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 * http://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 package org.apache.commons.math.geometry;
19
20 import java.io.Serializable;
21
22 /**
23 * This class implements rotations in a three-dimensional space.
24 *
25 * <p>Rotations can be represented by several different mathematical
26 * entities (matrices, axe and angle, Cardan or Euler angles,
27 * quaternions). This class presents an higher level abstraction, more
28 * user-oriented and hiding this implementation details. Well, for the
29 * curious, we use quaternions for the internal representation. The
30 * user can build a rotation from any of these representations, and
31 * any of these representations can be retrieved from a
32 * <code>Rotation</code> instance (see the various constructors and
33 * getters). In addition, a rotation can also be built implicitely
34 * from a set of vectors and their image.</p>
35 * <p>This implies that this class can be used to convert from one
36 * representation to another one. For example, converting a rotation
37 * matrix into a set of Cardan angles from can be done using the
38 * followong single line of code:</p>
39 * <pre>
40 * double[] angles = new Rotation(matrix, 1.0e-10).getAngles(RotationOrder.XYZ);
41 * </pre>
42 * <p>Focus is oriented on what a rotation <em>do</em> rather than on its
43 * underlying representation. Once it has been built, and regardless of its
44 * internal representation, a rotation is an <em>operator</em> which basically
45 * transforms three dimensional {@link Vector3D vectors} into other three
46 * dimensional {@link Vector3D vectors}. Depending on the application, the
47 * meaning of these vectors may vary and the semantics of the rotation also.</p>
48 * <p>For example in an spacecraft attitude simulation tool, users will often
49 * consider the vectors are fixed (say the Earth direction for example) and the
50 * rotation transforms the coordinates coordinates of this vector in inertial
51 * frame into the coordinates of the same vector in satellite frame. In this
52 * case, the rotation implicitely defines the relation between the two frames.
53 * Another example could be a telescope control application, where the rotation
54 * would transform the sighting direction at rest into the desired observing
55 * direction when the telescope is pointed towards an object of interest. In this
56 * case the rotation transforms the directionf at rest in a topocentric frame
57 * into the sighting direction in the same topocentric frame. In many case, both
58 * approaches will be combined, in our telescope example, we will probably also
59 * need to transform the observing direction in the topocentric frame into the
60 * observing direction in inertial frame taking into account the observatory
61 * location and the Earth rotation.</p>
62 *
63 * <p>These examples show that a rotation is what the user wants it to be, so this
64 * class does not push the user towards one specific definition and hence does not
65 * provide methods like <code>projectVectorIntoDestinationFrame</code> or
66 * <code>computeTransformedDirection</code>. It provides simpler and more generic
67 * methods: {@link #applyTo(Vector3D) applyTo(Vector3D)} and {@link
68 * #applyInverseTo(Vector3D) applyInverseTo(Vector3D)}.</p>
69 *
70 * <p>Since a rotation is basically a vectorial operator, several rotations can be
71 * composed together and the composite operation <code>r = r<sub>1</sub> o
72 * r<sub>2</sub></code> (which means that for each vector <code>u</code>,
73 * <code>r(u) = r<sub>1</sub>(r<sub>2</sub>(u))</code>) is also a rotation. Hence
74 * we can consider that in addition to vectors, a rotation can be applied to other
75 * rotations as well (or to itself). With our previous notations, we would say we
76 * can apply <code>r<sub>1</sub></code> to <code>r<sub>2</sub></code> and the result
77 * we get is <code>r = r<sub>1</sub> o r<sub>2</sub></code>. For this purpose, the
78 * class provides the methods: {@link #applyTo(Rotation) applyTo(Rotation)} and
79 * {@link #applyInverseTo(Rotation) applyInverseTo(Rotation)}.</p>
80 *
81 * <p>Rotations are guaranteed to be immutable objects.</p>
82 *
83 * @version $Revision: 627994 $ $Date: 2008-02-15 03:16:05 -0700 (Fri, 15 Feb 2008) $
84 * @see Vector3D
85 * @see RotationOrder
86 * @since 1.2
87 */
88
89 public class Rotation implements Serializable {
90
91 /** Build the identity rotation.
92 */
93 public Rotation() {
94 q0 = 1;
95 q1 = 0;
96 q2 = 0;
97 q3 = 0;
98 }
99
100 /** Build a rotation from the quaternion coordinates.
101 * <p>A rotation can be built from a <em>normalized</em> quaternion,
102 * i.e. a quaternion for which q<sub>0</sub><sup>2</sup> +
103 * q<sub>1</sub><sup>2</sup> + q<sub>2</sub><sup>2</sup> +
104 * q<sub>3</sub><sup>2</sup> = 1. If the quaternion is not normalized,
105 * the constructor can normalize it in a preprocessing step.</p>
106 * @param q0 scalar part of the quaternion
107 * @param q1 first coordinate of the vectorial part of the quaternion
108 * @param q2 second coordinate of the vectorial part of the quaternion
109 * @param q3 third coordinate of the vectorial part of the quaternion
110 * @param needsNormalization if true, the coordinates are considered
111 * not to be normalized, a normalization preprocessing step is performed
112 * before using them
113 */
114 public Rotation(double q0, double q1, double q2, double q3,
115 boolean needsNormalization) {
116
117 if (needsNormalization) {
118 // normalization preprocessing
119 double inv = 1.0 / Math.sqrt(q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3);
120 q0 *= inv;
121 q1 *= inv;
122 q2 *= inv;
123 q3 *= inv;
124 }
125
126 this.q0 = q0;
127 this.q1 = q1;
128 this.q2 = q2;
129 this.q3 = q3;
130
131 }
132
133 /** Build a rotation from an axis and an angle.
134 * <p>We use the convention that angles are oriented according to
135 * the effect of the rotation on vectors around the axis. That means
136 * that if (i, j, k) is a direct frame and if we first provide +k as
137 * the axis and PI/2 as the angle to this constructor, and then
138 * {@link #applyTo(Vector3D) apply} the instance to +i, we will get
139 * +j.</p>
140 * @param axis axis around which to rotate
141 * @param angle rotation angle.
142 * @exception ArithmeticException if the axis norm is zero
143 */
144 public Rotation(Vector3D axis, double angle) {
145
146 double norm = axis.getNorm();
147 if (norm == 0) {
148 throw new ArithmeticException("zero norm for rotation axis");
149 }
150
151 double halfAngle = -0.5 * angle;
152 double coeff = Math.sin(halfAngle) / norm;
153
154 q0 = Math.cos (halfAngle);
155 q1 = coeff * axis.getX();
156 q2 = coeff * axis.getY();
157 q3 = coeff * axis.getZ();
158
159 }
160
161 /** Build a rotation from a 3X3 matrix.
162
163 * <p>Rotation matrices are orthogonal matrices, i.e. unit matrices
164 * (which are matrices for which m.m<sup>T</sup> = I) with real
165 * coefficients. The module of the determinant of unit matrices is
166 * 1, among the orthogonal 3X3 matrices, only the ones having a
167 * positive determinant (+1) are rotation matrices.</p>
168
169 * <p>When a rotation is defined by a matrix with truncated values
170 * (typically when it is extracted from a technical sheet where only
171 * four to five significant digits are available), the matrix is not
172 * orthogonal anymore. This constructor handles this case
173 * transparently by using a copy of the given matrix and applying a
174 * correction to the copy in order to perfect its orthogonality. If
175 * the Frobenius norm of the correction needed is above the given
176 * threshold, then the matrix is considered to be too far from a
177 * true rotation matrix and an exception is thrown.<p>
178
179 * @param m rotation matrix
180 * @param threshold convergence threshold for the iterative
181 * orthogonality correction (convergence is reached when the
182 * difference between two steps of the Frobenius norm of the
183 * correction is below this threshold)
184
185 * @exception NotARotationMatrixException if the matrix is not a 3X3
186 * matrix, or if it cannot be transformed into an orthogonal matrix
187 * with the given threshold, or if the determinant of the resulting
188 * orthogonal matrix is negative
189
190 */
191 public Rotation(double[][] m, double threshold)
192 throws NotARotationMatrixException {
193
194 // dimension check
195 if ((m.length != 3) || (m[0].length != 3) ||
196 (m[1].length != 3) || (m[2].length != 3)) {
197 throw new NotARotationMatrixException("a {0}x{1} matrix" +
198 " cannot be a rotation matrix",
199 new Object[] {
200 Integer.toString(m.length),
201 Integer.toString(m[0].length)
202 });
203 }
204
205 // compute a "close" orthogonal matrix
206 double[][] ort = orthogonalizeMatrix(m, threshold);
207
208 // check the sign of the determinant
209 double det = ort[0][0] * (ort[1][1] * ort[2][2] - ort[2][1] * ort[1][2]) -
210 ort[1][0] * (ort[0][1] * ort[2][2] - ort[2][1] * ort[0][2]) +
211 ort[2][0] * (ort[0][1] * ort[1][2] - ort[1][1] * ort[0][2]);
212 if (det < 0.0) {
213 throw new NotARotationMatrixException("the closest orthogonal matrix" +
214 " has a negative determinant {0}",
215 new Object[] {
216 Double.toString(det)
217 });
218 }
219
220 // There are different ways to compute the quaternions elements
221 // from the matrix. They all involve computing one element from
222 // the diagonal of the matrix, and computing the three other ones
223 // using a formula involving a division by the first element,
224 // which unfortunately can be zero. Since the norm of the
225 // quaternion is 1, we know at least one element has an absolute
226 // value greater or equal to 0.5, so it is always possible to
227 // select the right formula and avoid division by zero and even
228 // numerical inaccuracy. Checking the elements in turn and using
229 // the first one greater than 0.45 is safe (this leads to a simple
230 // test since qi = 0.45 implies 4 qi^2 - 1 = -0.19)
231 double s = ort[0][0] + ort[1][1] + ort[2][2];
232 if (s > -0.19) {
233 // compute q0 and deduce q1, q2 and q3
234 q0 = 0.5 * Math.sqrt(s + 1.0);
235 double inv = 0.25 / q0;
236 q1 = inv * (ort[1][2] - ort[2][1]);
237 q2 = inv * (ort[2][0] - ort[0][2]);
238 q3 = inv * (ort[0][1] - ort[1][0]);
239 } else {
240 s = ort[0][0] - ort[1][1] - ort[2][2];
241 if (s > -0.19) {
242 // compute q1 and deduce q0, q2 and q3
243 q1 = 0.5 * Math.sqrt(s + 1.0);
244 double inv = 0.25 / q1;
245 q0 = inv * (ort[1][2] - ort[2][1]);
246 q2 = inv * (ort[0][1] + ort[1][0]);
247 q3 = inv * (ort[0][2] + ort[2][0]);
248 } else {
249 s = ort[1][1] - ort[0][0] - ort[2][2];
250 if (s > -0.19) {
251 // compute q2 and deduce q0, q1 and q3
252 q2 = 0.5 * Math.sqrt(s + 1.0);
253 double inv = 0.25 / q2;
254 q0 = inv * (ort[2][0] - ort[0][2]);
255 q1 = inv * (ort[0][1] + ort[1][0]);
256 q3 = inv * (ort[2][1] + ort[1][2]);
257 } else {
258 // compute q3 and deduce q0, q1 and q2
259 s = ort[2][2] - ort[0][0] - ort[1][1];
260 q3 = 0.5 * Math.sqrt(s + 1.0);
261 double inv = 0.25 / q3;
262 q0 = inv * (ort[0][1] - ort[1][0]);
263 q1 = inv * (ort[0][2] + ort[2][0]);
264 q2 = inv * (ort[2][1] + ort[1][2]);
265 }
266 }
267 }
268
269 }
270
271 /** Build the rotation that transforms a pair of vector into another pair.
272
273 * <p>Except for possible scale factors, if the instance were applied to
274 * the pair (u<sub>1</sub>, u<sub>2</sub>) it will produce the pair
275 * (v<sub>1</sub>, v<sub>2</sub>).</p>
276
277 * <p>If the angular separation between u<sub>1</sub> and u<sub>2</sub> is
278 * not the same as the angular separation between v<sub>1</sub> and
279 * v<sub>2</sub>, then a corrected v'<sub>2</sub> will be used rather than
280 * v<sub>2</sub>, the corrected vector will be in the (v<sub>1</sub>,
281 * v<sub>2</sub>) plane.</p>
282
283 * @param u1 first vector of the origin pair
284 * @param u2 second vector of the origin pair
285 * @param v1 desired image of u1 by the rotation
286 * @param v2 desired image of u2 by the rotation
287 * @exception IllegalArgumentException if the norm of one of the vectors is zero
288 */
289 public Rotation(Vector3D u1, Vector3D u2, Vector3D v1, Vector3D v2) {
290
291 // norms computation
292 double u1u1 = Vector3D.dotProduct(u1, u1);
293 double u2u2 = Vector3D.dotProduct(u2, u2);
294 double v1v1 = Vector3D.dotProduct(v1, v1);
295 double v2v2 = Vector3D.dotProduct(v2, v2);
296 if ((u1u1 == 0) || (u2u2 == 0) || (v1v1 == 0) || (v2v2 == 0)) {
297 throw new IllegalArgumentException("zero norm for rotation defining vector");
298 }
299
300 double u1x = u1.getX();
301 double u1y = u1.getY();
302 double u1z = u1.getZ();
303
304 double u2x = u2.getX();
305 double u2y = u2.getY();
306 double u2z = u2.getZ();
307
308 // normalize v1 in order to have (v1'|v1') = (u1|u1)
309 double coeff = Math.sqrt (u1u1 / v1v1);
310 double v1x = coeff * v1.getX();
311 double v1y = coeff * v1.getY();
312 double v1z = coeff * v1.getZ();
313 v1 = new Vector3D(v1x, v1y, v1z);
314
315 // adjust v2 in order to have (u1|u2) = (v1|v2) and (v2'|v2') = (u2|u2)
316 double u1u2 = Vector3D.dotProduct(u1, u2);
317 double v1v2 = Vector3D.dotProduct(v1, v2);
318 double coeffU = u1u2 / u1u1;
319 double coeffV = v1v2 / u1u1;
320 double beta = Math.sqrt((u2u2 - u1u2 * coeffU) / (v2v2 - v1v2 * coeffV));
321 double alpha = coeffU - beta * coeffV;
322 double v2x = alpha * v1x + beta * v2.getX();
323 double v2y = alpha * v1y + beta * v2.getY();
324 double v2z = alpha * v1z + beta * v2.getZ();
325 v2 = new Vector3D(v2x, v2y, v2z);
326
327 // preliminary computation (we use explicit formulation instead
328 // of relying on the Vector3D class in order to avoid building lots
329 // of temporary objects)
330 Vector3D uRef = u1;
331 Vector3D vRef = v1;
332 double dx1 = v1x - u1.getX();
333 double dy1 = v1y - u1.getY();
334 double dz1 = v1z - u1.getZ();
335 double dx2 = v2x - u2.getX();
336 double dy2 = v2y - u2.getY();
337 double dz2 = v2z - u2.getZ();
338 Vector3D k = new Vector3D(dy1 * dz2 - dz1 * dy2,
339 dz1 * dx2 - dx1 * dz2,
340 dx1 * dy2 - dy1 * dx2);
341 double c = k.getX() * (u1y * u2z - u1z * u2y) +
342 k.getY() * (u1z * u2x - u1x * u2z) +
343 k.getZ() * (u1x * u2y - u1y * u2x);
344
345 if (c == 0) {
346 // the (q1, q2, q3) vector is in the (u1, u2) plane
347 // we try other vectors
348 Vector3D u3 = Vector3D.crossProduct(u1, u2);
349 Vector3D v3 = Vector3D.crossProduct(v1, v2);
350 double u3x = u3.getX();
351 double u3y = u3.getY();
352 double u3z = u3.getZ();
353 double v3x = v3.getX();
354 double v3y = v3.getY();
355 double v3z = v3.getZ();
356
357 double dx3 = v3x - u3x;
358 double dy3 = v3y - u3y;
359 double dz3 = v3z - u3z;
360 k = new Vector3D(dy1 * dz3 - dz1 * dy3,
361 dz1 * dx3 - dx1 * dz3,
362 dx1 * dy3 - dy1 * dx3);
363 c = k.getX() * (u1y * u3z - u1z * u3y) +
364 k.getY() * (u1z * u3x - u1x * u3z) +
365 k.getZ() * (u1x * u3y - u1y * u3x);
366
367 if (c == 0) {
368 // the (q1, q2, q3) vector is aligned with u1:
369 // we try (u2, u3) and (v2, v3)
370 k = new Vector3D(dy2 * dz3 - dz2 * dy3,
371 dz2 * dx3 - dx2 * dz3,
372 dx2 * dy3 - dy2 * dx3);
373 c = k.getX() * (u2y * u3z - u2z * u3y) +
374 k.getY() * (u2z * u3x - u2x * u3z) +
375 k.getZ() * (u2x * u3y - u2y * u3x);
376
377 if (c == 0) {
378 // the (q1, q2, q3) vector is aligned with everything
379 // this is really the identity rotation
380 q0 = 1.0;
381 q1 = 0.0;
382 q2 = 0.0;
383 q3 = 0.0;
384 return;
385 }
386
387 // we will have to use u2 and v2 to compute the scalar part
388 uRef = u2;
389 vRef = v2;
390
391 }
392
393 }
394
395 // compute the vectorial part
396 c = Math.sqrt(c);
397 double inv = 1.0 / (c + c);
398 q1 = inv * k.getX();
399 q2 = inv * k.getY();
400 q3 = inv * k.getZ();
401
402 // compute the scalar part
403 k = new Vector3D(uRef.getY() * q3 - uRef.getZ() * q2,
404 uRef.getZ() * q1 - uRef.getX() * q3,
405 uRef.getX() * q2 - uRef.getY() * q1);
406 c = Vector3D.dotProduct(k, k);
407 q0 = Vector3D.dotProduct(vRef, k) / (c + c);
408
409 }
410
411 /** Build one of the rotations that transform one vector into another one.
412
413 * <p>Except for a possible scale factor, if the instance were
414 * applied to the vector u it will produce the vector v. There is an
415 * infinite number of such rotations, this constructor choose the
416 * one with the smallest associated angle (i.e. the one whose axis
417 * is orthogonal to the (u, v) plane). If u and v are colinear, an
418 * arbitrary rotation axis is chosen.</p>
419
420 * @param u origin vector
421 * @param v desired image of u by the rotation
422 * @exception IllegalArgumentException if the norm of one of the vectors is zero
423 */
424 public Rotation(Vector3D u, Vector3D v) {
425
426 double normProduct = u.getNorm() * v.getNorm();
427 if (normProduct == 0) {
428 throw new IllegalArgumentException("zero norm for rotation defining vector");
429 }
430
431 double dot = Vector3D.dotProduct(u, v);
432
433 if (dot < ((2.0e-15 - 1.0) * normProduct)) {
434 // special case u = -v: we select a PI angle rotation around
435 // an arbitrary vector orthogonal to u
436 Vector3D w = u.orthogonal();
437 q0 = 0.0;
438 q1 = -w.getX();
439 q2 = -w.getY();
440 q3 = -w.getZ();
441 } else {
442 // general case: (u, v) defines a plane, we select
443 // the shortest possible rotation: axis orthogonal to this plane
444 q0 = Math.sqrt(0.5 * (1.0 + dot / normProduct));
445 double coeff = 1.0 / (2.0 * q0 * normProduct);
446 q1 = coeff * (v.getY() * u.getZ() - v.getZ() * u.getY());
447 q2 = coeff * (v.getZ() * u.getX() - v.getX() * u.getZ());
448 q3 = coeff * (v.getX() * u.getY() - v.getY() * u.getX());
449 }
450
451 }
452
453 /** Build a rotation from three Cardan or Euler elementary rotations.
454
455 * <p>Cardan rotations are three successive rotations around the
456 * canonical axes X, Y and Z, each axis beeing used once. There are
457 * 6 such sets of rotations (XYZ, XZY, YXZ, YZX, ZXY and ZYX). Euler
458 * rotations are three successive rotations around the canonical
459 * axes X, Y and Z, the first and last rotations beeing around the
460 * same axis. There are 6 such sets of rotations (XYX, XZX, YXY,
461 * YZY, ZXZ and ZYZ), the most popular one being ZXZ.</p>
462 * <p>Beware that many people routinely use the term Euler angles even
463 * for what really are Cardan angles (this confusion is especially
464 * widespread in the aerospace business where Roll, Pitch and Yaw angles
465 * are often wrongly tagged as Euler angles).</p>
466
467 * @param order order of rotations to use
468 * @param alpha1 angle of the first elementary rotation
469 * @param alpha2 angle of the second elementary rotation
470 * @param alpha3 angle of the third elementary rotation
471 */
472 public Rotation(RotationOrder order,
473 double alpha1, double alpha2, double alpha3) {
474 Rotation r1 = new Rotation(order.getA1(), alpha1);
475 Rotation r2 = new Rotation(order.getA2(), alpha2);
476 Rotation r3 = new Rotation(order.getA3(), alpha3);
477 Rotation composed = r1.applyTo(r2.applyTo(r3));
478 q0 = composed.q0;
479 q1 = composed.q1;
480 q2 = composed.q2;
481 q3 = composed.q3;
482 }
483
484 /** Revert a rotation.
485 * Build a rotation which reverse the effect of another
486 * rotation. This means that if r(u) = v, then r.revert(v) = u. The
487 * instance is not changed.
488 * @return a new rotation whose effect is the reverse of the effect
489 * of the instance
490 */
491 public Rotation revert() {
492 return new Rotation(-q0, q1, q2, q3, false);
493 }
494
495 /** Get the scalar coordinate of the quaternion.
496 * @return scalar coordinate of the quaternion
497 */
498 public double getQ0() {
499 return q0;
500 }
501
502 /** Get the first coordinate of the vectorial part of the quaternion.
503 * @return first coordinate of the vectorial part of the quaternion
504 */
505 public double getQ1() {
506 return q1;
507 }
508
509 /** Get the second coordinate of the vectorial part of the quaternion.
510 * @return second coordinate of the vectorial part of the quaternion
511 */
512 public double getQ2() {
513 return q2;
514 }
515
516 /** Get the third coordinate of the vectorial part of the quaternion.
517 * @return third coordinate of the vectorial part of the quaternion
518 */
519 public double getQ3() {
520 return q3;
521 }
522
523 /** Get the normalized axis of the rotation.
524 * @return normalized axis of the rotation
525 */
526 public Vector3D getAxis() {
527 double squaredSine = q1 * q1 + q2 * q2 + q3 * q3;
528 if (squaredSine == 0) {
529 return new Vector3D(1, 0, 0);
530 } else if (q0 < 0) {
531 double inverse = 1 / Math.sqrt(squaredSine);
532 return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
533 }
534 double inverse = -1 / Math.sqrt(squaredSine);
535 return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
536 }
537
538 /** Get the angle of the rotation.
539 * @return angle of the rotation (between 0 and π)
540 */
541 public double getAngle() {
542 if ((q0 < -0.1) || (q0 > 0.1)) {
543 return 2 * Math.asin(Math.sqrt(q1 * q1 + q2 * q2 + q3 * q3));
544 } else if (q0 < 0) {
545 return 2 * Math.acos(-q0);
546 }
547 return 2 * Math.acos(q0);
548 }
549
550 /** Get the Cardan or Euler angles corresponding to the instance.
551
552 * <p>The equations show that each rotation can be defined by two
553 * different values of the Cardan or Euler angles set. For example
554 * if Cardan angles are used, the rotation defined by the angles
555 * a<sub>1</sub>, a<sub>2</sub> and a<sub>3</sub> is the same as
556 * the rotation defined by the angles π + a<sub>1</sub>, π
557 * - a<sub>2</sub> and π + a<sub>3</sub>. This method implements
558 * the following arbitrary choices:</p>
559 * <ul>
560 * <li>for Cardan angles, the chosen set is the one for which the
561 * second angle is between -π/2 and π/2 (i.e its cosine is
562 * positive),</li>
563 * <li>for Euler angles, the chosen set is the one for which the
564 * second angle is between 0 and π (i.e its sine is positive).</li>
565 * </ul>
566
567 * <p>Cardan and Euler angle have a very disappointing drawback: all
568 * of them have singularities. This means that if the instance is
569 * too close to the singularities corresponding to the given
570 * rotation order, it will be impossible to retrieve the angles. For
571 * Cardan angles, this is often called gimbal lock. There is
572 * <em>nothing</em> to do to prevent this, it is an intrinsic problem
573 * with Cardan and Euler representation (but not a problem with the
574 * rotation itself, which is perfectly well defined). For Cardan
575 * angles, singularities occur when the second angle is close to
576 * -π/2 or +π/2, for Euler angle singularities occur when the
577 * second angle is close to 0 or π, this implies that the identity
578 * rotation is always singular for Euler angles!</p>
579
580 * @param order rotation order to use
581 * @return an array of three angles, in the order specified by the set
582 * @exception CardanEulerSingularityException if the rotation is
583 * singular with respect to the angles set specified
584 */
585 public double[] getAngles(RotationOrder order)
586 throws CardanEulerSingularityException {
587
588 if (order == RotationOrder.XYZ) {
589
590 // r (Vector3D.plusK) coordinates are :
591 // sin (theta), -cos (theta) sin (phi), cos (theta) cos (phi)
592 // (-r) (Vector3D.plusI) coordinates are :
593 // cos (psi) cos (theta), -sin (psi) cos (theta), sin (theta)
594 // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
595 Vector3D v1 = applyTo(Vector3D.plusK);
596 Vector3D v2 = applyInverseTo(Vector3D.plusI);
597 if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
598 throw new CardanEulerSingularityException(true);
599 }
600 return new double[] {
601 Math.atan2(-(v1.getY()), v1.getZ()),
602 Math.asin(v2.getZ()),
603 Math.atan2(-(v2.getY()), v2.getX())
604 };
605
606 } else if (order == RotationOrder.XZY) {
607
608 // r (Vector3D.plusJ) coordinates are :
609 // -sin (psi), cos (psi) cos (phi), cos (psi) sin (phi)
610 // (-r) (Vector3D.plusI) coordinates are :
611 // cos (theta) cos (psi), -sin (psi), sin (theta) cos (psi)
612 // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
613 Vector3D v1 = applyTo(Vector3D.plusJ);
614 Vector3D v2 = applyInverseTo(Vector3D.plusI);
615 if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
616 throw new CardanEulerSingularityException(true);
617 }
618 return new double[] {
619 Math.atan2(v1.getZ(), v1.getY()),
620 -Math.asin(v2.getY()),
621 Math.atan2(v2.getZ(), v2.getX())
622 };
623
624 } else if (order == RotationOrder.YXZ) {
625
626 // r (Vector3D.plusK) coordinates are :
627 // cos (phi) sin (theta), -sin (phi), cos (phi) cos (theta)
628 // (-r) (Vector3D.plusJ) coordinates are :
629 // sin (psi) cos (phi), cos (psi) cos (phi), -sin (phi)
630 // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
631 Vector3D v1 = applyTo(Vector3D.plusK);
632 Vector3D v2 = applyInverseTo(Vector3D.plusJ);
633 if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
634 throw new CardanEulerSingularityException(true);
635 }
636 return new double[] {
637 Math.atan2(v1.getX(), v1.getZ()),
638 -Math.asin(v2.getZ()),
639 Math.atan2(v2.getX(), v2.getY())
640 };
641
642 } else if (order == RotationOrder.YZX) {
643
644 // r (Vector3D.plusI) coordinates are :
645 // cos (psi) cos (theta), sin (psi), -cos (psi) sin (theta)
646 // (-r) (Vector3D.plusJ) coordinates are :
647 // sin (psi), cos (phi) cos (psi), -sin (phi) cos (psi)
648 // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
649 Vector3D v1 = applyTo(Vector3D.plusI);
650 Vector3D v2 = applyInverseTo(Vector3D.plusJ);
651 if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
652 throw new CardanEulerSingularityException(true);
653 }
654 return new double[] {
655 Math.atan2(-(v1.getZ()), v1.getX()),
656 Math.asin(v2.getX()),
657 Math.atan2(-(v2.getZ()), v2.getY())
658 };
659
660 } else if (order == RotationOrder.ZXY) {
661
662 // r (Vector3D.plusJ) coordinates are :
663 // -cos (phi) sin (psi), cos (phi) cos (psi), sin (phi)
664 // (-r) (Vector3D.plusK) coordinates are :
665 // -sin (theta) cos (phi), sin (phi), cos (theta) cos (phi)
666 // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
667 Vector3D v1 = applyTo(Vector3D.plusJ);
668 Vector3D v2 = applyInverseTo(Vector3D.plusK);
669 if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
670 throw new CardanEulerSingularityException(true);
671 }
672 return new double[] {
673 Math.atan2(-(v1.getX()), v1.getY()),
674 Math.asin(v2.getY()),
675 Math.atan2(-(v2.getX()), v2.getZ())
676 };
677
678 } else if (order == RotationOrder.ZYX) {
679
680 // r (Vector3D.plusI) coordinates are :
681 // cos (theta) cos (psi), cos (theta) sin (psi), -sin (theta)
682 // (-r) (Vector3D.plusK) coordinates are :
683 // -sin (theta), sin (phi) cos (theta), cos (phi) cos (theta)
684 // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
685 Vector3D v1 = applyTo(Vector3D.plusI);
686 Vector3D v2 = applyInverseTo(Vector3D.plusK);
687 if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
688 throw new CardanEulerSingularityException(true);
689 }
690 return new double[] {
691 Math.atan2(v1.getY(), v1.getX()),
692 -Math.asin(v2.getX()),
693 Math.atan2(v2.getY(), v2.getZ())
694 };
695
696 } else if (order == RotationOrder.XYX) {
697
698 // r (Vector3D.plusI) coordinates are :
699 // cos (theta), sin (phi1) sin (theta), -cos (phi1) sin (theta)
700 // (-r) (Vector3D.plusI) coordinates are :
701 // cos (theta), sin (theta) sin (phi2), sin (theta) cos (phi2)
702 // and we can choose to have theta in the interval [0 ; PI]
703 Vector3D v1 = applyTo(Vector3D.plusI);
704 Vector3D v2 = applyInverseTo(Vector3D.plusI);
705 if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
706 throw new CardanEulerSingularityException(false);
707 }
708 return new double[] {
709 Math.atan2(v1.getY(), -v1.getZ()),
710 Math.acos(v2.getX()),
711 Math.atan2(v2.getY(), v2.getZ())
712 };
713
714 } else if (order == RotationOrder.XZX) {
715
716 // r (Vector3D.plusI) coordinates are :
717 // cos (psi), cos (phi1) sin (psi), sin (phi1) sin (psi)
718 // (-r) (Vector3D.plusI) coordinates are :
719 // cos (psi), -sin (psi) cos (phi2), sin (psi) sin (phi2)
720 // and we can choose to have psi in the interval [0 ; PI]
721 Vector3D v1 = applyTo(Vector3D.plusI);
722 Vector3D v2 = applyInverseTo(Vector3D.plusI);
723 if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
724 throw new CardanEulerSingularityException(false);
725 }
726 return new double[] {
727 Math.atan2(v1.getZ(), v1.getY()),
728 Math.acos(v2.getX()),
729 Math.atan2(v2.getZ(), -v2.getY())
730 };
731
732 } else if (order == RotationOrder.YXY) {
733
734 // r (Vector3D.plusJ) coordinates are :
735 // sin (theta1) sin (phi), cos (phi), cos (theta1) sin (phi)
736 // (-r) (Vector3D.plusJ) coordinates are :
737 // sin (phi) sin (theta2), cos (phi), -sin (phi) cos (theta2)
738 // and we can choose to have phi in the interval [0 ; PI]
739 Vector3D v1 = applyTo(Vector3D.plusJ);
740 Vector3D v2 = applyInverseTo(Vector3D.plusJ);
741 if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
742 throw new CardanEulerSingularityException(false);
743 }
744 return new double[] {
745 Math.atan2(v1.getX(), v1.getZ()),
746 Math.acos(v2.getY()),
747 Math.atan2(v2.getX(), -v2.getZ())
748 };
749
750 } else if (order == RotationOrder.YZY) {
751
752 // r (Vector3D.plusJ) coordinates are :
753 // -cos (theta1) sin (psi), cos (psi), sin (theta1) sin (psi)
754 // (-r) (Vector3D.plusJ) coordinates are :
755 // sin (psi) cos (theta2), cos (psi), sin (psi) sin (theta2)
756 // and we can choose to have psi in the interval [0 ; PI]
757 Vector3D v1 = applyTo(Vector3D.plusJ);
758 Vector3D v2 = applyInverseTo(Vector3D.plusJ);
759 if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
760 throw new CardanEulerSingularityException(false);
761 }
762 return new double[] {
763 Math.atan2(v1.getZ(), -v1.getX()),
764 Math.acos(v2.getY()),
765 Math.atan2(v2.getZ(), v2.getX())
766 };
767
768 } else if (order == RotationOrder.ZXZ) {
769
770 // r (Vector3D.plusK) coordinates are :
771 // sin (psi1) sin (phi), -cos (psi1) sin (phi), cos (phi)
772 // (-r) (Vector3D.plusK) coordinates are :
773 // sin (phi) sin (psi2), sin (phi) cos (psi2), cos (phi)
774 // and we can choose to have phi in the interval [0 ; PI]
775 Vector3D v1 = applyTo(Vector3D.plusK);
776 Vector3D v2 = applyInverseTo(Vector3D.plusK);
777 if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
778 throw new CardanEulerSingularityException(false);
779 }
780 return new double[] {
781 Math.atan2(v1.getX(), -v1.getY()),
782 Math.acos(v2.getZ()),
783 Math.atan2(v2.getX(), v2.getY())
784 };
785
786 } else { // last possibility is ZYZ
787
788 // r (Vector3D.plusK) coordinates are :
789 // cos (psi1) sin (theta), sin (psi1) sin (theta), cos (theta)
790 // (-r) (Vector3D.plusK) coordinates are :
791 // -sin (theta) cos (psi2), sin (theta) sin (psi2), cos (theta)
792 // and we can choose to have theta in the interval [0 ; PI]
793 Vector3D v1 = applyTo(Vector3D.plusK);
794 Vector3D v2 = applyInverseTo(Vector3D.plusK);
795 if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
796 throw new CardanEulerSingularityException(false);
797 }
798 return new double[] {
799 Math.atan2(v1.getY(), v1.getX()),
800 Math.acos(v2.getZ()),
801 Math.atan2(v2.getY(), -v2.getX())
802 };
803
804 }
805
806 }
807
808 /** Get the 3X3 matrix corresponding to the instance
809 * @return the matrix corresponding to the instance
810 */
811 public double[][] getMatrix() {
812
813 // products
814 double q0q0 = q0 * q0;
815 double q0q1 = q0 * q1;
816 double q0q2 = q0 * q2;
817 double q0q3 = q0 * q3;
818 double q1q1 = q1 * q1;
819 double q1q2 = q1 * q2;
820 double q1q3 = q1 * q3;
821 double q2q2 = q2 * q2;
822 double q2q3 = q2 * q3;
823 double q3q3 = q3 * q3;
824
825 // create the matrix
826 double[][] m = new double[3][];
827 m[0] = new double[3];
828 m[1] = new double[3];
829 m[2] = new double[3];
830
831 m [0][0] = 2.0 * (q0q0 + q1q1) - 1.0;
832 m [1][0] = 2.0 * (q1q2 - q0q3);
833 m [2][0] = 2.0 * (q1q3 + q0q2);
834
835 m [0][1] = 2.0 * (q1q2 + q0q3);
836 m [1][1] = 2.0 * (q0q0 + q2q2) - 1.0;
837 m [2][1] = 2.0 * (q2q3 - q0q1);
838
839 m [0][2] = 2.0 * (q1q3 - q0q2);
840 m [1][2] = 2.0 * (q2q3 + q0q1);
841 m [2][2] = 2.0 * (q0q0 + q3q3) - 1.0;
842
843 return m;
844
845 }
846
847 /** Apply the rotation to a vector.
848 * @param u vector to apply the rotation to
849 * @return a new vector which is the image of u by the rotation
850 */
851 public Vector3D applyTo(Vector3D u) {
852
853 double x = u.getX();
854 double y = u.getY();
855 double z = u.getZ();
856
857 double s = q1 * x + q2 * y + q3 * z;
858
859 return new Vector3D(2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x,
860 2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y,
861 2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - z);
862
863 }
864
865 /** Apply the inverse of the rotation to a vector.
866 * @param u vector to apply the inverse of the rotation to
867 * @return a new vector which such that u is its image by the rotation
868 */
869 public Vector3D applyInverseTo(Vector3D u) {
870
871 double x = u.getX();
872 double y = u.getY();
873 double z = u.getZ();
874
875 double s = q1 * x + q2 * y + q3 * z;
876 double m0 = -q0;
877
878 return new Vector3D(2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x,
879 2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y,
880 2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - z);
881
882 }
883
884 /** Apply the instance to another rotation.
885 * Applying the instance to a rotation is computing the composition
886 * in an order compliant with the following rule : let u be any
887 * vector and v its image by r (i.e. r.applyTo(u) = v), let w be the image
888 * of v by the instance (i.e. applyTo(v) = w), then w = comp.applyTo(u),
889 * where comp = applyTo(r).
890 * @param r rotation to apply the rotation to
891 * @return a new rotation which is the composition of r by the instance
892 */
893 public Rotation applyTo(Rotation r) {
894 return new Rotation(r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),
895 r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),
896 r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),
897 r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),
898 false);
899 }
900
901 /** Apply the inverse of the instance to another rotation.
902 * Applying the inverse of the instance to a rotation is computing
903 * the composition in an order compliant with the following rule :
904 * let u be any vector and v its image by r (i.e. r.applyTo(u) = v),
905 * let w be the inverse image of v by the instance
906 * (i.e. applyInverseTo(v) = w), then w = comp.applyTo(u), where
907 * comp = applyInverseTo(r).
908 * @param r rotation to apply the rotation to
909 * @return a new rotation which is the composition of r by the inverse
910 * of the instance
911 */
912 public Rotation applyInverseTo(Rotation r) {
913 return new Rotation(-r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),
914 -r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),
915 -r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),
916 -r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),
917 false);
918 }
919
920 /** Perfect orthogonality on a 3X3 matrix.
921 * @param m initial matrix (not exactly orthogonal)
922 * @param threshold convergence threshold for the iterative
923 * orthogonality correction (convergence is reached when the
924 * difference between two steps of the Frobenius norm of the
925 * correction is below this threshold)
926 * @return an orthogonal matrix close to m
927 * @exception NotARotationMatrixException if the matrix cannot be
928 * orthogonalized with the given threshold after 10 iterations
929 */
930 private double[][] orthogonalizeMatrix(double[][] m, double threshold)
931 throws NotARotationMatrixException {
932 double[] m0 = m[0];
933 double[] m1 = m[1];
934 double[] m2 = m[2];
935 double x00 = m0[0];
936 double x01 = m0[1];
937 double x02 = m0[2];
938 double x10 = m1[0];
939 double x11 = m1[1];
940 double x12 = m1[2];
941 double x20 = m2[0];
942 double x21 = m2[1];
943 double x22 = m2[2];
944 double fn = 0;
945 double fn1;
946
947 double[][] o = new double[3][3];
948 double[] o0 = o[0];
949 double[] o1 = o[1];
950 double[] o2 = o[2];
951
952 // iterative correction: Xn+1 = Xn - 0.5 * (Xn.Mt.Xn - M)
953 int i = 0;
954 while (++i < 11) {
955
956 // Mt.Xn
957 double mx00 = m0[0] * x00 + m1[0] * x10 + m2[0] * x20;
958 double mx10 = m0[1] * x00 + m1[1] * x10 + m2[1] * x20;
959 double mx20 = m0[2] * x00 + m1[2] * x10 + m2[2] * x20;
960 double mx01 = m0[0] * x01 + m1[0] * x11 + m2[0] * x21;
961 double mx11 = m0[1] * x01 + m1[1] * x11 + m2[1] * x21;
962 double mx21 = m0[2] * x01 + m1[2] * x11 + m2[2] * x21;
963 double mx02 = m0[0] * x02 + m1[0] * x12 + m2[0] * x22;
964 double mx12 = m0[1] * x02 + m1[1] * x12 + m2[1] * x22;
965 double mx22 = m0[2] * x02 + m1[2] * x12 + m2[2] * x22;
966
967 // Xn+1
968 o0[0] = x00 - 0.5 * (x00 * mx00 + x01 * mx10 + x02 * mx20 - m0[0]);
969 o0[1] = x01 - 0.5 * (x00 * mx01 + x01 * mx11 + x02 * mx21 - m0[1]);
970 o0[2] = x02 - 0.5 * (x00 * mx02 + x01 * mx12 + x02 * mx22 - m0[2]);
971 o1[0] = x10 - 0.5 * (x10 * mx00 + x11 * mx10 + x12 * mx20 - m1[0]);
972 o1[1] = x11 - 0.5 * (x10 * mx01 + x11 * mx11 + x12 * mx21 - m1[1]);
973 o1[2] = x12 - 0.5 * (x10 * mx02 + x11 * mx12 + x12 * mx22 - m1[2]);
974 o2[0] = x20 - 0.5 * (x20 * mx00 + x21 * mx10 + x22 * mx20 - m2[0]);
975 o2[1] = x21 - 0.5 * (x20 * mx01 + x21 * mx11 + x22 * mx21 - m2[1]);
976 o2[2] = x22 - 0.5 * (x20 * mx02 + x21 * mx12 + x22 * mx22 - m2[2]);
977
978 // correction on each elements
979 double corr00 = o0[0] - m0[0];
980 double corr01 = o0[1] - m0[1];
981 double corr02 = o0[2] - m0[2];
982 double corr10 = o1[0] - m1[0];
983 double corr11 = o1[1] - m1[1];
984 double corr12 = o1[2] - m1[2];
985 double corr20 = o2[0] - m2[0];
986 double corr21 = o2[1] - m2[1];
987 double corr22 = o2[2] - m2[2];
988
989 // Frobenius norm of the correction
990 fn1 = corr00 * corr00 + corr01 * corr01 + corr02 * corr02 +
991 corr10 * corr10 + corr11 * corr11 + corr12 * corr12 +
992 corr20 * corr20 + corr21 * corr21 + corr22 * corr22;
993
994 // convergence test
995 if (Math.abs(fn1 - fn) <= threshold)
996 return o;
997
998 // prepare next iteration
999 x00 = o0[0];
1000 x01 = o0[1];
1001 x02 = o0[2];
1002 x10 = o1[0];
1003 x11 = o1[1];
1004 x12 = o1[2];
1005 x20 = o2[0];
1006 x21 = o2[1];
1007 x22 = o2[2];
1008 fn = fn1;
1009
1010 }
1011
1012 // the algorithm did not converge after 10 iterations
1013 throw new NotARotationMatrixException("unable to orthogonalize matrix" +
1014 " in {0} iterations",
1015 new Object[] {
1016 Integer.toString(i - 1)
1017 });
1018 }
1019
1020 /** Scalar coordinate of the quaternion. */
1021 private final double q0;
1022
1023 /** First coordinate of the vectorial part of the quaternion. */
1024 private final double q1;
1025
1026 /** Second coordinate of the vectorial part of the quaternion. */
1027 private final double q2;
1028
1029 /** Third coordinate of the vectorial part of the quaternion. */
1030 private final double q3;
1031
1032 /** Serializable version identifier */
1033 private static final long serialVersionUID = 8225864499430109352L;
1034
1035 }