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.euclidean.threed;
19
20 import java.io.Serializable;
21
22 import org.apache.commons.math.exception.MathIllegalArgumentException;
23 import org.apache.commons.math.exception.util.LocalizedFormats;
24 import org.apache.commons.math.util.FastMath;
25
26 /**
27 * This class implements rotations in a three-dimensional space.
28 *
29 * <p>Rotations can be represented by several different mathematical
30 * entities (matrices, axe and angle, Cardan or Euler angles,
31 * quaternions). This class presents an higher level abstraction, more
32 * user-oriented and hiding this implementation details. Well, for the
33 * curious, we use quaternions for the internal representation. The
34 * user can build a rotation from any of these representations, and
35 * any of these representations can be retrieved from a
36 * <code>Rotation</code> instance (see the various constructors and
37 * getters). In addition, a rotation can also be built implicitly
38 * from a set of vectors and their image.</p>
39 * <p>This implies that this class can be used to convert from one
40 * representation to another one. For example, converting a rotation
41 * matrix into a set of Cardan angles from can be done using the
42 * following single line of code:</p>
43 * <pre>
44 * double[] angles = new Rotation(matrix, 1.0e-10).getAngles(RotationOrder.XYZ);
45 * </pre>
46 * <p>Focus is oriented on what a rotation <em>do</em> rather than on its
47 * underlying representation. Once it has been built, and regardless of its
48 * internal representation, a rotation is an <em>operator</em> which basically
49 * transforms three dimensional {@link Vector3D vectors} into other three
50 * dimensional {@link Vector3D vectors}. Depending on the application, the
51 * meaning of these vectors may vary and the semantics of the rotation also.</p>
52 * <p>For example in an spacecraft attitude simulation tool, users will often
53 * consider the vectors are fixed (say the Earth direction for example) and the
54 * frames change. The rotation transforms the coordinates of the vector in inertial
55 * frame into the coordinates of the same vector in satellite frame. In this
56 * case, the rotation implicitly defines the relation between the two frames.</p>
57 * <p>Another example could be a telescope control application, where the rotation
58 * would transform the sighting direction at rest into the desired observing
59 * direction when the telescope is pointed towards an object of interest. In this
60 * case the rotation transforms the direction at rest in a topocentric frame
61 * into the sighting direction in the same topocentric frame. This implies in this
62 * case the frame is fixed and the vector moves.</p>
63 * <p>In many case, both approaches will be combined. In our telescope example,
64 * we will probably also need to transform the observing direction in the topocentric
65 * frame into the observing direction in inertial frame taking into account the observatory
66 * location and the Earth rotation, which would essentially be an application of the
67 * first approach.</p>
68 *
69 * <p>These examples show that a rotation is what the user wants it to be. This
70 * class does not push the user towards one specific definition and hence does not
71 * provide methods like <code>projectVectorIntoDestinationFrame</code> or
72 * <code>computeTransformedDirection</code>. It provides simpler and more generic
73 * methods: {@link #applyTo(Vector3D) applyTo(Vector3D)} and {@link
74 * #applyInverseTo(Vector3D) applyInverseTo(Vector3D)}.</p>
75 *
76 * <p>Since a rotation is basically a vectorial operator, several rotations can be
77 * composed together and the composite operation <code>r = r<sub>1</sub> o
78 * r<sub>2</sub></code> (which means that for each vector <code>u</code>,
79 * <code>r(u) = r<sub>1</sub>(r<sub>2</sub>(u))</code>) is also a rotation. Hence
80 * we can consider that in addition to vectors, a rotation can be applied to other
81 * rotations as well (or to itself). With our previous notations, we would say we
82 * can apply <code>r<sub>1</sub></code> to <code>r<sub>2</sub></code> and the result
83 * we get is <code>r = r<sub>1</sub> o r<sub>2</sub></code>. For this purpose, the
84 * class provides the methods: {@link #applyTo(Rotation) applyTo(Rotation)} and
85 * {@link #applyInverseTo(Rotation) applyInverseTo(Rotation)}.</p>
86 *
87 * <p>Rotations are guaranteed to be immutable objects.</p>
88 *
89 * @version $Id: Rotation.java 1178082 2011-10-01 20:01:22Z luc $
90 * @see Vector3D
91 * @see RotationOrder
92 * @since 1.2
93 */
94
95 public class Rotation implements Serializable {
96
97 /** Identity rotation. */
98 public static final Rotation IDENTITY = new Rotation(1.0, 0.0, 0.0, 0.0, false);
99
100 /** Serializable version identifier */
101 private static final long serialVersionUID = -2153622329907944313L;
102
103 /** Scalar coordinate of the quaternion. */
104 private final double q0;
105
106 /** First coordinate of the vectorial part of the quaternion. */
107 private final double q1;
108
109 /** Second coordinate of the vectorial part of the quaternion. */
110 private final double q2;
111
112 /** Third coordinate of the vectorial part of the quaternion. */
113 private final double q3;
114
115 /** Build a rotation from the quaternion coordinates.
116 * <p>A rotation can be built from a <em>normalized</em> quaternion,
117 * i.e. a quaternion for which q<sub>0</sub><sup>2</sup> +
118 * q<sub>1</sub><sup>2</sup> + q<sub>2</sub><sup>2</sup> +
119 * q<sub>3</sub><sup>2</sup> = 1. If the quaternion is not normalized,
120 * the constructor can normalize it in a preprocessing step.</p>
121 * <p>Note that some conventions put the scalar part of the quaternion
122 * as the 4<sup>th</sup> component and the vector part as the first three
123 * components. This is <em>not</em> our convention. We put the scalar part
124 * as the first component.</p>
125 * @param q0 scalar part of the quaternion
126 * @param q1 first coordinate of the vectorial part of the quaternion
127 * @param q2 second coordinate of the vectorial part of the quaternion
128 * @param q3 third coordinate of the vectorial part of the quaternion
129 * @param needsNormalization if true, the coordinates are considered
130 * not to be normalized, a normalization preprocessing step is performed
131 * before using them
132 */
133 public Rotation(double q0, double q1, double q2, double q3,
134 boolean needsNormalization) {
135
136 if (needsNormalization) {
137 // normalization preprocessing
138 double inv = 1.0 / FastMath.sqrt(q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3);
139 q0 *= inv;
140 q1 *= inv;
141 q2 *= inv;
142 q3 *= inv;
143 }
144
145 this.q0 = q0;
146 this.q1 = q1;
147 this.q2 = q2;
148 this.q3 = q3;
149
150 }
151
152 /** Build a rotation from an axis and an angle.
153 * <p>We use the convention that angles are oriented according to
154 * the effect of the rotation on vectors around the axis. That means
155 * that if (i, j, k) is a direct frame and if we first provide +k as
156 * the axis and π/2 as the angle to this constructor, and then
157 * {@link #applyTo(Vector3D) apply} the instance to +i, we will get
158 * +j.</p>
159 * <p>Another way to represent our convention is to say that a rotation
160 * of angle θ about the unit vector (x, y, z) is the same as the
161 * rotation build from quaternion components { cos(-θ/2),
162 * x * sin(-θ/2), y * sin(-θ/2), z * sin(-θ/2) }.
163 * Note the minus sign on the angle!</p>
164 * <p>On the one hand this convention is consistent with a vectorial
165 * perspective (moving vectors in fixed frames), on the other hand it
166 * is different from conventions with a frame perspective (fixed vectors
167 * viewed from different frames) like the ones used for example in spacecraft
168 * attitude community or in the graphics community.</p>
169 * @param axis axis around which to rotate
170 * @param angle rotation angle.
171 * @exception MathIllegalArgumentException if the axis norm is zero
172 */
173 public Rotation(Vector3D axis, double angle) {
174
175 double norm = axis.getNorm();
176 if (norm == 0) {
177 throw new MathIllegalArgumentException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_AXIS);
178 }
179
180 double halfAngle = -0.5 * angle;
181 double coeff = FastMath.sin(halfAngle) / norm;
182
183 q0 = FastMath.cos (halfAngle);
184 q1 = coeff * axis.getX();
185 q2 = coeff * axis.getY();
186 q3 = coeff * axis.getZ();
187
188 }
189
190 /** Build a rotation from a 3X3 matrix.
191
192 * <p>Rotation matrices are orthogonal matrices, i.e. unit matrices
193 * (which are matrices for which m.m<sup>T</sup> = I) with real
194 * coefficients. The module of the determinant of unit matrices is
195 * 1, among the orthogonal 3X3 matrices, only the ones having a
196 * positive determinant (+1) are rotation matrices.</p>
197
198 * <p>When a rotation is defined by a matrix with truncated values
199 * (typically when it is extracted from a technical sheet where only
200 * four to five significant digits are available), the matrix is not
201 * orthogonal anymore. This constructor handles this case
202 * transparently by using a copy of the given matrix and applying a
203 * correction to the copy in order to perfect its orthogonality. If
204 * the Frobenius norm of the correction needed is above the given
205 * threshold, then the matrix is considered to be too far from a
206 * true rotation matrix and an exception is thrown.<p>
207
208 * @param m rotation matrix
209 * @param threshold convergence threshold for the iterative
210 * orthogonality correction (convergence is reached when the
211 * difference between two steps of the Frobenius norm of the
212 * correction is below this threshold)
213
214 * @exception NotARotationMatrixException if the matrix is not a 3X3
215 * matrix, or if it cannot be transformed into an orthogonal matrix
216 * with the given threshold, or if the determinant of the resulting
217 * orthogonal matrix is negative
218
219 */
220 public Rotation(double[][] m, double threshold)
221 throws NotARotationMatrixException {
222
223 // dimension check
224 if ((m.length != 3) || (m[0].length != 3) ||
225 (m[1].length != 3) || (m[2].length != 3)) {
226 throw new NotARotationMatrixException(
227 LocalizedFormats.ROTATION_MATRIX_DIMENSIONS,
228 m.length, m[0].length);
229 }
230
231 // compute a "close" orthogonal matrix
232 double[][] ort = orthogonalizeMatrix(m, threshold);
233
234 // check the sign of the determinant
235 double det = ort[0][0] * (ort[1][1] * ort[2][2] - ort[2][1] * ort[1][2]) -
236 ort[1][0] * (ort[0][1] * ort[2][2] - ort[2][1] * ort[0][2]) +
237 ort[2][0] * (ort[0][1] * ort[1][2] - ort[1][1] * ort[0][2]);
238 if (det < 0.0) {
239 throw new NotARotationMatrixException(
240 LocalizedFormats.CLOSEST_ORTHOGONAL_MATRIX_HAS_NEGATIVE_DETERMINANT,
241 det);
242 }
243
244 // There are different ways to compute the quaternions elements
245 // from the matrix. They all involve computing one element from
246 // the diagonal of the matrix, and computing the three other ones
247 // using a formula involving a division by the first element,
248 // which unfortunately can be zero. Since the norm of the
249 // quaternion is 1, we know at least one element has an absolute
250 // value greater or equal to 0.5, so it is always possible to
251 // select the right formula and avoid division by zero and even
252 // numerical inaccuracy. Checking the elements in turn and using
253 // the first one greater than 0.45 is safe (this leads to a simple
254 // test since qi = 0.45 implies 4 qi^2 - 1 = -0.19)
255 double s = ort[0][0] + ort[1][1] + ort[2][2];
256 if (s > -0.19) {
257 // compute q0 and deduce q1, q2 and q3
258 q0 = 0.5 * FastMath.sqrt(s + 1.0);
259 double inv = 0.25 / q0;
260 q1 = inv * (ort[1][2] - ort[2][1]);
261 q2 = inv * (ort[2][0] - ort[0][2]);
262 q3 = inv * (ort[0][1] - ort[1][0]);
263 } else {
264 s = ort[0][0] - ort[1][1] - ort[2][2];
265 if (s > -0.19) {
266 // compute q1 and deduce q0, q2 and q3
267 q1 = 0.5 * FastMath.sqrt(s + 1.0);
268 double inv = 0.25 / q1;
269 q0 = inv * (ort[1][2] - ort[2][1]);
270 q2 = inv * (ort[0][1] + ort[1][0]);
271 q3 = inv * (ort[0][2] + ort[2][0]);
272 } else {
273 s = ort[1][1] - ort[0][0] - ort[2][2];
274 if (s > -0.19) {
275 // compute q2 and deduce q0, q1 and q3
276 q2 = 0.5 * FastMath.sqrt(s + 1.0);
277 double inv = 0.25 / q2;
278 q0 = inv * (ort[2][0] - ort[0][2]);
279 q1 = inv * (ort[0][1] + ort[1][0]);
280 q3 = inv * (ort[2][1] + ort[1][2]);
281 } else {
282 // compute q3 and deduce q0, q1 and q2
283 s = ort[2][2] - ort[0][0] - ort[1][1];
284 q3 = 0.5 * FastMath.sqrt(s + 1.0);
285 double inv = 0.25 / q3;
286 q0 = inv * (ort[0][1] - ort[1][0]);
287 q1 = inv * (ort[0][2] + ort[2][0]);
288 q2 = inv * (ort[2][1] + ort[1][2]);
289 }
290 }
291 }
292
293 }
294
295 /** Build the rotation that transforms a pair of vector into another pair.
296
297 * <p>Except for possible scale factors, if the instance were applied to
298 * the pair (u<sub>1</sub>, u<sub>2</sub>) it will produce the pair
299 * (v<sub>1</sub>, v<sub>2</sub>).</p>
300
301 * <p>If the angular separation between u<sub>1</sub> and u<sub>2</sub> is
302 * not the same as the angular separation between v<sub>1</sub> and
303 * v<sub>2</sub>, then a corrected v'<sub>2</sub> will be used rather than
304 * v<sub>2</sub>, the corrected vector will be in the (v<sub>1</sub>,
305 * v<sub>2</sub>) plane.</p>
306
307 * @param u1 first vector of the origin pair
308 * @param u2 second vector of the origin pair
309 * @param v1 desired image of u1 by the rotation
310 * @param v2 desired image of u2 by the rotation
311 * @exception MathIllegalArgumentException if the norm of one of the vectors is zero
312 */
313 public Rotation(Vector3D u1, Vector3D u2, Vector3D v1, Vector3D v2) {
314
315 // norms computation
316 double u1u1 = u1.getNormSq();
317 double u2u2 = u2.getNormSq();
318 double v1v1 = v1.getNormSq();
319 double v2v2 = v2.getNormSq();
320 if ((u1u1 == 0) || (u2u2 == 0) || (v1v1 == 0) || (v2v2 == 0)) {
321 throw new MathIllegalArgumentException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_DEFINING_VECTOR);
322 }
323
324 // normalize v1 in order to have (v1'|v1') = (u1|u1)
325 v1 = new Vector3D(FastMath.sqrt(u1u1 / v1v1), v1);
326
327 // adjust v2 in order to have (u1|u2) = (v1'|v2') and (v2'|v2') = (u2|u2)
328 double u1u2 = u1.dotProduct(u2);
329 double v1v2 = v1.dotProduct(v2);
330 double coeffU = u1u2 / u1u1;
331 double coeffV = v1v2 / u1u1;
332 double beta = FastMath.sqrt((u2u2 - u1u2 * coeffU) / (v2v2 - v1v2 * coeffV));
333 double alpha = coeffU - beta * coeffV;
334 v2 = new Vector3D(alpha, v1, beta, v2);
335
336 // preliminary computation
337 Vector3D uRef = u1;
338 Vector3D vRef = v1;
339 Vector3D v1Su1 = v1.subtract(u1);
340 Vector3D v2Su2 = v2.subtract(u2);
341 Vector3D k = v1Su1.crossProduct(v2Su2);
342 Vector3D u3 = u1.crossProduct(u2);
343 double c = k.dotProduct(u3);
344 final double inPlaneThreshold = 0.001;
345 if (c <= inPlaneThreshold * k.getNorm() * u3.getNorm()) {
346 // the (q1, q2, q3) vector is close to the (u1, u2) plane
347 // we try other vectors
348 Vector3D v3 = Vector3D.crossProduct(v1, v2);
349 Vector3D v3Su3 = v3.subtract(u3);
350 k = v1Su1.crossProduct(v3Su3);
351 Vector3D u2Prime = u1.crossProduct(u3);
352 c = k.dotProduct(u2Prime);
353
354 if (c <= inPlaneThreshold * k.getNorm() * u2Prime.getNorm()) {
355 // the (q1, q2, q3) vector is also close to the (u1, u3) plane,
356 // it is almost aligned with u1: we try (u2, u3) and (v2, v3)
357 k = v2Su2.crossProduct(v3Su3);;
358 c = k.dotProduct(u2.crossProduct(u3));;
359
360 if (c <= 0) {
361 // the (q1, q2, q3) vector is aligned with everything
362 // this is really the identity rotation
363 q0 = 1.0;
364 q1 = 0.0;
365 q2 = 0.0;
366 q3 = 0.0;
367 return;
368 }
369
370 // we will have to use u2 and v2 to compute the scalar part
371 uRef = u2;
372 vRef = v2;
373
374 }
375
376 }
377
378 // compute the vectorial part
379 c = FastMath.sqrt(c);
380 double inv = 1.0 / (c + c);
381 q1 = inv * k.getX();
382 q2 = inv * k.getY();
383 q3 = inv * k.getZ();
384
385 // compute the scalar part
386 k = new Vector3D(uRef.getY() * q3 - uRef.getZ() * q2,
387 uRef.getZ() * q1 - uRef.getX() * q3,
388 uRef.getX() * q2 - uRef.getY() * q1);
389 q0 = vRef.dotProduct(k) / (2 * k.getNormSq());
390
391 }
392
393 /** Build one of the rotations that transform one vector into another one.
394
395 * <p>Except for a possible scale factor, if the instance were
396 * applied to the vector u it will produce the vector v. There is an
397 * infinite number of such rotations, this constructor choose the
398 * one with the smallest associated angle (i.e. the one whose axis
399 * is orthogonal to the (u, v) plane). If u and v are colinear, an
400 * arbitrary rotation axis is chosen.</p>
401
402 * @param u origin vector
403 * @param v desired image of u by the rotation
404 * @exception MathIllegalArgumentException if the norm of one of the vectors is zero
405 */
406 public Rotation(Vector3D u, Vector3D v) {
407
408 double normProduct = u.getNorm() * v.getNorm();
409 if (normProduct == 0) {
410 throw new MathIllegalArgumentException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_DEFINING_VECTOR);
411 }
412
413 double dot = u.dotProduct(v);
414
415 if (dot < ((2.0e-15 - 1.0) * normProduct)) {
416 // special case u = -v: we select a PI angle rotation around
417 // an arbitrary vector orthogonal to u
418 Vector3D w = u.orthogonal();
419 q0 = 0.0;
420 q1 = -w.getX();
421 q2 = -w.getY();
422 q3 = -w.getZ();
423 } else {
424 // general case: (u, v) defines a plane, we select
425 // the shortest possible rotation: axis orthogonal to this plane
426 q0 = FastMath.sqrt(0.5 * (1.0 + dot / normProduct));
427 double coeff = 1.0 / (2.0 * q0 * normProduct);
428 Vector3D q = v.crossProduct(u);
429 q1 = coeff * q.getX();
430 q2 = coeff * q.getY();
431 q3 = coeff * q.getZ();
432 }
433
434 }
435
436 /** Build a rotation from three Cardan or Euler elementary rotations.
437
438 * <p>Cardan rotations are three successive rotations around the
439 * canonical axes X, Y and Z, each axis being used once. There are
440 * 6 such sets of rotations (XYZ, XZY, YXZ, YZX, ZXY and ZYX). Euler
441 * rotations are three successive rotations around the canonical
442 * axes X, Y and Z, the first and last rotations being around the
443 * same axis. There are 6 such sets of rotations (XYX, XZX, YXY,
444 * YZY, ZXZ and ZYZ), the most popular one being ZXZ.</p>
445 * <p>Beware that many people routinely use the term Euler angles even
446 * for what really are Cardan angles (this confusion is especially
447 * widespread in the aerospace business where Roll, Pitch and Yaw angles
448 * are often wrongly tagged as Euler angles).</p>
449
450 * @param order order of rotations to use
451 * @param alpha1 angle of the first elementary rotation
452 * @param alpha2 angle of the second elementary rotation
453 * @param alpha3 angle of the third elementary rotation
454 */
455 public Rotation(RotationOrder order,
456 double alpha1, double alpha2, double alpha3) {
457 Rotation r1 = new Rotation(order.getA1(), alpha1);
458 Rotation r2 = new Rotation(order.getA2(), alpha2);
459 Rotation r3 = new Rotation(order.getA3(), alpha3);
460 Rotation composed = r1.applyTo(r2.applyTo(r3));
461 q0 = composed.q0;
462 q1 = composed.q1;
463 q2 = composed.q2;
464 q3 = composed.q3;
465 }
466
467 /** Revert a rotation.
468 * Build a rotation which reverse the effect of another
469 * rotation. This means that if r(u) = v, then r.revert(v) = u. The
470 * instance is not changed.
471 * @return a new rotation whose effect is the reverse of the effect
472 * of the instance
473 */
474 public Rotation revert() {
475 return new Rotation(-q0, q1, q2, q3, false);
476 }
477
478 /** Get the scalar coordinate of the quaternion.
479 * @return scalar coordinate of the quaternion
480 */
481 public double getQ0() {
482 return q0;
483 }
484
485 /** Get the first coordinate of the vectorial part of the quaternion.
486 * @return first coordinate of the vectorial part of the quaternion
487 */
488 public double getQ1() {
489 return q1;
490 }
491
492 /** Get the second coordinate of the vectorial part of the quaternion.
493 * @return second coordinate of the vectorial part of the quaternion
494 */
495 public double getQ2() {
496 return q2;
497 }
498
499 /** Get the third coordinate of the vectorial part of the quaternion.
500 * @return third coordinate of the vectorial part of the quaternion
501 */
502 public double getQ3() {
503 return q3;
504 }
505
506 /** Get the normalized axis of the rotation.
507 * @return normalized axis of the rotation
508 * @see #Rotation(Vector3D, double)
509 */
510 public Vector3D getAxis() {
511 double squaredSine = q1 * q1 + q2 * q2 + q3 * q3;
512 if (squaredSine == 0) {
513 return new Vector3D(1, 0, 0);
514 } else if (q0 < 0) {
515 double inverse = 1 / FastMath.sqrt(squaredSine);
516 return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
517 }
518 double inverse = -1 / FastMath.sqrt(squaredSine);
519 return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
520 }
521
522 /** Get the angle of the rotation.
523 * @return angle of the rotation (between 0 and π)
524 * @see #Rotation(Vector3D, double)
525 */
526 public double getAngle() {
527 if ((q0 < -0.1) || (q0 > 0.1)) {
528 return 2 * FastMath.asin(FastMath.sqrt(q1 * q1 + q2 * q2 + q3 * q3));
529 } else if (q0 < 0) {
530 return 2 * FastMath.acos(-q0);
531 }
532 return 2 * FastMath.acos(q0);
533 }
534
535 /** Get the Cardan or Euler angles corresponding to the instance.
536
537 * <p>The equations show that each rotation can be defined by two
538 * different values of the Cardan or Euler angles set. For example
539 * if Cardan angles are used, the rotation defined by the angles
540 * a<sub>1</sub>, a<sub>2</sub> and a<sub>3</sub> is the same as
541 * the rotation defined by the angles π + a<sub>1</sub>, π
542 * - a<sub>2</sub> and π + a<sub>3</sub>. This method implements
543 * the following arbitrary choices:</p>
544 * <ul>
545 * <li>for Cardan angles, the chosen set is the one for which the
546 * second angle is between -π/2 and π/2 (i.e its cosine is
547 * positive),</li>
548 * <li>for Euler angles, the chosen set is the one for which the
549 * second angle is between 0 and π (i.e its sine is positive).</li>
550 * </ul>
551
552 * <p>Cardan and Euler angle have a very disappointing drawback: all
553 * of them have singularities. This means that if the instance is
554 * too close to the singularities corresponding to the given
555 * rotation order, it will be impossible to retrieve the angles. For
556 * Cardan angles, this is often called gimbal lock. There is
557 * <em>nothing</em> to do to prevent this, it is an intrinsic problem
558 * with Cardan and Euler representation (but not a problem with the
559 * rotation itself, which is perfectly well defined). For Cardan
560 * angles, singularities occur when the second angle is close to
561 * -π/2 or +π/2, for Euler angle singularities occur when the
562 * second angle is close to 0 or π, this implies that the identity
563 * rotation is always singular for Euler angles!</p>
564
565 * @param order rotation order to use
566 * @return an array of three angles, in the order specified by the set
567 * @exception CardanEulerSingularityException if the rotation is
568 * singular with respect to the angles set specified
569 */
570 public double[] getAngles(RotationOrder order)
571 throws CardanEulerSingularityException {
572
573 if (order == RotationOrder.XYZ) {
574
575 // r (Vector3D.plusK) coordinates are :
576 // sin (theta), -cos (theta) sin (phi), cos (theta) cos (phi)
577 // (-r) (Vector3D.plusI) coordinates are :
578 // cos (psi) cos (theta), -sin (psi) cos (theta), sin (theta)
579 // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
580 Vector3D v1 = applyTo(Vector3D.PLUS_K);
581 Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
582 if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
583 throw new CardanEulerSingularityException(true);
584 }
585 return new double[] {
586 FastMath.atan2(-(v1.getY()), v1.getZ()),
587 FastMath.asin(v2.getZ()),
588 FastMath.atan2(-(v2.getY()), v2.getX())
589 };
590
591 } else if (order == RotationOrder.XZY) {
592
593 // r (Vector3D.plusJ) coordinates are :
594 // -sin (psi), cos (psi) cos (phi), cos (psi) sin (phi)
595 // (-r) (Vector3D.plusI) coordinates are :
596 // cos (theta) cos (psi), -sin (psi), sin (theta) cos (psi)
597 // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
598 Vector3D v1 = applyTo(Vector3D.PLUS_J);
599 Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
600 if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
601 throw new CardanEulerSingularityException(true);
602 }
603 return new double[] {
604 FastMath.atan2(v1.getZ(), v1.getY()),
605 -FastMath.asin(v2.getY()),
606 FastMath.atan2(v2.getZ(), v2.getX())
607 };
608
609 } else if (order == RotationOrder.YXZ) {
610
611 // r (Vector3D.plusK) coordinates are :
612 // cos (phi) sin (theta), -sin (phi), cos (phi) cos (theta)
613 // (-r) (Vector3D.plusJ) coordinates are :
614 // sin (psi) cos (phi), cos (psi) cos (phi), -sin (phi)
615 // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
616 Vector3D v1 = applyTo(Vector3D.PLUS_K);
617 Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
618 if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
619 throw new CardanEulerSingularityException(true);
620 }
621 return new double[] {
622 FastMath.atan2(v1.getX(), v1.getZ()),
623 -FastMath.asin(v2.getZ()),
624 FastMath.atan2(v2.getX(), v2.getY())
625 };
626
627 } else if (order == RotationOrder.YZX) {
628
629 // r (Vector3D.plusI) coordinates are :
630 // cos (psi) cos (theta), sin (psi), -cos (psi) sin (theta)
631 // (-r) (Vector3D.plusJ) coordinates are :
632 // sin (psi), cos (phi) cos (psi), -sin (phi) cos (psi)
633 // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
634 Vector3D v1 = applyTo(Vector3D.PLUS_I);
635 Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
636 if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
637 throw new CardanEulerSingularityException(true);
638 }
639 return new double[] {
640 FastMath.atan2(-(v1.getZ()), v1.getX()),
641 FastMath.asin(v2.getX()),
642 FastMath.atan2(-(v2.getZ()), v2.getY())
643 };
644
645 } else if (order == RotationOrder.ZXY) {
646
647 // r (Vector3D.plusJ) coordinates are :
648 // -cos (phi) sin (psi), cos (phi) cos (psi), sin (phi)
649 // (-r) (Vector3D.plusK) coordinates are :
650 // -sin (theta) cos (phi), sin (phi), cos (theta) cos (phi)
651 // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
652 Vector3D v1 = applyTo(Vector3D.PLUS_J);
653 Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
654 if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
655 throw new CardanEulerSingularityException(true);
656 }
657 return new double[] {
658 FastMath.atan2(-(v1.getX()), v1.getY()),
659 FastMath.asin(v2.getY()),
660 FastMath.atan2(-(v2.getX()), v2.getZ())
661 };
662
663 } else if (order == RotationOrder.ZYX) {
664
665 // r (Vector3D.plusI) coordinates are :
666 // cos (theta) cos (psi), cos (theta) sin (psi), -sin (theta)
667 // (-r) (Vector3D.plusK) coordinates are :
668 // -sin (theta), sin (phi) cos (theta), cos (phi) cos (theta)
669 // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
670 Vector3D v1 = applyTo(Vector3D.PLUS_I);
671 Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
672 if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
673 throw new CardanEulerSingularityException(true);
674 }
675 return new double[] {
676 FastMath.atan2(v1.getY(), v1.getX()),
677 -FastMath.asin(v2.getX()),
678 FastMath.atan2(v2.getY(), v2.getZ())
679 };
680
681 } else if (order == RotationOrder.XYX) {
682
683 // r (Vector3D.plusI) coordinates are :
684 // cos (theta), sin (phi1) sin (theta), -cos (phi1) sin (theta)
685 // (-r) (Vector3D.plusI) coordinates are :
686 // cos (theta), sin (theta) sin (phi2), sin (theta) cos (phi2)
687 // and we can choose to have theta in the interval [0 ; PI]
688 Vector3D v1 = applyTo(Vector3D.PLUS_I);
689 Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
690 if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
691 throw new CardanEulerSingularityException(false);
692 }
693 return new double[] {
694 FastMath.atan2(v1.getY(), -v1.getZ()),
695 FastMath.acos(v2.getX()),
696 FastMath.atan2(v2.getY(), v2.getZ())
697 };
698
699 } else if (order == RotationOrder.XZX) {
700
701 // r (Vector3D.plusI) coordinates are :
702 // cos (psi), cos (phi1) sin (psi), sin (phi1) sin (psi)
703 // (-r) (Vector3D.plusI) coordinates are :
704 // cos (psi), -sin (psi) cos (phi2), sin (psi) sin (phi2)
705 // and we can choose to have psi in the interval [0 ; PI]
706 Vector3D v1 = applyTo(Vector3D.PLUS_I);
707 Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
708 if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
709 throw new CardanEulerSingularityException(false);
710 }
711 return new double[] {
712 FastMath.atan2(v1.getZ(), v1.getY()),
713 FastMath.acos(v2.getX()),
714 FastMath.atan2(v2.getZ(), -v2.getY())
715 };
716
717 } else if (order == RotationOrder.YXY) {
718
719 // r (Vector3D.plusJ) coordinates are :
720 // sin (theta1) sin (phi), cos (phi), cos (theta1) sin (phi)
721 // (-r) (Vector3D.plusJ) coordinates are :
722 // sin (phi) sin (theta2), cos (phi), -sin (phi) cos (theta2)
723 // and we can choose to have phi in the interval [0 ; PI]
724 Vector3D v1 = applyTo(Vector3D.PLUS_J);
725 Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
726 if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
727 throw new CardanEulerSingularityException(false);
728 }
729 return new double[] {
730 FastMath.atan2(v1.getX(), v1.getZ()),
731 FastMath.acos(v2.getY()),
732 FastMath.atan2(v2.getX(), -v2.getZ())
733 };
734
735 } else if (order == RotationOrder.YZY) {
736
737 // r (Vector3D.plusJ) coordinates are :
738 // -cos (theta1) sin (psi), cos (psi), sin (theta1) sin (psi)
739 // (-r) (Vector3D.plusJ) coordinates are :
740 // sin (psi) cos (theta2), cos (psi), sin (psi) sin (theta2)
741 // and we can choose to have psi in the interval [0 ; PI]
742 Vector3D v1 = applyTo(Vector3D.PLUS_J);
743 Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
744 if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
745 throw new CardanEulerSingularityException(false);
746 }
747 return new double[] {
748 FastMath.atan2(v1.getZ(), -v1.getX()),
749 FastMath.acos(v2.getY()),
750 FastMath.atan2(v2.getZ(), v2.getX())
751 };
752
753 } else if (order == RotationOrder.ZXZ) {
754
755 // r (Vector3D.plusK) coordinates are :
756 // sin (psi1) sin (phi), -cos (psi1) sin (phi), cos (phi)
757 // (-r) (Vector3D.plusK) coordinates are :
758 // sin (phi) sin (psi2), sin (phi) cos (psi2), cos (phi)
759 // and we can choose to have phi in the interval [0 ; PI]
760 Vector3D v1 = applyTo(Vector3D.PLUS_K);
761 Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
762 if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
763 throw new CardanEulerSingularityException(false);
764 }
765 return new double[] {
766 FastMath.atan2(v1.getX(), -v1.getY()),
767 FastMath.acos(v2.getZ()),
768 FastMath.atan2(v2.getX(), v2.getY())
769 };
770
771 } else { // last possibility is ZYZ
772
773 // r (Vector3D.plusK) coordinates are :
774 // cos (psi1) sin (theta), sin (psi1) sin (theta), cos (theta)
775 // (-r) (Vector3D.plusK) coordinates are :
776 // -sin (theta) cos (psi2), sin (theta) sin (psi2), cos (theta)
777 // and we can choose to have theta in the interval [0 ; PI]
778 Vector3D v1 = applyTo(Vector3D.PLUS_K);
779 Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
780 if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
781 throw new CardanEulerSingularityException(false);
782 }
783 return new double[] {
784 FastMath.atan2(v1.getY(), v1.getX()),
785 FastMath.acos(v2.getZ()),
786 FastMath.atan2(v2.getY(), -v2.getX())
787 };
788
789 }
790
791 }
792
793 /** Get the 3X3 matrix corresponding to the instance
794 * @return the matrix corresponding to the instance
795 */
796 public double[][] getMatrix() {
797
798 // products
799 double q0q0 = q0 * q0;
800 double q0q1 = q0 * q1;
801 double q0q2 = q0 * q2;
802 double q0q3 = q0 * q3;
803 double q1q1 = q1 * q1;
804 double q1q2 = q1 * q2;
805 double q1q3 = q1 * q3;
806 double q2q2 = q2 * q2;
807 double q2q3 = q2 * q3;
808 double q3q3 = q3 * q3;
809
810 // create the matrix
811 double[][] m = new double[3][];
812 m[0] = new double[3];
813 m[1] = new double[3];
814 m[2] = new double[3];
815
816 m [0][0] = 2.0 * (q0q0 + q1q1) - 1.0;
817 m [1][0] = 2.0 * (q1q2 - q0q3);
818 m [2][0] = 2.0 * (q1q3 + q0q2);
819
820 m [0][1] = 2.0 * (q1q2 + q0q3);
821 m [1][1] = 2.0 * (q0q0 + q2q2) - 1.0;
822 m [2][1] = 2.0 * (q2q3 - q0q1);
823
824 m [0][2] = 2.0 * (q1q3 - q0q2);
825 m [1][2] = 2.0 * (q2q3 + q0q1);
826 m [2][2] = 2.0 * (q0q0 + q3q3) - 1.0;
827
828 return m;
829
830 }
831
832 /** Apply the rotation to a vector.
833 * @param u vector to apply the rotation to
834 * @return a new vector which is the image of u by the rotation
835 */
836 public Vector3D applyTo(Vector3D u) {
837
838 double x = u.getX();
839 double y = u.getY();
840 double z = u.getZ();
841
842 double s = q1 * x + q2 * y + q3 * z;
843
844 return new Vector3D(2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x,
845 2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y,
846 2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - z);
847
848 }
849
850 /** Apply the inverse of the rotation to a vector.
851 * @param u vector to apply the inverse of the rotation to
852 * @return a new vector which such that u is its image by the rotation
853 */
854 public Vector3D applyInverseTo(Vector3D u) {
855
856 double x = u.getX();
857 double y = u.getY();
858 double z = u.getZ();
859
860 double s = q1 * x + q2 * y + q3 * z;
861 double m0 = -q0;
862
863 return new Vector3D(2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x,
864 2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y,
865 2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - z);
866
867 }
868
869 /** Apply the instance to another rotation.
870 * Applying the instance to a rotation is computing the composition
871 * in an order compliant with the following rule : let u be any
872 * vector and v its image by r (i.e. r.applyTo(u) = v), let w be the image
873 * of v by the instance (i.e. applyTo(v) = w), then w = comp.applyTo(u),
874 * where comp = applyTo(r).
875 * @param r rotation to apply the rotation to
876 * @return a new rotation which is the composition of r by the instance
877 */
878 public Rotation applyTo(Rotation r) {
879 return new Rotation(r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),
880 r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),
881 r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),
882 r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),
883 false);
884 }
885
886 /** Apply the inverse of the instance to another rotation.
887 * Applying the inverse of the instance to a rotation is computing
888 * the composition in an order compliant with the following rule :
889 * let u be any vector and v its image by r (i.e. r.applyTo(u) = v),
890 * let w be the inverse image of v by the instance
891 * (i.e. applyInverseTo(v) = w), then w = comp.applyTo(u), where
892 * comp = applyInverseTo(r).
893 * @param r rotation to apply the rotation to
894 * @return a new rotation which is the composition of r by the inverse
895 * of the instance
896 */
897 public Rotation applyInverseTo(Rotation r) {
898 return new Rotation(-r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),
899 -r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),
900 -r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),
901 -r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),
902 false);
903 }
904
905 /** Perfect orthogonality on a 3X3 matrix.
906 * @param m initial matrix (not exactly orthogonal)
907 * @param threshold convergence threshold for the iterative
908 * orthogonality correction (convergence is reached when the
909 * difference between two steps of the Frobenius norm of the
910 * correction is below this threshold)
911 * @return an orthogonal matrix close to m
912 * @exception NotARotationMatrixException if the matrix cannot be
913 * orthogonalized with the given threshold after 10 iterations
914 */
915 private double[][] orthogonalizeMatrix(double[][] m, double threshold)
916 throws NotARotationMatrixException {
917 double[] m0 = m[0];
918 double[] m1 = m[1];
919 double[] m2 = m[2];
920 double x00 = m0[0];
921 double x01 = m0[1];
922 double x02 = m0[2];
923 double x10 = m1[0];
924 double x11 = m1[1];
925 double x12 = m1[2];
926 double x20 = m2[0];
927 double x21 = m2[1];
928 double x22 = m2[2];
929 double fn = 0;
930 double fn1;
931
932 double[][] o = new double[3][3];
933 double[] o0 = o[0];
934 double[] o1 = o[1];
935 double[] o2 = o[2];
936
937 // iterative correction: Xn+1 = Xn - 0.5 * (Xn.Mt.Xn - M)
938 int i = 0;
939 while (++i < 11) {
940
941 // Mt.Xn
942 double mx00 = m0[0] * x00 + m1[0] * x10 + m2[0] * x20;
943 double mx10 = m0[1] * x00 + m1[1] * x10 + m2[1] * x20;
944 double mx20 = m0[2] * x00 + m1[2] * x10 + m2[2] * x20;
945 double mx01 = m0[0] * x01 + m1[0] * x11 + m2[0] * x21;
946 double mx11 = m0[1] * x01 + m1[1] * x11 + m2[1] * x21;
947 double mx21 = m0[2] * x01 + m1[2] * x11 + m2[2] * x21;
948 double mx02 = m0[0] * x02 + m1[0] * x12 + m2[0] * x22;
949 double mx12 = m0[1] * x02 + m1[1] * x12 + m2[1] * x22;
950 double mx22 = m0[2] * x02 + m1[2] * x12 + m2[2] * x22;
951
952 // Xn+1
953 o0[0] = x00 - 0.5 * (x00 * mx00 + x01 * mx10 + x02 * mx20 - m0[0]);
954 o0[1] = x01 - 0.5 * (x00 * mx01 + x01 * mx11 + x02 * mx21 - m0[1]);
955 o0[2] = x02 - 0.5 * (x00 * mx02 + x01 * mx12 + x02 * mx22 - m0[2]);
956 o1[0] = x10 - 0.5 * (x10 * mx00 + x11 * mx10 + x12 * mx20 - m1[0]);
957 o1[1] = x11 - 0.5 * (x10 * mx01 + x11 * mx11 + x12 * mx21 - m1[1]);
958 o1[2] = x12 - 0.5 * (x10 * mx02 + x11 * mx12 + x12 * mx22 - m1[2]);
959 o2[0] = x20 - 0.5 * (x20 * mx00 + x21 * mx10 + x22 * mx20 - m2[0]);
960 o2[1] = x21 - 0.5 * (x20 * mx01 + x21 * mx11 + x22 * mx21 - m2[1]);
961 o2[2] = x22 - 0.5 * (x20 * mx02 + x21 * mx12 + x22 * mx22 - m2[2]);
962
963 // correction on each elements
964 double corr00 = o0[0] - m0[0];
965 double corr01 = o0[1] - m0[1];
966 double corr02 = o0[2] - m0[2];
967 double corr10 = o1[0] - m1[0];
968 double corr11 = o1[1] - m1[1];
969 double corr12 = o1[2] - m1[2];
970 double corr20 = o2[0] - m2[0];
971 double corr21 = o2[1] - m2[1];
972 double corr22 = o2[2] - m2[2];
973
974 // Frobenius norm of the correction
975 fn1 = corr00 * corr00 + corr01 * corr01 + corr02 * corr02 +
976 corr10 * corr10 + corr11 * corr11 + corr12 * corr12 +
977 corr20 * corr20 + corr21 * corr21 + corr22 * corr22;
978
979 // convergence test
980 if (FastMath.abs(fn1 - fn) <= threshold) {
981 return o;
982 }
983
984 // prepare next iteration
985 x00 = o0[0];
986 x01 = o0[1];
987 x02 = o0[2];
988 x10 = o1[0];
989 x11 = o1[1];
990 x12 = o1[2];
991 x20 = o2[0];
992 x21 = o2[1];
993 x22 = o2[2];
994 fn = fn1;
995
996 }
997
998 // the algorithm did not converge after 10 iterations
999 throw new NotARotationMatrixException(
1000 LocalizedFormats.UNABLE_TO_ORTHOGONOLIZE_MATRIX,
1001 i - 1);
1002 }
1003
1004 /** Compute the <i>distance</i> between two rotations.
1005 * <p>The <i>distance</i> is intended here as a way to check if two
1006 * rotations are almost similar (i.e. they transform vectors the same way)
1007 * or very different. It is mathematically defined as the angle of
1008 * the rotation r that prepended to one of the rotations gives the other
1009 * one:</p>
1010 * <pre>
1011 * r<sub>1</sub>(r) = r<sub>2</sub>
1012 * </pre>
1013 * <p>This distance is an angle between 0 and π. Its value is the smallest
1014 * possible upper bound of the angle in radians between r<sub>1</sub>(v)
1015 * and r<sub>2</sub>(v) for all possible vectors v. This upper bound is
1016 * reached for some v. The distance is equal to 0 if and only if the two
1017 * rotations are identical.</p>
1018 * <p>Comparing two rotations should always be done using this value rather
1019 * than for example comparing the components of the quaternions. It is much
1020 * more stable, and has a geometric meaning. Also comparing quaternions
1021 * components is error prone since for example quaternions (0.36, 0.48, -0.48, -0.64)
1022 * and (-0.36, -0.48, 0.48, 0.64) represent exactly the same rotation despite
1023 * their components are different (they are exact opposites).</p>
1024 * @param r1 first rotation
1025 * @param r2 second rotation
1026 * @return <i>distance</i> between r1 and r2
1027 */
1028 public static double distance(Rotation r1, Rotation r2) {
1029 return r1.applyInverseTo(r2).getAngle();
1030 }
1031
1032 }