1 /*
2 * Licensed to the Apache Software Foundation (ASF) under one or more
3 * contributor license agreements. See the NOTICE file distributed with
4 * this work for additional information regarding copyright ownership.
5 * The ASF licenses this file to You under the Apache License, Version 2.0
6 * (the "License"); you may not use this file except in compliance with
7 * the License. You may obtain a copy of the License at
8 *
9 * https://www.apache.org/licenses/LICENSE-2.0
10 *
11 * Unless required by applicable law or agreed to in writing, software
12 * distributed under the License is distributed on an "AS IS" BASIS,
13 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
14 * See the License for the specific language governing permissions and
15 * limitations under the License.
16 */
17
18 /*
19 * This is not the original file distributed by the Apache Software Foundation
20 * It has been modified by the Hipparchus project
21 */
22
23 package org.hipparchus.geometry.euclidean.threed;
24
25 import java.io.Serializable;
26
27 import org.hipparchus.exception.MathIllegalArgumentException;
28 import org.hipparchus.exception.MathRuntimeException;
29 import org.hipparchus.geometry.LocalizedGeometryFormats;
30 import org.hipparchus.util.FastMath;
31 import org.hipparchus.util.MathArrays;
32 import org.hipparchus.util.SinCos;
33
34 /**
35 * This class implements rotations in a three-dimensional space.
36 *
37 * <p>Rotations can be represented by several different mathematical
38 * entities (matrices, axe and angle, Cardan or Euler angles,
39 * quaternions). This class presents an higher level abstraction, more
40 * user-oriented and hiding this implementation details. Well, for the
41 * curious, we use quaternions for the internal representation. The
42 * user can build a rotation from any of these representations, and
43 * any of these representations can be retrieved from a
44 * <code>Rotation</code> instance (see the various constructors and
45 * getters). In addition, a rotation can also be built implicitly
46 * from a set of vectors and their image.</p>
47 * <p>This implies that this class can be used to convert from one
48 * representation to another one. For example, converting a rotation
49 * matrix into a set of Cardan angles from can be done using the
50 * following single line of code:</p>
51 * <pre>
52 * double[] angles = new Rotation(matrix, 1.0e-10).getAngles(RotationOrder.XYZ);
53 * </pre>
54 * <p>Focus is oriented on what a rotation <em>do</em> rather than on its
55 * underlying representation. Once it has been built, and regardless of its
56 * internal representation, a rotation is an <em>operator</em> which basically
57 * transforms three dimensional {@link Vector3D vectors} into other three
58 * dimensional {@link Vector3D vectors}. Depending on the application, the
59 * meaning of these vectors may vary and the semantics of the rotation also.</p>
60 * <p>For example in an spacecraft attitude simulation tool, users will often
61 * consider the vectors are fixed (say the Earth direction for example) and the
62 * frames change. The rotation transforms the coordinates of the vector in inertial
63 * frame into the coordinates of the same vector in satellite frame. In this
64 * case, the rotation implicitly defines the relation between the two frames.</p>
65 * <p>Another example could be a telescope control application, where the rotation
66 * would transform the sighting direction at rest into the desired observing
67 * direction when the telescope is pointed towards an object of interest. In this
68 * case the rotation transforms the direction at rest in a topocentric frame
69 * into the sighting direction in the same topocentric frame. This implies in this
70 * case the frame is fixed and the vector moves.</p>
71 * <p>In many case, both approaches will be combined. In our telescope example,
72 * we will probably also need to transform the observing direction in the topocentric
73 * frame into the observing direction in inertial frame taking into account the observatory
74 * location and the Earth rotation, which would essentially be an application of the
75 * first approach.</p>
76 *
77 * <p>These examples show that a rotation is what the user wants it to be. This
78 * class does not push the user towards one specific definition and hence does not
79 * provide methods like <code>projectVectorIntoDestinationFrame</code> or
80 * <code>computeTransformedDirection</code>. It provides simpler and more generic
81 * methods: {@link #applyTo(Vector3D) applyTo(Vector3D)} and {@link
82 * #applyInverseTo(Vector3D) applyInverseTo(Vector3D)}.</p>
83 *
84 * <p>Since a rotation is basically a vectorial operator, several rotations can be
85 * composed together and the composite operation <code>r = r<sub>1</sub> o
86 * r<sub>2</sub></code> (which means that for each vector <code>u</code>,
87 * <code>r(u) = r<sub>1</sub>(r<sub>2</sub>(u))</code>) is also a rotation. Hence
88 * we can consider that in addition to vectors, a rotation can be applied to other
89 * rotations as well (or to itself). With our previous notations, we would say we
90 * can apply <code>r<sub>1</sub></code> to <code>r<sub>2</sub></code> and the result
91 * we get is <code>r = r<sub>1</sub> o r<sub>2</sub></code>. For this purpose, the
92 * class provides the methods: {@link #applyTo(Rotation) applyTo(Rotation)} and
93 * {@link #applyInverseTo(Rotation) applyInverseTo(Rotation)}.</p>
94 *
95 * <p>Rotations are guaranteed to be immutable objects.</p>
96 *
97 * @see Vector3D
98 * @see RotationOrder
99 */
100
101 public class Rotation implements Serializable {
102
103 /** Identity rotation. */
104 public static final Rotation IDENTITY = new Rotation(1.0, 0.0, 0.0, 0.0, false);
105
106 /** Serializable version identifier */
107 private static final long serialVersionUID = -2153622329907944313L;
108
109 /** Scalar coordinate of the quaternion. */
110 private final double q0;
111
112 /** First coordinate of the vectorial part of the quaternion. */
113 private final double q1;
114
115 /** Second coordinate of the vectorial part of the quaternion. */
116 private final double q2;
117
118 /** Third coordinate of the vectorial part of the quaternion. */
119 private final double q3;
120
121 /** Build a rotation from the quaternion coordinates.
122 * <p>A rotation can be built from a <em>normalized</em> quaternion,
123 * i.e. a quaternion for which q<sub>0</sub><sup>2</sup> +
124 * q<sub>1</sub><sup>2</sup> + q<sub>2</sub><sup>2</sup> +
125 * q<sub>3</sub><sup>2</sup> = 1. If the quaternion is not normalized,
126 * the constructor can normalize it in a preprocessing step.</p>
127 * <p>Note that some conventions put the scalar part of the quaternion
128 * as the 4<sup>th</sup> component and the vector part as the first three
129 * components. This is <em>not</em> our convention. We put the scalar part
130 * as the first component.</p>
131 * @param q0 scalar part of the quaternion
132 * @param q1 first coordinate of the vectorial part of the quaternion
133 * @param q2 second coordinate of the vectorial part of the quaternion
134 * @param q3 third coordinate of the vectorial part of the quaternion
135 * @param needsNormalization if true, the coordinates are considered
136 * not to be normalized, a normalization preprocessing step is performed
137 * before using them
138 */
139 public Rotation(double q0, double q1, double q2, double q3,
140 boolean needsNormalization) {
141
142 if (needsNormalization) {
143 // normalization preprocessing
144 double inv = 1.0 / FastMath.sqrt(q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3);
145 q0 *= inv;
146 q1 *= inv;
147 q2 *= inv;
148 q3 *= inv;
149 }
150
151 this.q0 = q0;
152 this.q1 = q1;
153 this.q2 = q2;
154 this.q3 = q3;
155
156 }
157
158 /** Build a rotation from an axis and an angle.
159 * @param axis axis around which to rotate
160 * @param angle rotation angle
161 * @param convention convention to use for the semantics of the angle
162 * @exception MathIllegalArgumentException if the axis norm is zero
163 */
164 public Rotation(final Vector3D axis, final double angle, final RotationConvention convention)
165 throws MathIllegalArgumentException {
166
167 double norm = axis.getNorm();
168 if (norm == 0) {
169 throw new MathIllegalArgumentException(LocalizedGeometryFormats.ZERO_NORM_FOR_ROTATION_AXIS);
170 }
171
172 double halfAngle = convention == RotationConvention.VECTOR_OPERATOR ? -0.5 * angle : 0.5 * angle;
173 SinCos sinCos = FastMath.sinCos(halfAngle);
174 double coeff = sinCos.sin() / norm;
175
176 q0 = sinCos.cos();
177 q1 = coeff * axis.getX();
178 q2 = coeff * axis.getY();
179 q3 = coeff * axis.getZ();
180
181 }
182
183 /** Build a rotation from a 3X3 matrix.
184
185 * <p>Rotation matrices are orthogonal matrices, i.e. unit matrices
186 * (which are matrices for which m.m<sup>T</sup> = I) with real
187 * coefficients. The module of the determinant of unit matrices is
188 * 1, among the orthogonal 3X3 matrices, only the ones having a
189 * positive determinant (+1) are rotation matrices.</p>
190
191 * <p>When a rotation is defined by a matrix with truncated values
192 * (typically when it is extracted from a technical sheet where only
193 * four to five significant digits are available), the matrix is not
194 * orthogonal anymore. This constructor handles this case
195 * transparently by using a copy of the given matrix and applying a
196 * correction to the copy in order to perfect its orthogonality. If
197 * the Frobenius norm of the correction needed is above the given
198 * threshold, then the matrix is considered to be too far from a
199 * true rotation matrix and an exception is thrown.</p>
200
201 * @param m rotation matrix
202 * @param threshold convergence threshold for the iterative
203 * orthogonality correction (convergence is reached when the
204 * difference between two steps of the Frobenius norm of the
205 * correction is below this threshold)
206
207 * @exception MathIllegalArgumentException if the matrix is not a 3X3
208 * matrix, or if it cannot be transformed into an orthogonal matrix
209 * with the given threshold, or if the determinant of the resulting
210 * orthogonal matrix is negative
211
212 */
213 public Rotation(double[][] m, double threshold)
214 throws MathIllegalArgumentException {
215
216 // dimension check
217 if ((m.length != 3) || (m[0].length != 3) ||
218 (m[1].length != 3) || (m[2].length != 3)) {
219 throw new MathIllegalArgumentException(LocalizedGeometryFormats.ROTATION_MATRIX_DIMENSIONS,
220 m.length, m[0].length);
221 }
222
223 // compute a "close" orthogonal matrix
224 double[][] ort = orthogonalizeMatrix(m, threshold);
225
226 // check the sign of the determinant
227 double det = ort[0][0] * (ort[1][1] * ort[2][2] - ort[2][1] * ort[1][2]) -
228 ort[1][0] * (ort[0][1] * ort[2][2] - ort[2][1] * ort[0][2]) +
229 ort[2][0] * (ort[0][1] * ort[1][2] - ort[1][1] * ort[0][2]);
230 if (det < 0.0) {
231 throw new MathIllegalArgumentException(LocalizedGeometryFormats.CLOSEST_ORTHOGONAL_MATRIX_HAS_NEGATIVE_DETERMINANT,
232 det);
233 }
234
235 double[] quat = mat2quat(ort);
236 q0 = quat[0];
237 q1 = quat[1];
238 q2 = quat[2];
239 q3 = quat[3];
240
241 }
242
243 /** Build the rotation that transforms a pair of vectors into another pair.
244
245 * <p>Except for possible scale factors, if the instance were applied to
246 * the pair (u<sub>1</sub>, u<sub>2</sub>) it will produce the pair
247 * (v<sub>1</sub>, v<sub>2</sub>).</p>
248
249 * <p>If the angular separation between u<sub>1</sub> and u<sub>2</sub> is
250 * not the same as the angular separation between v<sub>1</sub> and
251 * v<sub>2</sub>, then a corrected v'<sub>2</sub> will be used rather than
252 * v<sub>2</sub>, the corrected vector will be in the (±v<sub>1</sub>,
253 * +v<sub>2</sub>) half-plane.</p>
254 * @param u1 first vector of the origin pair
255 * @param u2 second vector of the origin pair
256 * @param v1 desired image of u1 by the rotation
257 * @param v2 desired image of u2 by the rotation
258 * @exception MathRuntimeException if the norm of one of the vectors is zero,
259 * or if one of the pair is degenerated (i.e. the vectors of the pair are collinear)
260 */
261 public Rotation(Vector3D u1, Vector3D u2, Vector3D v1, Vector3D v2)
262 throws MathRuntimeException {
263
264 // build orthonormalized base from u1, u2
265 // this fails when vectors are null or collinear, which is forbidden to define a rotation
266 final Vector3D u3 = u1.crossProduct(u2).normalize();
267 u2 = u3.crossProduct(u1).normalize();
268 u1 = u1.normalize();
269
270 // build an orthonormalized base from v1, v2
271 // this fails when vectors are null or collinear, which is forbidden to define a rotation
272 final Vector3D v3 = v1.crossProduct(v2).normalize();
273 v2 = v3.crossProduct(v1).normalize();
274 v1 = v1.normalize();
275
276 // buid a matrix transforming the first base into the second one
277 final double[][] m = {
278 {
279 MathArrays.linearCombination(u1.getX(), v1.getX(), u2.getX(), v2.getX(), u3.getX(), v3.getX()),
280 MathArrays.linearCombination(u1.getY(), v1.getX(), u2.getY(), v2.getX(), u3.getY(), v3.getX()),
281 MathArrays.linearCombination(u1.getZ(), v1.getX(), u2.getZ(), v2.getX(), u3.getZ(), v3.getX())
282 },
283 {
284 MathArrays.linearCombination(u1.getX(), v1.getY(), u2.getX(), v2.getY(), u3.getX(), v3.getY()),
285 MathArrays.linearCombination(u1.getY(), v1.getY(), u2.getY(), v2.getY(), u3.getY(), v3.getY()),
286 MathArrays.linearCombination(u1.getZ(), v1.getY(), u2.getZ(), v2.getY(), u3.getZ(), v3.getY())
287 },
288 {
289 MathArrays.linearCombination(u1.getX(), v1.getZ(), u2.getX(), v2.getZ(), u3.getX(), v3.getZ()),
290 MathArrays.linearCombination(u1.getY(), v1.getZ(), u2.getY(), v2.getZ(), u3.getY(), v3.getZ()),
291 MathArrays.linearCombination(u1.getZ(), v1.getZ(), u2.getZ(), v2.getZ(), u3.getZ(), v3.getZ())
292 }
293 };
294
295 double[] quat = mat2quat(m);
296 q0 = quat[0];
297 q1 = quat[1];
298 q2 = quat[2];
299 q3 = quat[3];
300
301 }
302
303 /** Build one of the rotations that transform one vector into another one.
304
305 * <p>Except for a possible scale factor, if the instance were
306 * applied to the vector u it will produce the vector v. There is an
307 * infinite number of such rotations, this constructor choose the
308 * one with the smallest associated angle (i.e. the one whose axis
309 * is orthogonal to the (u, v) plane). If u and v are collinear, an
310 * arbitrary rotation axis is chosen.</p>
311
312 * @param u origin vector
313 * @param v desired image of u by the rotation
314 * @exception MathRuntimeException if the norm of one of the vectors is zero
315 */
316 public Rotation(Vector3D u, Vector3D v) throws MathRuntimeException {
317
318 double normProduct = u.getNorm() * v.getNorm();
319 if (normProduct == 0) {
320 throw new MathRuntimeException(LocalizedGeometryFormats.ZERO_NORM_FOR_ROTATION_DEFINING_VECTOR);
321 }
322
323 double dot = u.dotProduct(v);
324
325 if (dot < ((2.0e-15 - 1.0) * normProduct)) {
326 // special case u = -v: we select a PI angle rotation around
327 // an arbitrary vector orthogonal to u
328 Vector3D w = u.orthogonal();
329 q0 = 0.0;
330 q1 = -w.getX();
331 q2 = -w.getY();
332 q3 = -w.getZ();
333 } else {
334 // general case: (u, v) defines a plane, we select
335 // the shortest possible rotation: axis orthogonal to this plane
336 q0 = FastMath.sqrt(0.5 * (1.0 + dot / normProduct));
337 double coeff = 1.0 / (2.0 * q0 * normProduct);
338 Vector3D q = v.crossProduct(u);
339 q1 = coeff * q.getX();
340 q2 = coeff * q.getY();
341 q3 = coeff * q.getZ();
342 }
343
344 }
345
346 /** Build a rotation from three Cardan or Euler elementary rotations.
347
348 * <p>Cardan rotations are three successive rotations around the
349 * canonical axes X, Y and Z, each axis being used once. There are
350 * 6 such sets of rotations (XYZ, XZY, YXZ, YZX, ZXY and ZYX). Euler
351 * rotations are three successive rotations around the canonical
352 * axes X, Y and Z, the first and last rotations being around the
353 * same axis. There are 6 such sets of rotations (XYX, XZX, YXY,
354 * YZY, ZXZ and ZYZ), the most popular one being ZXZ.</p>
355 * <p>Beware that many people routinely use the term Euler angles even
356 * for what really are Cardan angles (this confusion is especially
357 * widespread in the aerospace business where Roll, Pitch and Yaw angles
358 * are often wrongly tagged as Euler angles).</p>
359
360 * @param order order of rotations to compose, from left to right
361 * (i.e. we will use {@code r1.compose(r2.compose(r3, convention), convention)})
362 * @param convention convention to use for the semantics of the angle
363 * @param alpha1 angle of the first elementary rotation
364 * @param alpha2 angle of the second elementary rotation
365 * @param alpha3 angle of the third elementary rotation
366 */
367 public Rotation(RotationOrder order, RotationConvention convention,
368 double alpha1, double alpha2, double alpha3) {
369 Rotation r1 = new Rotation(order.getA1(), alpha1, convention);
370 Rotation r2 = new Rotation(order.getA2(), alpha2, convention);
371 Rotation r3 = new Rotation(order.getA3(), alpha3, convention);
372 Rotation composed = r1.compose(r2.compose(r3, convention), convention);
373 q0 = composed.q0;
374 q1 = composed.q1;
375 q2 = composed.q2;
376 q3 = composed.q3;
377 }
378
379 /** Convert an orthogonal rotation matrix to a quaternion.
380 * @param ort orthogonal rotation matrix
381 * @return quaternion corresponding to the matrix
382 */
383 private static double[] mat2quat(final double[][] ort) {
384
385 final double[] quat = new double[4];
386
387 // There are different ways to compute the quaternions elements
388 // from the matrix. They all involve computing one element from
389 // the diagonal of the matrix, and computing the three other ones
390 // using a formula involving a division by the first element,
391 // which unfortunately can be zero. Since the norm of the
392 // quaternion is 1, we know at least one element has an absolute
393 // value greater or equal to 0.5, so it is always possible to
394 // select the right formula and avoid division by zero and even
395 // numerical inaccuracy. Checking the elements in turn and using
396 // the first one greater than 0.45 is safe (this leads to a simple
397 // test since qi = 0.45 implies 4 qi^2 - 1 = -0.19)
398 double s = ort[0][0] + ort[1][1] + ort[2][2];
399 if (s > -0.19) {
400 // compute q0 and deduce q1, q2 and q3
401 quat[0] = 0.5 * FastMath.sqrt(s + 1.0);
402 double inv = 0.25 / quat[0];
403 quat[1] = inv * (ort[1][2] - ort[2][1]);
404 quat[2] = inv * (ort[2][0] - ort[0][2]);
405 quat[3] = inv * (ort[0][1] - ort[1][0]);
406 } else {
407 s = ort[0][0] - ort[1][1] - ort[2][2];
408 if (s > -0.19) {
409 // compute q1 and deduce q0, q2 and q3
410 quat[1] = 0.5 * FastMath.sqrt(s + 1.0);
411 double inv = 0.25 / quat[1];
412 quat[0] = inv * (ort[1][2] - ort[2][1]);
413 quat[2] = inv * (ort[0][1] + ort[1][0]);
414 quat[3] = inv * (ort[0][2] + ort[2][0]);
415 } else {
416 s = ort[1][1] - ort[0][0] - ort[2][2];
417 if (s > -0.19) {
418 // compute q2 and deduce q0, q1 and q3
419 quat[2] = 0.5 * FastMath.sqrt(s + 1.0);
420 double inv = 0.25 / quat[2];
421 quat[0] = inv * (ort[2][0] - ort[0][2]);
422 quat[1] = inv * (ort[0][1] + ort[1][0]);
423 quat[3] = inv * (ort[2][1] + ort[1][2]);
424 } else {
425 // compute q3 and deduce q0, q1 and q2
426 s = ort[2][2] - ort[0][0] - ort[1][1];
427 quat[3] = 0.5 * FastMath.sqrt(s + 1.0);
428 double inv = 0.25 / quat[3];
429 quat[0] = inv * (ort[0][1] - ort[1][0]);
430 quat[1] = inv * (ort[0][2] + ort[2][0]);
431 quat[2] = inv * (ort[2][1] + ort[1][2]);
432 }
433 }
434 }
435
436 return quat;
437
438 }
439
440 /** Revert a rotation.
441 * Build a rotation which reverse the effect of another
442 * rotation. This means that if r(u) = v, then r.revert(v) = u. The
443 * instance is not changed.
444 * @return a new rotation whose effect is the reverse of the effect
445 * of the instance
446 */
447 public Rotation revert() {
448 return new Rotation(-q0, q1, q2, q3, false);
449 }
450
451 /** Get the scalar coordinate of the quaternion.
452 * @return scalar coordinate of the quaternion
453 */
454 public double getQ0() {
455 return q0;
456 }
457
458 /** Get the first coordinate of the vectorial part of the quaternion.
459 * @return first coordinate of the vectorial part of the quaternion
460 */
461 public double getQ1() {
462 return q1;
463 }
464
465 /** Get the second coordinate of the vectorial part of the quaternion.
466 * @return second coordinate of the vectorial part of the quaternion
467 */
468 public double getQ2() {
469 return q2;
470 }
471
472 /** Get the third coordinate of the vectorial part of the quaternion.
473 * @return third coordinate of the vectorial part of the quaternion
474 */
475 public double getQ3() {
476 return q3;
477 }
478
479 /** Get the normalized axis of the rotation.
480 * <p>
481 * Note that as {@link #getAngle()} always returns an angle
482 * between 0 and π, changing the convention changes the
483 * direction of the axis, not the sign of the angle.
484 * </p>
485 * @param convention convention to use for the semantics of the angle
486 * @return normalized axis of the rotation
487 * @see #Rotation(Vector3D, double, RotationConvention)
488 */
489 public Vector3D getAxis(final RotationConvention convention) {
490 final double squaredSine = q1 * q1 + q2 * q2 + q3 * q3;
491 if (squaredSine == 0) {
492 return convention == RotationConvention.VECTOR_OPERATOR ? Vector3D.PLUS_I : Vector3D.MINUS_I;
493 } else {
494 final double sgn = convention == RotationConvention.VECTOR_OPERATOR ? +1 : -1;
495 if (q0 < 0) {
496 final double inverse = sgn / FastMath.sqrt(squaredSine);
497 return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
498 }
499 final double inverse = -sgn / FastMath.sqrt(squaredSine);
500 return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
501 }
502 }
503
504 /** Get the angle of the rotation.
505 * @return angle of the rotation (between 0 and π)
506 * @see #Rotation(Vector3D, double, RotationConvention)
507 */
508 public double getAngle() {
509 if ((q0 < -0.1) || (q0 > 0.1)) {
510 return 2 * FastMath.asin(FastMath.sqrt(q1 * q1 + q2 * q2 + q3 * q3));
511 } else if (q0 < 0) {
512 return 2 * FastMath.acos(-q0);
513 }
514 return 2 * FastMath.acos(q0);
515 }
516
517 /** Get the Cardan or Euler angles corresponding to the instance.
518
519 * <p>The equations show that each rotation can be defined by two
520 * different values of the Cardan or Euler angles set. For example
521 * if Cardan angles are used, the rotation defined by the angles
522 * a<sub>1</sub>, a<sub>2</sub> and a<sub>3</sub> is the same as
523 * the rotation defined by the angles π + a<sub>1</sub>, π
524 * - a<sub>2</sub> and π + a<sub>3</sub>. This method implements
525 * the following arbitrary choices:</p>
526 * <ul>
527 * <li>for Cardan angles, the chosen set is the one for which the
528 * second angle is between -π/2 and π/2 (i.e its cosine is
529 * positive),</li>
530 * <li>for Euler angles, the chosen set is the one for which the
531 * second angle is between 0 and π (i.e its sine is positive).</li>
532 * </ul>
533
534 * <p>Cardan and Euler angle have a very disappointing drawback: all
535 * of them have singularities. This means that if the instance is
536 * too close to the singularities corresponding to the given
537 * rotation order, it will be impossible to retrieve the angles. For
538 * Cardan angles, this is often called gimbal lock. There is
539 * <em>nothing</em> to do to prevent this, it is an intrinsic problem
540 * with Cardan and Euler representation (but not a problem with the
541 * rotation itself, which is perfectly well defined). For Cardan
542 * angles, singularities occur when the second angle is close to
543 * -π/2 or +π/2, for Euler angle singularities occur when the
544 * second angle is close to 0 or π, this implies that the identity
545 * rotation is always singular for Euler angles!</p>
546
547 * @param order rotation order to use
548 * @param convention convention to use for the semantics of the angle
549 * @return an array of three angles, in the order specified by the set
550 */
551 public double[] getAngles(RotationOrder order, RotationConvention convention) {
552 return order.getAngles(this, convention);
553 }
554
555 /** Get the 3X3 matrix corresponding to the instance
556 * @return the matrix corresponding to the instance
557 */
558 public double[][] getMatrix() {
559
560 // products
561 double q0q0 = q0 * q0;
562 double q0q1 = q0 * q1;
563 double q0q2 = q0 * q2;
564 double q0q3 = q0 * q3;
565 double q1q1 = q1 * q1;
566 double q1q2 = q1 * q2;
567 double q1q3 = q1 * q3;
568 double q2q2 = q2 * q2;
569 double q2q3 = q2 * q3;
570 double q3q3 = q3 * q3;
571
572 // create the matrix
573 double[][] m = new double[3][];
574 m[0] = new double[3];
575 m[1] = new double[3];
576 m[2] = new double[3];
577
578 m [0][0] = 2.0 * (q0q0 + q1q1) - 1.0;
579 m [1][0] = 2.0 * (q1q2 - q0q3);
580 m [2][0] = 2.0 * (q1q3 + q0q2);
581
582 m [0][1] = 2.0 * (q1q2 + q0q3);
583 m [1][1] = 2.0 * (q0q0 + q2q2) - 1.0;
584 m [2][1] = 2.0 * (q2q3 - q0q1);
585
586 m [0][2] = 2.0 * (q1q3 - q0q2);
587 m [1][2] = 2.0 * (q2q3 + q0q1);
588 m [2][2] = 2.0 * (q0q0 + q3q3) - 1.0;
589
590 return m;
591
592 }
593
594 /** Apply the rotation to a vector.
595 * @param u vector to apply the rotation to
596 * @return a new vector which is the image of u by the rotation
597 */
598 public Vector3D applyTo(Vector3D u) {
599
600 double x = u.getX();
601 double y = u.getY();
602 double z = u.getZ();
603
604 double s = q1 * x + q2 * y + q3 * z;
605
606 return new Vector3D(2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x,
607 2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y,
608 2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - z);
609
610 }
611
612 /** Apply the rotation to a vector stored in an array.
613 * @param in an array with three items which stores vector to rotate
614 * @param out an array with three items to put result to (it can be the same
615 * array as in)
616 */
617 public void applyTo(final double[] in, final double[] out) {
618
619 final double x = in[0];
620 final double y = in[1];
621 final double z = in[2];
622
623 final double s = q1 * x + q2 * y + q3 * z;
624
625 out[0] = 2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x;
626 out[1] = 2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y;
627 out[2] = 2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - z;
628
629 }
630
631 /** Apply the inverse of the rotation to a vector.
632 * @param u vector to apply the inverse of the rotation to
633 * @return a new vector which such that u is its image by the rotation
634 */
635 public Vector3D applyInverseTo(Vector3D u) {
636
637 double x = u.getX();
638 double y = u.getY();
639 double z = u.getZ();
640
641 double s = q1 * x + q2 * y + q3 * z;
642 double m0 = -q0;
643
644 return new Vector3D(2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x,
645 2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y,
646 2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - z);
647
648 }
649
650 /** Apply the inverse of the rotation to a vector stored in an array.
651 * @param in an array with three items which stores vector to rotate
652 * @param out an array with three items to put result to (it can be the same
653 * array as in)
654 */
655 public void applyInverseTo(final double[] in, final double[] out) {
656
657 final double x = in[0];
658 final double y = in[1];
659 final double z = in[2];
660
661 final double s = q1 * x + q2 * y + q3 * z;
662 final double m0 = -q0;
663
664 out[0] = 2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x;
665 out[1] = 2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y;
666 out[2] = 2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - z;
667
668 }
669
670 /** Apply the instance to another rotation.
671 * <p>
672 * Calling this method is equivalent to call
673 * {@link #compose(Rotation, RotationConvention)
674 * compose(r, RotationConvention.VECTOR_OPERATOR)}.
675 * </p>
676 * @param r rotation to apply the rotation to
677 * @return a new rotation which is the composition of r by the instance
678 */
679 public Rotation applyTo(Rotation r) {
680 return compose(r, RotationConvention.VECTOR_OPERATOR);
681 }
682
683 /** Compose the instance with another rotation.
684 * <p>
685 * If the semantics of the rotations composition corresponds to a
686 * {@link RotationConvention#VECTOR_OPERATOR vector operator} convention,
687 * applying the instance to a rotation is computing the composition
688 * in an order compliant with the following rule : let {@code u} be any
689 * vector and {@code v} its image by {@code r1} (i.e.
690 * {@code r1.applyTo(u) = v}). Let {@code w} be the image of {@code v} by
691 * rotation {@code r2} (i.e. {@code r2.applyTo(v) = w}). Then
692 * {@code w = comp.applyTo(u)}, where
693 * {@code comp = r2.compose(r1, RotationConvention.VECTOR_OPERATOR)}.
694 * </p>
695 * <p>
696 * If the semantics of the rotations composition corresponds to a
697 * {@link RotationConvention#FRAME_TRANSFORM frame transform} convention,
698 * the application order will be reversed. So keeping the exact same
699 * meaning of all {@code r1}, {@code r2}, {@code u}, {@code v}, {@code w}
700 * and {@code comp} as above, {@code comp} could also be computed as
701 * {@code comp = r1.compose(r2, RotationConvention.FRAME_TRANSFORM)}.
702 * </p>
703 * @param r rotation to apply the rotation to
704 * @param convention convention to use for the semantics of the angle
705 * @return a new rotation which is the composition of r by the instance
706 */
707 public Rotation compose(final Rotation r, final RotationConvention convention) {
708 return convention == RotationConvention.VECTOR_OPERATOR ?
709 composeInternal(r) : r.composeInternal(this);
710 }
711
712 /** Compose the instance with another rotation using vector operator convention.
713 * @param r rotation to apply the rotation to
714 * @return a new rotation which is the composition of r by the instance
715 * using vector operator convention
716 */
717 private Rotation composeInternal(final Rotation r) {
718 return new Rotation(r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),
719 r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),
720 r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),
721 r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),
722 false);
723 }
724
725 /** Apply the inverse of the instance to another rotation.
726 * <p>
727 * Calling this method is equivalent to call
728 * {@link #composeInverse(Rotation, RotationConvention)
729 * composeInverse(r, RotationConvention.VECTOR_OPERATOR)}.
730 * </p>
731 * @param r rotation to apply the rotation to
732 * @return a new rotation which is the composition of r by the inverse
733 * of the instance
734 */
735 public Rotation applyInverseTo(Rotation r) {
736 return composeInverse(r, RotationConvention.VECTOR_OPERATOR);
737 }
738
739 /** Compose the inverse of the instance with another rotation.
740 * <p>
741 * If the semantics of the rotations composition corresponds to a
742 * {@link RotationConvention#VECTOR_OPERATOR vector operator} convention,
743 * applying the inverse of the instance to a rotation is computing
744 * the composition in an order compliant with the following rule :
745 * let {@code u} be any vector and {@code v} its image by {@code r1}
746 * (i.e. {@code r1.applyTo(u) = v}). Let {@code w} be the inverse image
747 * of {@code v} by {@code r2} (i.e. {@code r2.applyInverseTo(v) = w}).
748 * Then {@code w = comp.applyTo(u)}, where
749 * {@code comp = r2.composeInverse(r1)}.
750 * </p>
751 * <p>
752 * If the semantics of the rotations composition corresponds to a
753 * {@link RotationConvention#FRAME_TRANSFORM frame transform} convention,
754 * the application order will be reversed, which means it is the
755 * <em>innermost</em> rotation that will be reversed. So keeping the exact same
756 * meaning of all {@code r1}, {@code r2}, {@code u}, {@code v}, {@code w}
757 * and {@code comp} as above, {@code comp} could also be computed as
758 * {@code comp = r1.revert().composeInverse(r2.revert(), RotationConvention.FRAME_TRANSFORM)}.
759 * </p>
760 * @param r rotation to apply the rotation to
761 * @param convention convention to use for the semantics of the angle
762 * @return a new rotation which is the composition of r by the inverse
763 * of the instance
764 */
765 public Rotation composeInverse(final Rotation r, final RotationConvention convention) {
766 return convention == RotationConvention.VECTOR_OPERATOR ?
767 composeInverseInternal(r) : r.composeInternal(revert());
768 }
769
770 /** Compose the inverse of the instance with another rotation
771 * using vector operator convention.
772 * @param r rotation to apply the rotation to
773 * @return a new rotation which is the composition of r by the inverse
774 * of the instance using vector operator convention
775 */
776 private Rotation composeInverseInternal(Rotation r) {
777 return new Rotation(-r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),
778 -r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),
779 -r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),
780 -r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),
781 false);
782 }
783
784 /** Perfect orthogonality on a 3X3 matrix.
785 * @param m initial matrix (not exactly orthogonal)
786 * @param threshold convergence threshold for the iterative
787 * orthogonality correction (convergence is reached when the
788 * difference between two steps of the Frobenius norm of the
789 * correction is below this threshold)
790 * @return an orthogonal matrix close to m
791 * @exception MathIllegalArgumentException if the matrix cannot be
792 * orthogonalized with the given threshold after 10 iterations
793 */
794 private double[][] orthogonalizeMatrix(double[][] m, double threshold)
795 throws MathIllegalArgumentException {
796 double[] m0 = m[0];
797 double[] m1 = m[1];
798 double[] m2 = m[2];
799 double x00 = m0[0];
800 double x01 = m0[1];
801 double x02 = m0[2];
802 double x10 = m1[0];
803 double x11 = m1[1];
804 double x12 = m1[2];
805 double x20 = m2[0];
806 double x21 = m2[1];
807 double x22 = m2[2];
808 double fn = 0;
809 double fn1;
810
811 double[][] o = new double[3][3];
812 double[] o0 = o[0];
813 double[] o1 = o[1];
814 double[] o2 = o[2];
815
816 // iterative correction: Xn+1 = Xn - 0.5 * (Xn.Mt.Xn - M)
817 int i;
818 for (i = 0; i < 11; ++i) {
819
820 // Mt.Xn
821 double mx00 = m0[0] * x00 + m1[0] * x10 + m2[0] * x20;
822 double mx10 = m0[1] * x00 + m1[1] * x10 + m2[1] * x20;
823 double mx20 = m0[2] * x00 + m1[2] * x10 + m2[2] * x20;
824 double mx01 = m0[0] * x01 + m1[0] * x11 + m2[0] * x21;
825 double mx11 = m0[1] * x01 + m1[1] * x11 + m2[1] * x21;
826 double mx21 = m0[2] * x01 + m1[2] * x11 + m2[2] * x21;
827 double mx02 = m0[0] * x02 + m1[0] * x12 + m2[0] * x22;
828 double mx12 = m0[1] * x02 + m1[1] * x12 + m2[1] * x22;
829 double mx22 = m0[2] * x02 + m1[2] * x12 + m2[2] * x22;
830
831 // Xn+1
832 o0[0] = x00 - 0.5 * (x00 * mx00 + x01 * mx10 + x02 * mx20 - m0[0]);
833 o0[1] = x01 - 0.5 * (x00 * mx01 + x01 * mx11 + x02 * mx21 - m0[1]);
834 o0[2] = x02 - 0.5 * (x00 * mx02 + x01 * mx12 + x02 * mx22 - m0[2]);
835 o1[0] = x10 - 0.5 * (x10 * mx00 + x11 * mx10 + x12 * mx20 - m1[0]);
836 o1[1] = x11 - 0.5 * (x10 * mx01 + x11 * mx11 + x12 * mx21 - m1[1]);
837 o1[2] = x12 - 0.5 * (x10 * mx02 + x11 * mx12 + x12 * mx22 - m1[2]);
838 o2[0] = x20 - 0.5 * (x20 * mx00 + x21 * mx10 + x22 * mx20 - m2[0]);
839 o2[1] = x21 - 0.5 * (x20 * mx01 + x21 * mx11 + x22 * mx21 - m2[1]);
840 o2[2] = x22 - 0.5 * (x20 * mx02 + x21 * mx12 + x22 * mx22 - m2[2]);
841
842 // correction on each elements
843 double corr00 = o0[0] - m0[0];
844 double corr01 = o0[1] - m0[1];
845 double corr02 = o0[2] - m0[2];
846 double corr10 = o1[0] - m1[0];
847 double corr11 = o1[1] - m1[1];
848 double corr12 = o1[2] - m1[2];
849 double corr20 = o2[0] - m2[0];
850 double corr21 = o2[1] - m2[1];
851 double corr22 = o2[2] - m2[2];
852
853 // Frobenius norm of the correction
854 fn1 = corr00 * corr00 + corr01 * corr01 + corr02 * corr02 +
855 corr10 * corr10 + corr11 * corr11 + corr12 * corr12 +
856 corr20 * corr20 + corr21 * corr21 + corr22 * corr22;
857
858 // convergence test
859 if (FastMath.abs(fn1 - fn) <= threshold) {
860 return o;
861 }
862
863 // prepare next iteration
864 x00 = o0[0];
865 x01 = o0[1];
866 x02 = o0[2];
867 x10 = o1[0];
868 x11 = o1[1];
869 x12 = o1[2];
870 x20 = o2[0];
871 x21 = o2[1];
872 x22 = o2[2];
873 fn = fn1;
874
875 }
876
877 // the algorithm did not converge after 10 iterations
878 throw new MathIllegalArgumentException(LocalizedGeometryFormats.UNABLE_TO_ORTHOGONOLIZE_MATRIX,
879 i - 1);
880 }
881
882 /** Compute the <i>distance</i> between two rotations.
883 * <p>The <i>distance</i> is intended here as a way to check if two
884 * rotations are almost similar (i.e. they transform vectors the same way)
885 * or very different. It is mathematically defined as the angle of
886 * the rotation r that prepended to one of the rotations gives the other
887 * one: \(r_1(r) = r_2\)
888 * </p>
889 * <p>This distance is an angle between 0 and π. Its value is the smallest
890 * possible upper bound of the angle in radians between r<sub>1</sub>(v)
891 * and r<sub>2</sub>(v) for all possible vectors v. This upper bound is
892 * reached for some v. The distance is equal to 0 if and only if the two
893 * rotations are identical.</p>
894 * <p>Comparing two rotations should always be done using this value rather
895 * than for example comparing the components of the quaternions. It is much
896 * more stable, and has a geometric meaning. Also comparing quaternions
897 * components is error prone since for example quaternions (0.36, 0.48, -0.48, -0.64)
898 * and (-0.36, -0.48, 0.48, 0.64) represent exactly the same rotation despite
899 * their components are different (they are exact opposites).</p>
900 * @param r1 first rotation
901 * @param r2 second rotation
902 * @return <i>distance</i> between r1 and r2
903 */
904 public static double distance(Rotation r1, Rotation r2) {
905 return r1.composeInverseInternal(r2).getAngle();
906 }
907
908 }