001 /*
002 * Licensed to the Apache Software Foundation (ASF) under one or more
003 * contributor license agreements. See the NOTICE file distributed with
004 * this work for additional information regarding copyright ownership.
005 * The ASF licenses this file to You under the Apache License, Version 2.0
006 * (the "License"); you may not use this file except in compliance with
007 * the License. You may obtain a copy of the License at
008 *
009 * http://www.apache.org/licenses/LICENSE-2.0
010 *
011 * Unless required by applicable law or agreed to in writing, software
012 * distributed under the License is distributed on an "AS IS" BASIS,
013 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
014 * See the License for the specific language governing permissions and
015 * limitations under the License.
016 */
017
018 package org.apache.commons.math.geometry;
019
020 import java.io.Serializable;
021
022 import org.apache.commons.math.MathRuntimeException;
023
024 /**
025 * This class implements rotations in a three-dimensional space.
026 *
027 * <p>Rotations can be represented by several different mathematical
028 * entities (matrices, axe and angle, Cardan or Euler angles,
029 * quaternions). This class presents an higher level abstraction, more
030 * user-oriented and hiding this implementation details. Well, for the
031 * curious, we use quaternions for the internal representation. The
032 * user can build a rotation from any of these representations, and
033 * any of these representations can be retrieved from a
034 * <code>Rotation</code> instance (see the various constructors and
035 * getters). In addition, a rotation can also be built implicitely
036 * from a set of vectors and their image.</p>
037 * <p>This implies that this class can be used to convert from one
038 * representation to another one. For example, converting a rotation
039 * matrix into a set of Cardan angles from can be done using the
040 * followong single line of code:</p>
041 * <pre>
042 * double[] angles = new Rotation(matrix, 1.0e-10).getAngles(RotationOrder.XYZ);
043 * </pre>
044 * <p>Focus is oriented on what a rotation <em>do</em> rather than on its
045 * underlying representation. Once it has been built, and regardless of its
046 * internal representation, a rotation is an <em>operator</em> which basically
047 * transforms three dimensional {@link Vector3D vectors} into other three
048 * dimensional {@link Vector3D vectors}. Depending on the application, the
049 * meaning of these vectors may vary and the semantics of the rotation also.</p>
050 * <p>For example in an spacecraft attitude simulation tool, users will often
051 * consider the vectors are fixed (say the Earth direction for example) and the
052 * rotation transforms the coordinates coordinates of this vector in inertial
053 * frame into the coordinates of the same vector in satellite frame. In this
054 * case, the rotation implicitely defines the relation between the two frames.
055 * Another example could be a telescope control application, where the rotation
056 * would transform the sighting direction at rest into the desired observing
057 * direction when the telescope is pointed towards an object of interest. In this
058 * case the rotation transforms the directionf at rest in a topocentric frame
059 * into the sighting direction in the same topocentric frame. In many case, both
060 * approaches will be combined, in our telescope example, we will probably also
061 * need to transform the observing direction in the topocentric frame into the
062 * observing direction in inertial frame taking into account the observatory
063 * location and the Earth rotation.</p>
064 *
065 * <p>These examples show that a rotation is what the user wants it to be, so this
066 * class does not push the user towards one specific definition and hence does not
067 * provide methods like <code>projectVectorIntoDestinationFrame</code> or
068 * <code>computeTransformedDirection</code>. It provides simpler and more generic
069 * methods: {@link #applyTo(Vector3D) applyTo(Vector3D)} and {@link
070 * #applyInverseTo(Vector3D) applyInverseTo(Vector3D)}.</p>
071 *
072 * <p>Since a rotation is basically a vectorial operator, several rotations can be
073 * composed together and the composite operation <code>r = r<sub>1</sub> o
074 * r<sub>2</sub></code> (which means that for each vector <code>u</code>,
075 * <code>r(u) = r<sub>1</sub>(r<sub>2</sub>(u))</code>) is also a rotation. Hence
076 * we can consider that in addition to vectors, a rotation can be applied to other
077 * rotations as well (or to itself). With our previous notations, we would say we
078 * can apply <code>r<sub>1</sub></code> to <code>r<sub>2</sub></code> and the result
079 * we get is <code>r = r<sub>1</sub> o r<sub>2</sub></code>. For this purpose, the
080 * class provides the methods: {@link #applyTo(Rotation) applyTo(Rotation)} and
081 * {@link #applyInverseTo(Rotation) applyInverseTo(Rotation)}.</p>
082 *
083 * <p>Rotations are guaranteed to be immutable objects.</p>
084 *
085 * @version $Revision: 772119 $ $Date: 2009-05-06 05:43:28 -0400 (Wed, 06 May 2009) $
086 * @see Vector3D
087 * @see RotationOrder
088 * @since 1.2
089 */
090
091 public class Rotation implements Serializable {
092
093 /** Identity rotation. */
094 public static final Rotation IDENTITY = new Rotation(1.0, 0.0, 0.0, 0.0, false);
095
096 /** Serializable version identifier */
097 private static final long serialVersionUID = -2153622329907944313L;
098
099 /** Scalar coordinate of the quaternion. */
100 private final double q0;
101
102 /** First coordinate of the vectorial part of the quaternion. */
103 private final double q1;
104
105 /** Second coordinate of the vectorial part of the quaternion. */
106 private final double q2;
107
108 /** Third coordinate of the vectorial part of the quaternion. */
109 private final double q3;
110
111 /** Build a rotation from the quaternion coordinates.
112 * <p>A rotation can be built from a <em>normalized</em> quaternion,
113 * i.e. a quaternion for which q<sub>0</sub><sup>2</sup> +
114 * q<sub>1</sub><sup>2</sup> + q<sub>2</sub><sup>2</sup> +
115 * q<sub>3</sub><sup>2</sup> = 1. If the quaternion is not normalized,
116 * the constructor can normalize it in a preprocessing step.</p>
117 * @param q0 scalar part of the quaternion
118 * @param q1 first coordinate of the vectorial part of the quaternion
119 * @param q2 second coordinate of the vectorial part of the quaternion
120 * @param q3 third coordinate of the vectorial part of the quaternion
121 * @param needsNormalization if true, the coordinates are considered
122 * not to be normalized, a normalization preprocessing step is performed
123 * before using them
124 */
125 public Rotation(double q0, double q1, double q2, double q3,
126 boolean needsNormalization) {
127
128 if (needsNormalization) {
129 // normalization preprocessing
130 double inv = 1.0 / Math.sqrt(q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3);
131 q0 *= inv;
132 q1 *= inv;
133 q2 *= inv;
134 q3 *= inv;
135 }
136
137 this.q0 = q0;
138 this.q1 = q1;
139 this.q2 = q2;
140 this.q3 = q3;
141
142 }
143
144 /** Build a rotation from an axis and an angle.
145 * <p>We use the convention that angles are oriented according to
146 * the effect of the rotation on vectors around the axis. That means
147 * that if (i, j, k) is a direct frame and if we first provide +k as
148 * the axis and PI/2 as the angle to this constructor, and then
149 * {@link #applyTo(Vector3D) apply} the instance to +i, we will get
150 * +j.</p>
151 * @param axis axis around which to rotate
152 * @param angle rotation angle.
153 * @exception ArithmeticException if the axis norm is zero
154 */
155 public Rotation(Vector3D axis, double angle) {
156
157 double norm = axis.getNorm();
158 if (norm == 0) {
159 throw MathRuntimeException.createArithmeticException("zero norm for rotation axis");
160 }
161
162 double halfAngle = -0.5 * angle;
163 double coeff = Math.sin(halfAngle) / norm;
164
165 q0 = Math.cos (halfAngle);
166 q1 = coeff * axis.getX();
167 q2 = coeff * axis.getY();
168 q3 = coeff * axis.getZ();
169
170 }
171
172 /** Build a rotation from a 3X3 matrix.
173
174 * <p>Rotation matrices are orthogonal matrices, i.e. unit matrices
175 * (which are matrices for which m.m<sup>T</sup> = I) with real
176 * coefficients. The module of the determinant of unit matrices is
177 * 1, among the orthogonal 3X3 matrices, only the ones having a
178 * positive determinant (+1) are rotation matrices.</p>
179
180 * <p>When a rotation is defined by a matrix with truncated values
181 * (typically when it is extracted from a technical sheet where only
182 * four to five significant digits are available), the matrix is not
183 * orthogonal anymore. This constructor handles this case
184 * transparently by using a copy of the given matrix and applying a
185 * correction to the copy in order to perfect its orthogonality. If
186 * the Frobenius norm of the correction needed is above the given
187 * threshold, then the matrix is considered to be too far from a
188 * true rotation matrix and an exception is thrown.<p>
189
190 * @param m rotation matrix
191 * @param threshold convergence threshold for the iterative
192 * orthogonality correction (convergence is reached when the
193 * difference between two steps of the Frobenius norm of the
194 * correction is below this threshold)
195
196 * @exception NotARotationMatrixException if the matrix is not a 3X3
197 * matrix, or if it cannot be transformed into an orthogonal matrix
198 * with the given threshold, or if the determinant of the resulting
199 * orthogonal matrix is negative
200
201 */
202 public Rotation(double[][] m, double threshold)
203 throws NotARotationMatrixException {
204
205 // dimension check
206 if ((m.length != 3) || (m[0].length != 3) ||
207 (m[1].length != 3) || (m[2].length != 3)) {
208 throw new NotARotationMatrixException(
209 "a {0}x{1} matrix cannot be a rotation matrix",
210 m.length, m[0].length);
211 }
212
213 // compute a "close" orthogonal matrix
214 double[][] ort = orthogonalizeMatrix(m, threshold);
215
216 // check the sign of the determinant
217 double det = ort[0][0] * (ort[1][1] * ort[2][2] - ort[2][1] * ort[1][2]) -
218 ort[1][0] * (ort[0][1] * ort[2][2] - ort[2][1] * ort[0][2]) +
219 ort[2][0] * (ort[0][1] * ort[1][2] - ort[1][1] * ort[0][2]);
220 if (det < 0.0) {
221 throw new NotARotationMatrixException(
222 "the closest orthogonal matrix has a negative determinant {0}",
223 det);
224 }
225
226 // There are different ways to compute the quaternions elements
227 // from the matrix. They all involve computing one element from
228 // the diagonal of the matrix, and computing the three other ones
229 // using a formula involving a division by the first element,
230 // which unfortunately can be zero. Since the norm of the
231 // quaternion is 1, we know at least one element has an absolute
232 // value greater or equal to 0.5, so it is always possible to
233 // select the right formula and avoid division by zero and even
234 // numerical inaccuracy. Checking the elements in turn and using
235 // the first one greater than 0.45 is safe (this leads to a simple
236 // test since qi = 0.45 implies 4 qi^2 - 1 = -0.19)
237 double s = ort[0][0] + ort[1][1] + ort[2][2];
238 if (s > -0.19) {
239 // compute q0 and deduce q1, q2 and q3
240 q0 = 0.5 * Math.sqrt(s + 1.0);
241 double inv = 0.25 / q0;
242 q1 = inv * (ort[1][2] - ort[2][1]);
243 q2 = inv * (ort[2][0] - ort[0][2]);
244 q3 = inv * (ort[0][1] - ort[1][0]);
245 } else {
246 s = ort[0][0] - ort[1][1] - ort[2][2];
247 if (s > -0.19) {
248 // compute q1 and deduce q0, q2 and q3
249 q1 = 0.5 * Math.sqrt(s + 1.0);
250 double inv = 0.25 / q1;
251 q0 = inv * (ort[1][2] - ort[2][1]);
252 q2 = inv * (ort[0][1] + ort[1][0]);
253 q3 = inv * (ort[0][2] + ort[2][0]);
254 } else {
255 s = ort[1][1] - ort[0][0] - ort[2][2];
256 if (s > -0.19) {
257 // compute q2 and deduce q0, q1 and q3
258 q2 = 0.5 * Math.sqrt(s + 1.0);
259 double inv = 0.25 / q2;
260 q0 = inv * (ort[2][0] - ort[0][2]);
261 q1 = inv * (ort[0][1] + ort[1][0]);
262 q3 = inv * (ort[2][1] + ort[1][2]);
263 } else {
264 // compute q3 and deduce q0, q1 and q2
265 s = ort[2][2] - ort[0][0] - ort[1][1];
266 q3 = 0.5 * Math.sqrt(s + 1.0);
267 double inv = 0.25 / q3;
268 q0 = inv * (ort[0][1] - ort[1][0]);
269 q1 = inv * (ort[0][2] + ort[2][0]);
270 q2 = inv * (ort[2][1] + ort[1][2]);
271 }
272 }
273 }
274
275 }
276
277 /** Build the rotation that transforms a pair of vector into another pair.
278
279 * <p>Except for possible scale factors, if the instance were applied to
280 * the pair (u<sub>1</sub>, u<sub>2</sub>) it will produce the pair
281 * (v<sub>1</sub>, v<sub>2</sub>).</p>
282
283 * <p>If the angular separation between u<sub>1</sub> and u<sub>2</sub> is
284 * not the same as the angular separation between v<sub>1</sub> and
285 * v<sub>2</sub>, then a corrected v'<sub>2</sub> will be used rather than
286 * v<sub>2</sub>, the corrected vector will be in the (v<sub>1</sub>,
287 * v<sub>2</sub>) plane.</p>
288
289 * @param u1 first vector of the origin pair
290 * @param u2 second vector of the origin pair
291 * @param v1 desired image of u1 by the rotation
292 * @param v2 desired image of u2 by the rotation
293 * @exception IllegalArgumentException if the norm of one of the vectors is zero
294 */
295 public Rotation(Vector3D u1, Vector3D u2, Vector3D v1, Vector3D v2) {
296
297 // norms computation
298 double u1u1 = Vector3D.dotProduct(u1, u1);
299 double u2u2 = Vector3D.dotProduct(u2, u2);
300 double v1v1 = Vector3D.dotProduct(v1, v1);
301 double v2v2 = Vector3D.dotProduct(v2, v2);
302 if ((u1u1 == 0) || (u2u2 == 0) || (v1v1 == 0) || (v2v2 == 0)) {
303 throw MathRuntimeException.createIllegalArgumentException("zero norm for rotation defining vector");
304 }
305
306 double u1x = u1.getX();
307 double u1y = u1.getY();
308 double u1z = u1.getZ();
309
310 double u2x = u2.getX();
311 double u2y = u2.getY();
312 double u2z = u2.getZ();
313
314 // normalize v1 in order to have (v1'|v1') = (u1|u1)
315 double coeff = Math.sqrt (u1u1 / v1v1);
316 double v1x = coeff * v1.getX();
317 double v1y = coeff * v1.getY();
318 double v1z = coeff * v1.getZ();
319 v1 = new Vector3D(v1x, v1y, v1z);
320
321 // adjust v2 in order to have (u1|u2) = (v1|v2) and (v2'|v2') = (u2|u2)
322 double u1u2 = Vector3D.dotProduct(u1, u2);
323 double v1v2 = Vector3D.dotProduct(v1, v2);
324 double coeffU = u1u2 / u1u1;
325 double coeffV = v1v2 / u1u1;
326 double beta = Math.sqrt((u2u2 - u1u2 * coeffU) / (v2v2 - v1v2 * coeffV));
327 double alpha = coeffU - beta * coeffV;
328 double v2x = alpha * v1x + beta * v2.getX();
329 double v2y = alpha * v1y + beta * v2.getY();
330 double v2z = alpha * v1z + beta * v2.getZ();
331 v2 = new Vector3D(v2x, v2y, v2z);
332
333 // preliminary computation (we use explicit formulation instead
334 // of relying on the Vector3D class in order to avoid building lots
335 // of temporary objects)
336 Vector3D uRef = u1;
337 Vector3D vRef = v1;
338 double dx1 = v1x - u1.getX();
339 double dy1 = v1y - u1.getY();
340 double dz1 = v1z - u1.getZ();
341 double dx2 = v2x - u2.getX();
342 double dy2 = v2y - u2.getY();
343 double dz2 = v2z - u2.getZ();
344 Vector3D k = new Vector3D(dy1 * dz2 - dz1 * dy2,
345 dz1 * dx2 - dx1 * dz2,
346 dx1 * dy2 - dy1 * dx2);
347 double c = k.getX() * (u1y * u2z - u1z * u2y) +
348 k.getY() * (u1z * u2x - u1x * u2z) +
349 k.getZ() * (u1x * u2y - u1y * u2x);
350
351 if (c == 0) {
352 // the (q1, q2, q3) vector is in the (u1, u2) plane
353 // we try other vectors
354 Vector3D u3 = Vector3D.crossProduct(u1, u2);
355 Vector3D v3 = Vector3D.crossProduct(v1, v2);
356 double u3x = u3.getX();
357 double u3y = u3.getY();
358 double u3z = u3.getZ();
359 double v3x = v3.getX();
360 double v3y = v3.getY();
361 double v3z = v3.getZ();
362
363 double dx3 = v3x - u3x;
364 double dy3 = v3y - u3y;
365 double dz3 = v3z - u3z;
366 k = new Vector3D(dy1 * dz3 - dz1 * dy3,
367 dz1 * dx3 - dx1 * dz3,
368 dx1 * dy3 - dy1 * dx3);
369 c = k.getX() * (u1y * u3z - u1z * u3y) +
370 k.getY() * (u1z * u3x - u1x * u3z) +
371 k.getZ() * (u1x * u3y - u1y * u3x);
372
373 if (c == 0) {
374 // the (q1, q2, q3) vector is aligned with u1:
375 // we try (u2, u3) and (v2, v3)
376 k = new Vector3D(dy2 * dz3 - dz2 * dy3,
377 dz2 * dx3 - dx2 * dz3,
378 dx2 * dy3 - dy2 * dx3);
379 c = k.getX() * (u2y * u3z - u2z * u3y) +
380 k.getY() * (u2z * u3x - u2x * u3z) +
381 k.getZ() * (u2x * u3y - u2y * u3x);
382
383 if (c == 0) {
384 // the (q1, q2, q3) vector is aligned with everything
385 // this is really the identity rotation
386 q0 = 1.0;
387 q1 = 0.0;
388 q2 = 0.0;
389 q3 = 0.0;
390 return;
391 }
392
393 // we will have to use u2 and v2 to compute the scalar part
394 uRef = u2;
395 vRef = v2;
396
397 }
398
399 }
400
401 // compute the vectorial part
402 c = Math.sqrt(c);
403 double inv = 1.0 / (c + c);
404 q1 = inv * k.getX();
405 q2 = inv * k.getY();
406 q3 = inv * k.getZ();
407
408 // compute the scalar part
409 k = new Vector3D(uRef.getY() * q3 - uRef.getZ() * q2,
410 uRef.getZ() * q1 - uRef.getX() * q3,
411 uRef.getX() * q2 - uRef.getY() * q1);
412 c = Vector3D.dotProduct(k, k);
413 q0 = Vector3D.dotProduct(vRef, k) / (c + c);
414
415 }
416
417 /** Build one of the rotations that transform one vector into another one.
418
419 * <p>Except for a possible scale factor, if the instance were
420 * applied to the vector u it will produce the vector v. There is an
421 * infinite number of such rotations, this constructor choose the
422 * one with the smallest associated angle (i.e. the one whose axis
423 * is orthogonal to the (u, v) plane). If u and v are colinear, an
424 * arbitrary rotation axis is chosen.</p>
425
426 * @param u origin vector
427 * @param v desired image of u by the rotation
428 * @exception IllegalArgumentException if the norm of one of the vectors is zero
429 */
430 public Rotation(Vector3D u, Vector3D v) {
431
432 double normProduct = u.getNorm() * v.getNorm();
433 if (normProduct == 0) {
434 throw MathRuntimeException.createIllegalArgumentException("zero norm for rotation defining vector");
435 }
436
437 double dot = Vector3D.dotProduct(u, v);
438
439 if (dot < ((2.0e-15 - 1.0) * normProduct)) {
440 // special case u = -v: we select a PI angle rotation around
441 // an arbitrary vector orthogonal to u
442 Vector3D w = u.orthogonal();
443 q0 = 0.0;
444 q1 = -w.getX();
445 q2 = -w.getY();
446 q3 = -w.getZ();
447 } else {
448 // general case: (u, v) defines a plane, we select
449 // the shortest possible rotation: axis orthogonal to this plane
450 q0 = Math.sqrt(0.5 * (1.0 + dot / normProduct));
451 double coeff = 1.0 / (2.0 * q0 * normProduct);
452 q1 = coeff * (v.getY() * u.getZ() - v.getZ() * u.getY());
453 q2 = coeff * (v.getZ() * u.getX() - v.getX() * u.getZ());
454 q3 = coeff * (v.getX() * u.getY() - v.getY() * u.getX());
455 }
456
457 }
458
459 /** Build a rotation from three Cardan or Euler elementary rotations.
460
461 * <p>Cardan rotations are three successive rotations around the
462 * canonical axes X, Y and Z, each axis being used once. There are
463 * 6 such sets of rotations (XYZ, XZY, YXZ, YZX, ZXY and ZYX). Euler
464 * rotations are three successive rotations around the canonical
465 * axes X, Y and Z, the first and last rotations being around the
466 * same axis. There are 6 such sets of rotations (XYX, XZX, YXY,
467 * YZY, ZXZ and ZYZ), the most popular one being ZXZ.</p>
468 * <p>Beware that many people routinely use the term Euler angles even
469 * for what really are Cardan angles (this confusion is especially
470 * widespread in the aerospace business where Roll, Pitch and Yaw angles
471 * are often wrongly tagged as Euler angles).</p>
472
473 * @param order order of rotations to use
474 * @param alpha1 angle of the first elementary rotation
475 * @param alpha2 angle of the second elementary rotation
476 * @param alpha3 angle of the third elementary rotation
477 */
478 public Rotation(RotationOrder order,
479 double alpha1, double alpha2, double alpha3) {
480 Rotation r1 = new Rotation(order.getA1(), alpha1);
481 Rotation r2 = new Rotation(order.getA2(), alpha2);
482 Rotation r3 = new Rotation(order.getA3(), alpha3);
483 Rotation composed = r1.applyTo(r2.applyTo(r3));
484 q0 = composed.q0;
485 q1 = composed.q1;
486 q2 = composed.q2;
487 q3 = composed.q3;
488 }
489
490 /** Revert a rotation.
491 * Build a rotation which reverse the effect of another
492 * rotation. This means that if r(u) = v, then r.revert(v) = u. The
493 * instance is not changed.
494 * @return a new rotation whose effect is the reverse of the effect
495 * of the instance
496 */
497 public Rotation revert() {
498 return new Rotation(-q0, q1, q2, q3, false);
499 }
500
501 /** Get the scalar coordinate of the quaternion.
502 * @return scalar coordinate of the quaternion
503 */
504 public double getQ0() {
505 return q0;
506 }
507
508 /** Get the first coordinate of the vectorial part of the quaternion.
509 * @return first coordinate of the vectorial part of the quaternion
510 */
511 public double getQ1() {
512 return q1;
513 }
514
515 /** Get the second coordinate of the vectorial part of the quaternion.
516 * @return second coordinate of the vectorial part of the quaternion
517 */
518 public double getQ2() {
519 return q2;
520 }
521
522 /** Get the third coordinate of the vectorial part of the quaternion.
523 * @return third coordinate of the vectorial part of the quaternion
524 */
525 public double getQ3() {
526 return q3;
527 }
528
529 /** Get the normalized axis of the rotation.
530 * @return normalized axis of the rotation
531 */
532 public Vector3D getAxis() {
533 double squaredSine = q1 * q1 + q2 * q2 + q3 * q3;
534 if (squaredSine == 0) {
535 return new Vector3D(1, 0, 0);
536 } else if (q0 < 0) {
537 double inverse = 1 / Math.sqrt(squaredSine);
538 return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
539 }
540 double inverse = -1 / Math.sqrt(squaredSine);
541 return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
542 }
543
544 /** Get the angle of the rotation.
545 * @return angle of the rotation (between 0 and π)
546 */
547 public double getAngle() {
548 if ((q0 < -0.1) || (q0 > 0.1)) {
549 return 2 * Math.asin(Math.sqrt(q1 * q1 + q2 * q2 + q3 * q3));
550 } else if (q0 < 0) {
551 return 2 * Math.acos(-q0);
552 }
553 return 2 * Math.acos(q0);
554 }
555
556 /** Get the Cardan or Euler angles corresponding to the instance.
557
558 * <p>The equations show that each rotation can be defined by two
559 * different values of the Cardan or Euler angles set. For example
560 * if Cardan angles are used, the rotation defined by the angles
561 * a<sub>1</sub>, a<sub>2</sub> and a<sub>3</sub> is the same as
562 * the rotation defined by the angles π + a<sub>1</sub>, π
563 * - a<sub>2</sub> and π + a<sub>3</sub>. This method implements
564 * the following arbitrary choices:</p>
565 * <ul>
566 * <li>for Cardan angles, the chosen set is the one for which the
567 * second angle is between -π/2 and π/2 (i.e its cosine is
568 * positive),</li>
569 * <li>for Euler angles, the chosen set is the one for which the
570 * second angle is between 0 and π (i.e its sine is positive).</li>
571 * </ul>
572
573 * <p>Cardan and Euler angle have a very disappointing drawback: all
574 * of them have singularities. This means that if the instance is
575 * too close to the singularities corresponding to the given
576 * rotation order, it will be impossible to retrieve the angles. For
577 * Cardan angles, this is often called gimbal lock. There is
578 * <em>nothing</em> to do to prevent this, it is an intrinsic problem
579 * with Cardan and Euler representation (but not a problem with the
580 * rotation itself, which is perfectly well defined). For Cardan
581 * angles, singularities occur when the second angle is close to
582 * -π/2 or +π/2, for Euler angle singularities occur when the
583 * second angle is close to 0 or π, this implies that the identity
584 * rotation is always singular for Euler angles!</p>
585
586 * @param order rotation order to use
587 * @return an array of three angles, in the order specified by the set
588 * @exception CardanEulerSingularityException if the rotation is
589 * singular with respect to the angles set specified
590 */
591 public double[] getAngles(RotationOrder order)
592 throws CardanEulerSingularityException {
593
594 if (order == RotationOrder.XYZ) {
595
596 // r (Vector3D.plusK) coordinates are :
597 // sin (theta), -cos (theta) sin (phi), cos (theta) cos (phi)
598 // (-r) (Vector3D.plusI) coordinates are :
599 // cos (psi) cos (theta), -sin (psi) cos (theta), sin (theta)
600 // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
601 Vector3D v1 = applyTo(Vector3D.PLUS_K);
602 Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
603 if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
604 throw new CardanEulerSingularityException(true);
605 }
606 return new double[] {
607 Math.atan2(-(v1.getY()), v1.getZ()),
608 Math.asin(v2.getZ()),
609 Math.atan2(-(v2.getY()), v2.getX())
610 };
611
612 } else if (order == RotationOrder.XZY) {
613
614 // r (Vector3D.plusJ) coordinates are :
615 // -sin (psi), cos (psi) cos (phi), cos (psi) sin (phi)
616 // (-r) (Vector3D.plusI) coordinates are :
617 // cos (theta) cos (psi), -sin (psi), sin (theta) cos (psi)
618 // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
619 Vector3D v1 = applyTo(Vector3D.PLUS_J);
620 Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
621 if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
622 throw new CardanEulerSingularityException(true);
623 }
624 return new double[] {
625 Math.atan2(v1.getZ(), v1.getY()),
626 -Math.asin(v2.getY()),
627 Math.atan2(v2.getZ(), v2.getX())
628 };
629
630 } else if (order == RotationOrder.YXZ) {
631
632 // r (Vector3D.plusK) coordinates are :
633 // cos (phi) sin (theta), -sin (phi), cos (phi) cos (theta)
634 // (-r) (Vector3D.plusJ) coordinates are :
635 // sin (psi) cos (phi), cos (psi) cos (phi), -sin (phi)
636 // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
637 Vector3D v1 = applyTo(Vector3D.PLUS_K);
638 Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
639 if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
640 throw new CardanEulerSingularityException(true);
641 }
642 return new double[] {
643 Math.atan2(v1.getX(), v1.getZ()),
644 -Math.asin(v2.getZ()),
645 Math.atan2(v2.getX(), v2.getY())
646 };
647
648 } else if (order == RotationOrder.YZX) {
649
650 // r (Vector3D.plusI) coordinates are :
651 // cos (psi) cos (theta), sin (psi), -cos (psi) sin (theta)
652 // (-r) (Vector3D.plusJ) coordinates are :
653 // sin (psi), cos (phi) cos (psi), -sin (phi) cos (psi)
654 // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
655 Vector3D v1 = applyTo(Vector3D.PLUS_I);
656 Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
657 if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
658 throw new CardanEulerSingularityException(true);
659 }
660 return new double[] {
661 Math.atan2(-(v1.getZ()), v1.getX()),
662 Math.asin(v2.getX()),
663 Math.atan2(-(v2.getZ()), v2.getY())
664 };
665
666 } else if (order == RotationOrder.ZXY) {
667
668 // r (Vector3D.plusJ) coordinates are :
669 // -cos (phi) sin (psi), cos (phi) cos (psi), sin (phi)
670 // (-r) (Vector3D.plusK) coordinates are :
671 // -sin (theta) cos (phi), sin (phi), cos (theta) cos (phi)
672 // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
673 Vector3D v1 = applyTo(Vector3D.PLUS_J);
674 Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
675 if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
676 throw new CardanEulerSingularityException(true);
677 }
678 return new double[] {
679 Math.atan2(-(v1.getX()), v1.getY()),
680 Math.asin(v2.getY()),
681 Math.atan2(-(v2.getX()), v2.getZ())
682 };
683
684 } else if (order == RotationOrder.ZYX) {
685
686 // r (Vector3D.plusI) coordinates are :
687 // cos (theta) cos (psi), cos (theta) sin (psi), -sin (theta)
688 // (-r) (Vector3D.plusK) coordinates are :
689 // -sin (theta), sin (phi) cos (theta), cos (phi) cos (theta)
690 // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
691 Vector3D v1 = applyTo(Vector3D.PLUS_I);
692 Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
693 if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
694 throw new CardanEulerSingularityException(true);
695 }
696 return new double[] {
697 Math.atan2(v1.getY(), v1.getX()),
698 -Math.asin(v2.getX()),
699 Math.atan2(v2.getY(), v2.getZ())
700 };
701
702 } else if (order == RotationOrder.XYX) {
703
704 // r (Vector3D.plusI) coordinates are :
705 // cos (theta), sin (phi1) sin (theta), -cos (phi1) sin (theta)
706 // (-r) (Vector3D.plusI) coordinates are :
707 // cos (theta), sin (theta) sin (phi2), sin (theta) cos (phi2)
708 // and we can choose to have theta in the interval [0 ; PI]
709 Vector3D v1 = applyTo(Vector3D.PLUS_I);
710 Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
711 if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
712 throw new CardanEulerSingularityException(false);
713 }
714 return new double[] {
715 Math.atan2(v1.getY(), -v1.getZ()),
716 Math.acos(v2.getX()),
717 Math.atan2(v2.getY(), v2.getZ())
718 };
719
720 } else if (order == RotationOrder.XZX) {
721
722 // r (Vector3D.plusI) coordinates are :
723 // cos (psi), cos (phi1) sin (psi), sin (phi1) sin (psi)
724 // (-r) (Vector3D.plusI) coordinates are :
725 // cos (psi), -sin (psi) cos (phi2), sin (psi) sin (phi2)
726 // and we can choose to have psi in the interval [0 ; PI]
727 Vector3D v1 = applyTo(Vector3D.PLUS_I);
728 Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
729 if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
730 throw new CardanEulerSingularityException(false);
731 }
732 return new double[] {
733 Math.atan2(v1.getZ(), v1.getY()),
734 Math.acos(v2.getX()),
735 Math.atan2(v2.getZ(), -v2.getY())
736 };
737
738 } else if (order == RotationOrder.YXY) {
739
740 // r (Vector3D.plusJ) coordinates are :
741 // sin (theta1) sin (phi), cos (phi), cos (theta1) sin (phi)
742 // (-r) (Vector3D.plusJ) coordinates are :
743 // sin (phi) sin (theta2), cos (phi), -sin (phi) cos (theta2)
744 // and we can choose to have phi in the interval [0 ; PI]
745 Vector3D v1 = applyTo(Vector3D.PLUS_J);
746 Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
747 if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
748 throw new CardanEulerSingularityException(false);
749 }
750 return new double[] {
751 Math.atan2(v1.getX(), v1.getZ()),
752 Math.acos(v2.getY()),
753 Math.atan2(v2.getX(), -v2.getZ())
754 };
755
756 } else if (order == RotationOrder.YZY) {
757
758 // r (Vector3D.plusJ) coordinates are :
759 // -cos (theta1) sin (psi), cos (psi), sin (theta1) sin (psi)
760 // (-r) (Vector3D.plusJ) coordinates are :
761 // sin (psi) cos (theta2), cos (psi), sin (psi) sin (theta2)
762 // and we can choose to have psi in the interval [0 ; PI]
763 Vector3D v1 = applyTo(Vector3D.PLUS_J);
764 Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
765 if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
766 throw new CardanEulerSingularityException(false);
767 }
768 return new double[] {
769 Math.atan2(v1.getZ(), -v1.getX()),
770 Math.acos(v2.getY()),
771 Math.atan2(v2.getZ(), v2.getX())
772 };
773
774 } else if (order == RotationOrder.ZXZ) {
775
776 // r (Vector3D.plusK) coordinates are :
777 // sin (psi1) sin (phi), -cos (psi1) sin (phi), cos (phi)
778 // (-r) (Vector3D.plusK) coordinates are :
779 // sin (phi) sin (psi2), sin (phi) cos (psi2), cos (phi)
780 // and we can choose to have phi in the interval [0 ; PI]
781 Vector3D v1 = applyTo(Vector3D.PLUS_K);
782 Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
783 if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
784 throw new CardanEulerSingularityException(false);
785 }
786 return new double[] {
787 Math.atan2(v1.getX(), -v1.getY()),
788 Math.acos(v2.getZ()),
789 Math.atan2(v2.getX(), v2.getY())
790 };
791
792 } else { // last possibility is ZYZ
793
794 // r (Vector3D.plusK) coordinates are :
795 // cos (psi1) sin (theta), sin (psi1) sin (theta), cos (theta)
796 // (-r) (Vector3D.plusK) coordinates are :
797 // -sin (theta) cos (psi2), sin (theta) sin (psi2), cos (theta)
798 // and we can choose to have theta in the interval [0 ; PI]
799 Vector3D v1 = applyTo(Vector3D.PLUS_K);
800 Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
801 if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
802 throw new CardanEulerSingularityException(false);
803 }
804 return new double[] {
805 Math.atan2(v1.getY(), v1.getX()),
806 Math.acos(v2.getZ()),
807 Math.atan2(v2.getY(), -v2.getX())
808 };
809
810 }
811
812 }
813
814 /** Get the 3X3 matrix corresponding to the instance
815 * @return the matrix corresponding to the instance
816 */
817 public double[][] getMatrix() {
818
819 // products
820 double q0q0 = q0 * q0;
821 double q0q1 = q0 * q1;
822 double q0q2 = q0 * q2;
823 double q0q3 = q0 * q3;
824 double q1q1 = q1 * q1;
825 double q1q2 = q1 * q2;
826 double q1q3 = q1 * q3;
827 double q2q2 = q2 * q2;
828 double q2q3 = q2 * q3;
829 double q3q3 = q3 * q3;
830
831 // create the matrix
832 double[][] m = new double[3][];
833 m[0] = new double[3];
834 m[1] = new double[3];
835 m[2] = new double[3];
836
837 m [0][0] = 2.0 * (q0q0 + q1q1) - 1.0;
838 m [1][0] = 2.0 * (q1q2 - q0q3);
839 m [2][0] = 2.0 * (q1q3 + q0q2);
840
841 m [0][1] = 2.0 * (q1q2 + q0q3);
842 m [1][1] = 2.0 * (q0q0 + q2q2) - 1.0;
843 m [2][1] = 2.0 * (q2q3 - q0q1);
844
845 m [0][2] = 2.0 * (q1q3 - q0q2);
846 m [1][2] = 2.0 * (q2q3 + q0q1);
847 m [2][2] = 2.0 * (q0q0 + q3q3) - 1.0;
848
849 return m;
850
851 }
852
853 /** Apply the rotation to a vector.
854 * @param u vector to apply the rotation to
855 * @return a new vector which is the image of u by the rotation
856 */
857 public Vector3D applyTo(Vector3D u) {
858
859 double x = u.getX();
860 double y = u.getY();
861 double z = u.getZ();
862
863 double s = q1 * x + q2 * y + q3 * z;
864
865 return new Vector3D(2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x,
866 2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y,
867 2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - z);
868
869 }
870
871 /** Apply the inverse of the rotation to a vector.
872 * @param u vector to apply the inverse of the rotation to
873 * @return a new vector which such that u is its image by the rotation
874 */
875 public Vector3D applyInverseTo(Vector3D u) {
876
877 double x = u.getX();
878 double y = u.getY();
879 double z = u.getZ();
880
881 double s = q1 * x + q2 * y + q3 * z;
882 double m0 = -q0;
883
884 return new Vector3D(2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x,
885 2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y,
886 2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - z);
887
888 }
889
890 /** Apply the instance to another rotation.
891 * Applying the instance to a rotation is computing the composition
892 * in an order compliant with the following rule : let u be any
893 * vector and v its image by r (i.e. r.applyTo(u) = v), let w be the image
894 * of v by the instance (i.e. applyTo(v) = w), then w = comp.applyTo(u),
895 * where comp = applyTo(r).
896 * @param r rotation to apply the rotation to
897 * @return a new rotation which is the composition of r by the instance
898 */
899 public Rotation applyTo(Rotation r) {
900 return new Rotation(r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),
901 r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),
902 r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),
903 r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),
904 false);
905 }
906
907 /** Apply the inverse of the instance to another rotation.
908 * Applying the inverse of the instance to a rotation is computing
909 * the composition in an order compliant with the following rule :
910 * let u be any vector and v its image by r (i.e. r.applyTo(u) = v),
911 * let w be the inverse image of v by the instance
912 * (i.e. applyInverseTo(v) = w), then w = comp.applyTo(u), where
913 * comp = applyInverseTo(r).
914 * @param r rotation to apply the rotation to
915 * @return a new rotation which is the composition of r by the inverse
916 * of the instance
917 */
918 public Rotation applyInverseTo(Rotation r) {
919 return new Rotation(-r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),
920 -r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),
921 -r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),
922 -r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),
923 false);
924 }
925
926 /** Perfect orthogonality on a 3X3 matrix.
927 * @param m initial matrix (not exactly orthogonal)
928 * @param threshold convergence threshold for the iterative
929 * orthogonality correction (convergence is reached when the
930 * difference between two steps of the Frobenius norm of the
931 * correction is below this threshold)
932 * @return an orthogonal matrix close to m
933 * @exception NotARotationMatrixException if the matrix cannot be
934 * orthogonalized with the given threshold after 10 iterations
935 */
936 private double[][] orthogonalizeMatrix(double[][] m, double threshold)
937 throws NotARotationMatrixException {
938 double[] m0 = m[0];
939 double[] m1 = m[1];
940 double[] m2 = m[2];
941 double x00 = m0[0];
942 double x01 = m0[1];
943 double x02 = m0[2];
944 double x10 = m1[0];
945 double x11 = m1[1];
946 double x12 = m1[2];
947 double x20 = m2[0];
948 double x21 = m2[1];
949 double x22 = m2[2];
950 double fn = 0;
951 double fn1;
952
953 double[][] o = new double[3][3];
954 double[] o0 = o[0];
955 double[] o1 = o[1];
956 double[] o2 = o[2];
957
958 // iterative correction: Xn+1 = Xn - 0.5 * (Xn.Mt.Xn - M)
959 int i = 0;
960 while (++i < 11) {
961
962 // Mt.Xn
963 double mx00 = m0[0] * x00 + m1[0] * x10 + m2[0] * x20;
964 double mx10 = m0[1] * x00 + m1[1] * x10 + m2[1] * x20;
965 double mx20 = m0[2] * x00 + m1[2] * x10 + m2[2] * x20;
966 double mx01 = m0[0] * x01 + m1[0] * x11 + m2[0] * x21;
967 double mx11 = m0[1] * x01 + m1[1] * x11 + m2[1] * x21;
968 double mx21 = m0[2] * x01 + m1[2] * x11 + m2[2] * x21;
969 double mx02 = m0[0] * x02 + m1[0] * x12 + m2[0] * x22;
970 double mx12 = m0[1] * x02 + m1[1] * x12 + m2[1] * x22;
971 double mx22 = m0[2] * x02 + m1[2] * x12 + m2[2] * x22;
972
973 // Xn+1
974 o0[0] = x00 - 0.5 * (x00 * mx00 + x01 * mx10 + x02 * mx20 - m0[0]);
975 o0[1] = x01 - 0.5 * (x00 * mx01 + x01 * mx11 + x02 * mx21 - m0[1]);
976 o0[2] = x02 - 0.5 * (x00 * mx02 + x01 * mx12 + x02 * mx22 - m0[2]);
977 o1[0] = x10 - 0.5 * (x10 * mx00 + x11 * mx10 + x12 * mx20 - m1[0]);
978 o1[1] = x11 - 0.5 * (x10 * mx01 + x11 * mx11 + x12 * mx21 - m1[1]);
979 o1[2] = x12 - 0.5 * (x10 * mx02 + x11 * mx12 + x12 * mx22 - m1[2]);
980 o2[0] = x20 - 0.5 * (x20 * mx00 + x21 * mx10 + x22 * mx20 - m2[0]);
981 o2[1] = x21 - 0.5 * (x20 * mx01 + x21 * mx11 + x22 * mx21 - m2[1]);
982 o2[2] = x22 - 0.5 * (x20 * mx02 + x21 * mx12 + x22 * mx22 - m2[2]);
983
984 // correction on each elements
985 double corr00 = o0[0] - m0[0];
986 double corr01 = o0[1] - m0[1];
987 double corr02 = o0[2] - m0[2];
988 double corr10 = o1[0] - m1[0];
989 double corr11 = o1[1] - m1[1];
990 double corr12 = o1[2] - m1[2];
991 double corr20 = o2[0] - m2[0];
992 double corr21 = o2[1] - m2[1];
993 double corr22 = o2[2] - m2[2];
994
995 // Frobenius norm of the correction
996 fn1 = corr00 * corr00 + corr01 * corr01 + corr02 * corr02 +
997 corr10 * corr10 + corr11 * corr11 + corr12 * corr12 +
998 corr20 * corr20 + corr21 * corr21 + corr22 * corr22;
999
1000 // convergence test
1001 if (Math.abs(fn1 - fn) <= threshold)
1002 return o;
1003
1004 // prepare next iteration
1005 x00 = o0[0];
1006 x01 = o0[1];
1007 x02 = o0[2];
1008 x10 = o1[0];
1009 x11 = o1[1];
1010 x12 = o1[2];
1011 x20 = o2[0];
1012 x21 = o2[1];
1013 x22 = o2[2];
1014 fn = fn1;
1015
1016 }
1017
1018 // the algorithm did not converge after 10 iterations
1019 throw new NotARotationMatrixException(
1020 "unable to orthogonalize matrix in {0} iterations",
1021 i - 1);
1022 }
1023
1024 /** Compute the <i>distance</i> between two rotations.
1025 * <p>The <i>distance</i> is intended here as a way to check if two
1026 * rotations are almost similar (i.e. they transform vectors the same way)
1027 * or very different. It is mathematically defined as the angle of
1028 * the rotation r that prepended to one of the rotations gives the other
1029 * one:</p>
1030 * <pre>
1031 * r<sub>1</sub>(r) = r<sub>2</sub>
1032 * </pre>
1033 * <p>This distance is an angle between 0 and π. Its value is the smallest
1034 * possible upper bound of the angle in radians between r<sub>1</sub>(v)
1035 * and r<sub>2</sub>(v) for all possible vectors v. This upper bound is
1036 * reached for some v. The distance is equal to 0 if and only if the two
1037 * rotations are identical.</p>
1038 * <p>Comparing two rotations should always be done using this value rather
1039 * than for example comparing the components of the quaternions. It is much
1040 * more stable, and has a geometric meaning. Also comparing quaternions
1041 * components is error prone since for example quaternions (0.36, 0.48, -0.48, -0.64)
1042 * and (-0.36, -0.48, 0.48, 0.64) represent exactly the same rotation despite
1043 * their components are different (they are exact opposites).</p>
1044 * @param r1 first rotation
1045 * @param r2 second rotation
1046 * @return <i>distance</i> between r1 and r2
1047 */
1048 public static double distance(Rotation r1, Rotation r2) {
1049 return r1.applyInverseTo(r2).getAngle();
1050 }
1051
1052 }