(&self, v: &VectorN<N, D>) -> VectorN<N, D> {
        self.rotation.inverse_transform_vector(v)
    }

    /// Transform the given unit vector by the inverse of this isometry, ignoring the
    /// translation component of the isometry. This may be
    /// less expensive than computing the entire isometry inverse and then
    /// transforming the point.
    ///
    /// # Example
    ///
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use std::f32;
    /// # use nalgebra::{Isometry3, Translation3, UnitQuaternion, Vector3};
    /// let tra = Translation3::new(0.0, 0.0, 3.0);
    /// let rot = UnitQuaternion::from_scaled_axis(Vector3::z() * f32::consts::FRAC_PI_2);
    /// let iso = Isometry3::from_parts(tra, rot);
    ///
    /// let transformed_point = iso.inverse_transform_unit_vector(&Vector3::x_axis());
    /// assert_relative_eq!(transformed_point, -Vector3::y_axis(), epsilon = 1.0e-6);
    /// ```
    #[inline]
    pub fn inverse_transform_unit_vector(&self, v: &Unit<VectorN<N, D>>) -> Unit<VectorN<N, D>> {
        self.rotation.inverse_transform_unit_vector(v)
    }
}

// NOTE: we don't require `R: Rotation<...>` here because this is not useful for the implementation
// and makes it hard to use it, e.g., for Transform × Isometry implementation.
// This is OK since all constructors of the isometry enforce the Rotation bound already (and
// explicit struct construction is prevented by the dummy ZST field).
/// # Conversion to a matrix
impl<N: SimdRealField, D: DimName, R> Isometry<N, D, R>
where
    DefaultAllocator: Allocator<N, D>,
{
    /// Converts this isometry into its equivalent homogeneous transformation matrix.
    ///
    /// This is the same as `self.to_matrix()`.
    ///
    /// # Example
    ///
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use std::f32;
    /// # use nalgebra::{Isometry2, Vector2, Matrix3};
    /// let iso = Isometry2::new(Vector2::new(10.0, 20.0), f32::consts::FRAC_PI_6);
    /// let expected = Matrix3::new(0.8660254, -0.5,      10.0,
    ///                             0.5,       0.8660254, 20.0,
    ///                             0.0,       0.0,       1.0);
    ///
    /// assert_relative_eq!(iso.to_homogeneous(), expected, epsilon = 1.0e-6);
    /// ```
    #[inline]
    pub fn to_homogeneous(&self) -> MatrixN<N, DimNameSum<D, U1>>
    where
        D: DimNameAdd<U1>,
        R: SubsetOf<MatrixN<N, DimNameSum<D, U1>>>,
        DefaultAllocator: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1>>,
    {
        let mut res: MatrixN<N, _> = crate::convert_ref(&self.rotation);
        res.fixed_slice_mut::<D, U1>(0, D::dim())
            .copy_from(&self.translation.vector);

        res
    }

    /// Converts this isometry into its equivalent homogeneous transformation matrix.
    ///
    /// This is the same as `self.to_homogeneous()`.
    ///
    /// # Example
    ///
    /// ```
    /// # #[macro_use] extern crate approx;
    /// # use std::f32;
    /// # use nalgebra::{Isometry2, Vector2, Matrix3};
    /// let iso = Isometry2::new(Vector2::new(10.0, 20.0), f32::consts::FRAC_PI_6);
    /// let expected = Matrix3::new(0.8660254, -0.5,      10.0,
    ///                             0.5,       0.8660254, 20.0,
    ///                             0.0,       0.0,       1.0);
    ///
    /// assert_relative_eq!(iso.to_matrix(), expected, epsilon = 1.0e-6);
    /// ```
    #[inline]
    pub fn to_matrix(&self) -> MatrixN<N, DimNameSum<D, U1>>
    where
        D: DimNameAdd<U1>,
        R: SubsetOf<MatrixN<N, DimNameSum<D, U1>>>,
        DefaultAllocator: Allocator<N, DimNameSum<D, U1>, DimNameSum<D, U1>>,
    {
        self.to_homogeneous()
    }
}

impl<N: SimdRealField, D: DimName, R> Eq for Isometry<N, D, R>
where
    R: AbstractRotation<N, D> + Eq,
    DefaultAllocator: Allocator<N, D>,
{
}

impl<N: SimdRealField, D: DimName, R> PartialEq for Isometry<N, D, R>
where
    R: AbstractRotation<N, D> + PartialEq,
    DefaultAllocator: Allocator<N, D>,
{
    #[inline]
    fn eq(&self, right: &Self) -> bool {
        self.translation == right.translation && self.rotation == right.rotation
    }
}

impl<N: RealField, D: DimName, R> AbsDiffEq for Isometry<N, D, R>
where
    R: AbstractRotation<N, D> + AbsDiffEq<Epsilon = N::Epsilon>,
    DefaultAllocator: Allocator<N, D>,
    N::Epsilon: Copy,
{
    type Epsilon = N::Epsilon;

    #[inline]
    fn default_epsilon() -> Self::Epsilon {
        N::default_epsilon()
    }

    #[inline]
    fn abs_diff_eq(&self, other: &Self, epsilon: Self::Epsilon) -> bool {
        self.translation.abs_diff_eq(&other.translation, epsilon)
            && self.rotation.abs_diff_eq(&other.rotation, epsilon)
    }
}

impl<N: RealField, D: DimName, R> RelativeEq for Isometry<N, D, R>
where
    R: AbstractRotation<N, D> + RelativeEq<Epsilon = N::Epsilon>,
    DefaultAllocator: Allocator<N, D>,
    N::Epsilon: Copy,
{
    #[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.translation
            .relative_eq(&other.translation, epsilon, max_relative)
            && self
                .rotation
                .relative_eq(&other.rotation, epsilon, max_relative)
    }
}

impl<N: RealField, D: DimName, R> UlpsEq for Isometry<N, D, R>
where
    R: AbstractRotation<N, D> + UlpsEq<Epsilon = N::Epsilon>,
    DefaultAllocator: Allocator<N, D>,
    N::Epsilon: Copy,
{
    #[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.translation
            .ulps_eq(&other.translation, epsilon, max_ulps)
            && self.rotation.ulps_eq(&other.rotation, epsilon, max_ulps)
    }
}

/*
 *
 * Display
 *
 */
impl<N: RealField + fmt::Display, D: DimName, R> fmt::Display for Isometry<N, D, R>
where
    R: fmt::Display,
    DefaultAllocator: Allocator<N, D> + Allocator<usize, D>,
{
    fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
        let precision = f.precision().unwrap_or(3);

        writeln!(f, "Isometry {{")?;
        write!(f, "{:.*}", precision, self.translation)?;
        write!(f, "{:.*}", precision, self.rotation)?;
        writeln!(f, "}}")
    }
}