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.math3.geometry.euclidean.threed;
019
020 import java.io.Serializable;
021
022 import org.apache.commons.math3.exception.MathArithmeticException;
023 import org.apache.commons.math3.exception.MathIllegalArgumentException;
024 import org.apache.commons.math3.exception.util.LocalizedFormats;
025 import org.apache.commons.math3.util.FastMath;
026 import org.apache.commons.math3.util.MathArrays;
027
028 /**
029 * This class implements rotations in a three-dimensional space.
030 *
031 * <p>Rotations can be represented by several different mathematical
032 * entities (matrices, axe and angle, Cardan or Euler angles,
033 * quaternions). This class presents an higher level abstraction, more
034 * user-oriented and hiding this implementation details. Well, for the
035 * curious, we use quaternions for the internal representation. The
036 * user can build a rotation from any of these representations, and
037 * any of these representations can be retrieved from a
038 * <code>Rotation</code> instance (see the various constructors and
039 * getters). In addition, a rotation can also be built implicitly
040 * from a set of vectors and their image.</p>
041 * <p>This implies that this class can be used to convert from one
042 * representation to another one. For example, converting a rotation
043 * matrix into a set of Cardan angles from can be done using the
044 * following single line of code:</p>
045 * <pre>
046 * double[] angles = new Rotation(matrix, 1.0e-10).getAngles(RotationOrder.XYZ);
047 * </pre>
048 * <p>Focus is oriented on what a rotation <em>do</em> rather than on its
049 * underlying representation. Once it has been built, and regardless of its
050 * internal representation, a rotation is an <em>operator</em> which basically
051 * transforms three dimensional {@link Vector3D vectors} into other three
052 * dimensional {@link Vector3D vectors}. Depending on the application, the
053 * meaning of these vectors may vary and the semantics of the rotation also.</p>
054 * <p>For example in an spacecraft attitude simulation tool, users will often
055 * consider the vectors are fixed (say the Earth direction for example) and the
056 * frames change. The rotation transforms the coordinates of the vector in inertial
057 * frame into the coordinates of the same vector in satellite frame. In this
058 * case, the rotation implicitly defines the relation between the two frames.</p>
059 * <p>Another example could be a telescope control application, where the rotation
060 * would transform the sighting direction at rest into the desired observing
061 * direction when the telescope is pointed towards an object of interest. In this
062 * case the rotation transforms the direction at rest in a topocentric frame
063 * into the sighting direction in the same topocentric frame. This implies in this
064 * case the frame is fixed and the vector moves.</p>
065 * <p>In many case, both approaches will be combined. In our telescope example,
066 * we will probably also need to transform the observing direction in the topocentric
067 * frame into the observing direction in inertial frame taking into account the observatory
068 * location and the Earth rotation, which would essentially be an application of the
069 * first approach.</p>
070 *
071 * <p>These examples show that a rotation is what the user wants it to be. This
072 * class does not push the user towards one specific definition and hence does not
073 * provide methods like <code>projectVectorIntoDestinationFrame</code> or
074 * <code>computeTransformedDirection</code>. It provides simpler and more generic
075 * methods: {@link #applyTo(Vector3D) applyTo(Vector3D)} and {@link
076 * #applyInverseTo(Vector3D) applyInverseTo(Vector3D)}.</p>
077 *
078 * <p>Since a rotation is basically a vectorial operator, several rotations can be
079 * composed together and the composite operation <code>r = r<sub>1</sub> o
080 * r<sub>2</sub></code> (which means that for each vector <code>u</code>,
081 * <code>r(u) = r<sub>1</sub>(r<sub>2</sub>(u))</code>) is also a rotation. Hence
082 * we can consider that in addition to vectors, a rotation can be applied to other
083 * rotations as well (or to itself). With our previous notations, we would say we
084 * can apply <code>r<sub>1</sub></code> to <code>r<sub>2</sub></code> and the result
085 * we get is <code>r = r<sub>1</sub> o r<sub>2</sub></code>. For this purpose, the
086 * class provides the methods: {@link #applyTo(Rotation) applyTo(Rotation)} and
087 * {@link #applyInverseTo(Rotation) applyInverseTo(Rotation)}.</p>
088 *
089 * <p>Rotations are guaranteed to be immutable objects.</p>
090 *
091 * @version $Id: Rotation.java 1416643 2012-12-03 19:37:14Z tn $
092 * @see Vector3D
093 * @see RotationOrder
094 * @since 1.2
095 */
096
097 public class Rotation implements Serializable {
098
099 /** Identity rotation. */
100 public static final Rotation IDENTITY = new Rotation(1.0, 0.0, 0.0, 0.0, false);
101
102 /** Serializable version identifier */
103 private static final long serialVersionUID = -2153622329907944313L;
104
105 /** Scalar coordinate of the quaternion. */
106 private final double q0;
107
108 /** First coordinate of the vectorial part of the quaternion. */
109 private final double q1;
110
111 /** Second coordinate of the vectorial part of the quaternion. */
112 private final double q2;
113
114 /** Third coordinate of the vectorial part of the quaternion. */
115 private final double q3;
116
117 /** Build a rotation from the quaternion coordinates.
118 * <p>A rotation can be built from a <em>normalized</em> quaternion,
119 * i.e. a quaternion for which q<sub>0</sub><sup>2</sup> +
120 * q<sub>1</sub><sup>2</sup> + q<sub>2</sub><sup>2</sup> +
121 * q<sub>3</sub><sup>2</sup> = 1. If the quaternion is not normalized,
122 * the constructor can normalize it in a preprocessing step.</p>
123 * <p>Note that some conventions put the scalar part of the quaternion
124 * as the 4<sup>th</sup> component and the vector part as the first three
125 * components. This is <em>not</em> our convention. We put the scalar part
126 * as the first component.</p>
127 * @param q0 scalar part of the quaternion
128 * @param q1 first coordinate of the vectorial part of the quaternion
129 * @param q2 second coordinate of the vectorial part of the quaternion
130 * @param q3 third coordinate of the vectorial part of the quaternion
131 * @param needsNormalization if true, the coordinates are considered
132 * not to be normalized, a normalization preprocessing step is performed
133 * before using them
134 */
135 public Rotation(double q0, double q1, double q2, double q3,
136 boolean needsNormalization) {
137
138 if (needsNormalization) {
139 // normalization preprocessing
140 double inv = 1.0 / FastMath.sqrt(q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3);
141 q0 *= inv;
142 q1 *= inv;
143 q2 *= inv;
144 q3 *= inv;
145 }
146
147 this.q0 = q0;
148 this.q1 = q1;
149 this.q2 = q2;
150 this.q3 = q3;
151
152 }
153
154 /** Build a rotation from an axis and an angle.
155 * <p>We use the convention that angles are oriented according to
156 * the effect of the rotation on vectors around the axis. That means
157 * that if (i, j, k) is a direct frame and if we first provide +k as
158 * the axis and π/2 as the angle to this constructor, and then
159 * {@link #applyTo(Vector3D) apply} the instance to +i, we will get
160 * +j.</p>
161 * <p>Another way to represent our convention is to say that a rotation
162 * of angle θ about the unit vector (x, y, z) is the same as the
163 * rotation build from quaternion components { cos(-θ/2),
164 * x * sin(-θ/2), y * sin(-θ/2), z * sin(-θ/2) }.
165 * Note the minus sign on the angle!</p>
166 * <p>On the one hand this convention is consistent with a vectorial
167 * perspective (moving vectors in fixed frames), on the other hand it
168 * is different from conventions with a frame perspective (fixed vectors
169 * viewed from different frames) like the ones used for example in spacecraft
170 * attitude community or in the graphics community.</p>
171 * @param axis axis around which to rotate
172 * @param angle rotation angle.
173 * @exception MathIllegalArgumentException if the axis norm is zero
174 */
175 public Rotation(Vector3D axis, double angle) throws MathIllegalArgumentException {
176
177 double norm = axis.getNorm();
178 if (norm == 0) {
179 throw new MathIllegalArgumentException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_AXIS);
180 }
181
182 double halfAngle = -0.5 * angle;
183 double coeff = FastMath.sin(halfAngle) / norm;
184
185 q0 = FastMath.cos (halfAngle);
186 q1 = coeff * axis.getX();
187 q2 = coeff * axis.getY();
188 q3 = coeff * axis.getZ();
189
190 }
191
192 /** Build a rotation from a 3X3 matrix.
193
194 * <p>Rotation matrices are orthogonal matrices, i.e. unit matrices
195 * (which are matrices for which m.m<sup>T</sup> = I) with real
196 * coefficients. The module of the determinant of unit matrices is
197 * 1, among the orthogonal 3X3 matrices, only the ones having a
198 * positive determinant (+1) are rotation matrices.</p>
199
200 * <p>When a rotation is defined by a matrix with truncated values
201 * (typically when it is extracted from a technical sheet where only
202 * four to five significant digits are available), the matrix is not
203 * orthogonal anymore. This constructor handles this case
204 * transparently by using a copy of the given matrix and applying a
205 * correction to the copy in order to perfect its orthogonality. If
206 * the Frobenius norm of the correction needed is above the given
207 * threshold, then the matrix is considered to be too far from a
208 * true rotation matrix and an exception is thrown.<p>
209
210 * @param m rotation matrix
211 * @param threshold convergence threshold for the iterative
212 * orthogonality correction (convergence is reached when the
213 * difference between two steps of the Frobenius norm of the
214 * correction is below this threshold)
215
216 * @exception NotARotationMatrixException if the matrix is not a 3X3
217 * matrix, or if it cannot be transformed into an orthogonal matrix
218 * with the given threshold, or if the determinant of the resulting
219 * orthogonal matrix is negative
220
221 */
222 public Rotation(double[][] m, double threshold)
223 throws NotARotationMatrixException {
224
225 // dimension check
226 if ((m.length != 3) || (m[0].length != 3) ||
227 (m[1].length != 3) || (m[2].length != 3)) {
228 throw new NotARotationMatrixException(
229 LocalizedFormats.ROTATION_MATRIX_DIMENSIONS,
230 m.length, m[0].length);
231 }
232
233 // compute a "close" orthogonal matrix
234 double[][] ort = orthogonalizeMatrix(m, threshold);
235
236 // check the sign of the determinant
237 double det = ort[0][0] * (ort[1][1] * ort[2][2] - ort[2][1] * ort[1][2]) -
238 ort[1][0] * (ort[0][1] * ort[2][2] - ort[2][1] * ort[0][2]) +
239 ort[2][0] * (ort[0][1] * ort[1][2] - ort[1][1] * ort[0][2]);
240 if (det < 0.0) {
241 throw new NotARotationMatrixException(
242 LocalizedFormats.CLOSEST_ORTHOGONAL_MATRIX_HAS_NEGATIVE_DETERMINANT,
243 det);
244 }
245
246 double[] quat = mat2quat(ort);
247 q0 = quat[0];
248 q1 = quat[1];
249 q2 = quat[2];
250 q3 = quat[3];
251
252 }
253
254 /** Build the rotation that transforms a pair of vector into another pair.
255
256 * <p>Except for possible scale factors, if the instance were applied to
257 * the pair (u<sub>1</sub>, u<sub>2</sub>) it will produce the pair
258 * (v<sub>1</sub>, v<sub>2</sub>).</p>
259
260 * <p>If the angular separation between u<sub>1</sub> and u<sub>2</sub> is
261 * not the same as the angular separation between v<sub>1</sub> and
262 * v<sub>2</sub>, then a corrected v'<sub>2</sub> will be used rather than
263 * v<sub>2</sub>, the corrected vector will be in the (v<sub>1</sub>,
264 * v<sub>2</sub>) plane.</p>
265
266 * @param u1 first vector of the origin pair
267 * @param u2 second vector of the origin pair
268 * @param v1 desired image of u1 by the rotation
269 * @param v2 desired image of u2 by the rotation
270 * @exception MathArithmeticException if the norm of one of the vectors is zero,
271 * or if one of the pair is degenerated (i.e. the vectors of the pair are colinear)
272 */
273 public Rotation(Vector3D u1, Vector3D u2, Vector3D v1, Vector3D v2)
274 throws MathArithmeticException {
275
276 // build orthonormalized base from u1, u2
277 // this fails when vectors are null or colinear, which is forbidden to define a rotation
278 final Vector3D u3 = u1.crossProduct(u2).normalize();
279 u2 = u3.crossProduct(u1).normalize();
280 u1 = u1.normalize();
281
282 // build an orthonormalized base from v1, v2
283 // this fails when vectors are null or colinear, which is forbidden to define a rotation
284 final Vector3D v3 = v1.crossProduct(v2).normalize();
285 v2 = v3.crossProduct(v1).normalize();
286 v1 = v1.normalize();
287
288 // buid a matrix transforming the first base into the second one
289 final double[][] m = new double[][] {
290 {
291 MathArrays.linearCombination(u1.getX(), v1.getX(), u2.getX(), v2.getX(), u3.getX(), v3.getX()),
292 MathArrays.linearCombination(u1.getY(), v1.getX(), u2.getY(), v2.getX(), u3.getY(), v3.getX()),
293 MathArrays.linearCombination(u1.getZ(), v1.getX(), u2.getZ(), v2.getX(), u3.getZ(), v3.getX())
294 },
295 {
296 MathArrays.linearCombination(u1.getX(), v1.getY(), u2.getX(), v2.getY(), u3.getX(), v3.getY()),
297 MathArrays.linearCombination(u1.getY(), v1.getY(), u2.getY(), v2.getY(), u3.getY(), v3.getY()),
298 MathArrays.linearCombination(u1.getZ(), v1.getY(), u2.getZ(), v2.getY(), u3.getZ(), v3.getY())
299 },
300 {
301 MathArrays.linearCombination(u1.getX(), v1.getZ(), u2.getX(), v2.getZ(), u3.getX(), v3.getZ()),
302 MathArrays.linearCombination(u1.getY(), v1.getZ(), u2.getY(), v2.getZ(), u3.getY(), v3.getZ()),
303 MathArrays.linearCombination(u1.getZ(), v1.getZ(), u2.getZ(), v2.getZ(), u3.getZ(), v3.getZ())
304 }
305 };
306
307 double[] quat = mat2quat(m);
308 q0 = quat[0];
309 q1 = quat[1];
310 q2 = quat[2];
311 q3 = quat[3];
312
313 }
314
315 /** Build one of the rotations that transform one vector into another one.
316
317 * <p>Except for a possible scale factor, if the instance were
318 * applied to the vector u it will produce the vector v. There is an
319 * infinite number of such rotations, this constructor choose the
320 * one with the smallest associated angle (i.e. the one whose axis
321 * is orthogonal to the (u, v) plane). If u and v are colinear, an
322 * arbitrary rotation axis is chosen.</p>
323
324 * @param u origin vector
325 * @param v desired image of u by the rotation
326 * @exception MathArithmeticException if the norm of one of the vectors is zero
327 */
328 public Rotation(Vector3D u, Vector3D v) throws MathArithmeticException {
329
330 double normProduct = u.getNorm() * v.getNorm();
331 if (normProduct == 0) {
332 throw new MathArithmeticException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_DEFINING_VECTOR);
333 }
334
335 double dot = u.dotProduct(v);
336
337 if (dot < ((2.0e-15 - 1.0) * normProduct)) {
338 // special case u = -v: we select a PI angle rotation around
339 // an arbitrary vector orthogonal to u
340 Vector3D w = u.orthogonal();
341 q0 = 0.0;
342 q1 = -w.getX();
343 q2 = -w.getY();
344 q3 = -w.getZ();
345 } else {
346 // general case: (u, v) defines a plane, we select
347 // the shortest possible rotation: axis orthogonal to this plane
348 q0 = FastMath.sqrt(0.5 * (1.0 + dot / normProduct));
349 double coeff = 1.0 / (2.0 * q0 * normProduct);
350 Vector3D q = v.crossProduct(u);
351 q1 = coeff * q.getX();
352 q2 = coeff * q.getY();
353 q3 = coeff * q.getZ();
354 }
355
356 }
357
358 /** Build a rotation from three Cardan or Euler elementary rotations.
359
360 * <p>Cardan rotations are three successive rotations around the
361 * canonical axes X, Y and Z, each axis being used once. There are
362 * 6 such sets of rotations (XYZ, XZY, YXZ, YZX, ZXY and ZYX). Euler
363 * rotations are three successive rotations around the canonical
364 * axes X, Y and Z, the first and last rotations being around the
365 * same axis. There are 6 such sets of rotations (XYX, XZX, YXY,
366 * YZY, ZXZ and ZYZ), the most popular one being ZXZ.</p>
367 * <p>Beware that many people routinely use the term Euler angles even
368 * for what really are Cardan angles (this confusion is especially
369 * widespread in the aerospace business where Roll, Pitch and Yaw angles
370 * are often wrongly tagged as Euler angles).</p>
371
372 * @param order order of rotations to use
373 * @param alpha1 angle of the first elementary rotation
374 * @param alpha2 angle of the second elementary rotation
375 * @param alpha3 angle of the third elementary rotation
376 */
377 public Rotation(RotationOrder order,
378 double alpha1, double alpha2, double alpha3) {
379 Rotation r1 = new Rotation(order.getA1(), alpha1);
380 Rotation r2 = new Rotation(order.getA2(), alpha2);
381 Rotation r3 = new Rotation(order.getA3(), alpha3);
382 Rotation composed = r1.applyTo(r2.applyTo(r3));
383 q0 = composed.q0;
384 q1 = composed.q1;
385 q2 = composed.q2;
386 q3 = composed.q3;
387 }
388
389 /** Convert an orthogonal rotation matrix to a quaternion.
390 * @param ort orthogonal rotation matrix
391 * @return quaternion corresponding to the matrix
392 */
393 private static double[] mat2quat(final double[][] ort) {
394
395 final double[] quat = new double[4];
396
397 // There are different ways to compute the quaternions elements
398 // from the matrix. They all involve computing one element from
399 // the diagonal of the matrix, and computing the three other ones
400 // using a formula involving a division by the first element,
401 // which unfortunately can be zero. Since the norm of the
402 // quaternion is 1, we know at least one element has an absolute
403 // value greater or equal to 0.5, so it is always possible to
404 // select the right formula and avoid division by zero and even
405 // numerical inaccuracy. Checking the elements in turn and using
406 // the first one greater than 0.45 is safe (this leads to a simple
407 // test since qi = 0.45 implies 4 qi^2 - 1 = -0.19)
408 double s = ort[0][0] + ort[1][1] + ort[2][2];
409 if (s > -0.19) {
410 // compute q0 and deduce q1, q2 and q3
411 quat[0] = 0.5 * FastMath.sqrt(s + 1.0);
412 double inv = 0.25 / quat[0];
413 quat[1] = inv * (ort[1][2] - ort[2][1]);
414 quat[2] = inv * (ort[2][0] - ort[0][2]);
415 quat[3] = inv * (ort[0][1] - ort[1][0]);
416 } else {
417 s = ort[0][0] - ort[1][1] - ort[2][2];
418 if (s > -0.19) {
419 // compute q1 and deduce q0, q2 and q3
420 quat[1] = 0.5 * FastMath.sqrt(s + 1.0);
421 double inv = 0.25 / quat[1];
422 quat[0] = inv * (ort[1][2] - ort[2][1]);
423 quat[2] = inv * (ort[0][1] + ort[1][0]);
424 quat[3] = inv * (ort[0][2] + ort[2][0]);
425 } else {
426 s = ort[1][1] - ort[0][0] - ort[2][2];
427 if (s > -0.19) {
428 // compute q2 and deduce q0, q1 and q3
429 quat[2] = 0.5 * FastMath.sqrt(s + 1.0);
430 double inv = 0.25 / quat[2];
431 quat[0] = inv * (ort[2][0] - ort[0][2]);
432 quat[1] = inv * (ort[0][1] + ort[1][0]);
433 quat[3] = inv * (ort[2][1] + ort[1][2]);
434 } else {
435 // compute q3 and deduce q0, q1 and q2
436 s = ort[2][2] - ort[0][0] - ort[1][1];
437 quat[3] = 0.5 * FastMath.sqrt(s + 1.0);
438 double inv = 0.25 / quat[3];
439 quat[0] = inv * (ort[0][1] - ort[1][0]);
440 quat[1] = inv * (ort[0][2] + ort[2][0]);
441 quat[2] = inv * (ort[2][1] + ort[1][2]);
442 }
443 }
444 }
445
446 return quat;
447
448 }
449
450 /** Revert a rotation.
451 * Build a rotation which reverse the effect of another
452 * rotation. This means that if r(u) = v, then r.revert(v) = u. The
453 * instance is not changed.
454 * @return a new rotation whose effect is the reverse of the effect
455 * of the instance
456 */
457 public Rotation revert() {
458 return new Rotation(-q0, q1, q2, q3, false);
459 }
460
461 /** Get the scalar coordinate of the quaternion.
462 * @return scalar coordinate of the quaternion
463 */
464 public double getQ0() {
465 return q0;
466 }
467
468 /** Get the first coordinate of the vectorial part of the quaternion.
469 * @return first coordinate of the vectorial part of the quaternion
470 */
471 public double getQ1() {
472 return q1;
473 }
474
475 /** Get the second coordinate of the vectorial part of the quaternion.
476 * @return second coordinate of the vectorial part of the quaternion
477 */
478 public double getQ2() {
479 return q2;
480 }
481
482 /** Get the third coordinate of the vectorial part of the quaternion.
483 * @return third coordinate of the vectorial part of the quaternion
484 */
485 public double getQ3() {
486 return q3;
487 }
488
489 /** Get the normalized axis of the rotation.
490 * @return normalized axis of the rotation
491 * @see #Rotation(Vector3D, double)
492 */
493 public Vector3D getAxis() {
494 double squaredSine = q1 * q1 + q2 * q2 + q3 * q3;
495 if (squaredSine == 0) {
496 return new Vector3D(1, 0, 0);
497 } else if (q0 < 0) {
498 double inverse = 1 / FastMath.sqrt(squaredSine);
499 return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
500 }
501 double inverse = -1 / FastMath.sqrt(squaredSine);
502 return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
503 }
504
505 /** Get the angle of the rotation.
506 * @return angle of the rotation (between 0 and π)
507 * @see #Rotation(Vector3D, double)
508 */
509 public double getAngle() {
510 if ((q0 < -0.1) || (q0 > 0.1)) {
511 return 2 * FastMath.asin(FastMath.sqrt(q1 * q1 + q2 * q2 + q3 * q3));
512 } else if (q0 < 0) {
513 return 2 * FastMath.acos(-q0);
514 }
515 return 2 * FastMath.acos(q0);
516 }
517
518 /** Get the Cardan or Euler angles corresponding to the instance.
519
520 * <p>The equations show that each rotation can be defined by two
521 * different values of the Cardan or Euler angles set. For example
522 * if Cardan angles are used, the rotation defined by the angles
523 * a<sub>1</sub>, a<sub>2</sub> and a<sub>3</sub> is the same as
524 * the rotation defined by the angles π + a<sub>1</sub>, π
525 * - a<sub>2</sub> and π + a<sub>3</sub>. This method implements
526 * the following arbitrary choices:</p>
527 * <ul>
528 * <li>for Cardan angles, the chosen set is the one for which the
529 * second angle is between -π/2 and π/2 (i.e its cosine is
530 * positive),</li>
531 * <li>for Euler angles, the chosen set is the one for which the
532 * second angle is between 0 and π (i.e its sine is positive).</li>
533 * </ul>
534
535 * <p>Cardan and Euler angle have a very disappointing drawback: all
536 * of them have singularities. This means that if the instance is
537 * too close to the singularities corresponding to the given
538 * rotation order, it will be impossible to retrieve the angles. For
539 * Cardan angles, this is often called gimbal lock. There is
540 * <em>nothing</em> to do to prevent this, it is an intrinsic problem
541 * with Cardan and Euler representation (but not a problem with the
542 * rotation itself, which is perfectly well defined). For Cardan
543 * angles, singularities occur when the second angle is close to
544 * -π/2 or +π/2, for Euler angle singularities occur when the
545 * second angle is close to 0 or π, this implies that the identity
546 * rotation is always singular for Euler angles!</p>
547
548 * @param order rotation order to use
549 * @return an array of three angles, in the order specified by the set
550 * @exception CardanEulerSingularityException if the rotation is
551 * singular with respect to the angles set specified
552 */
553 public double[] getAngles(RotationOrder order)
554 throws CardanEulerSingularityException {
555
556 if (order == RotationOrder.XYZ) {
557
558 // r (Vector3D.plusK) coordinates are :
559 // sin (theta), -cos (theta) sin (phi), cos (theta) cos (phi)
560 // (-r) (Vector3D.plusI) coordinates are :
561 // cos (psi) cos (theta), -sin (psi) cos (theta), sin (theta)
562 // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
563 Vector3D v1 = applyTo(Vector3D.PLUS_K);
564 Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
565 if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
566 throw new CardanEulerSingularityException(true);
567 }
568 return new double[] {
569 FastMath.atan2(-(v1.getY()), v1.getZ()),
570 FastMath.asin(v2.getZ()),
571 FastMath.atan2(-(v2.getY()), v2.getX())
572 };
573
574 } else if (order == RotationOrder.XZY) {
575
576 // r (Vector3D.plusJ) coordinates are :
577 // -sin (psi), cos (psi) cos (phi), cos (psi) sin (phi)
578 // (-r) (Vector3D.plusI) coordinates are :
579 // cos (theta) cos (psi), -sin (psi), sin (theta) cos (psi)
580 // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
581 Vector3D v1 = applyTo(Vector3D.PLUS_J);
582 Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
583 if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
584 throw new CardanEulerSingularityException(true);
585 }
586 return new double[] {
587 FastMath.atan2(v1.getZ(), v1.getY()),
588 -FastMath.asin(v2.getY()),
589 FastMath.atan2(v2.getZ(), v2.getX())
590 };
591
592 } else if (order == RotationOrder.YXZ) {
593
594 // r (Vector3D.plusK) coordinates are :
595 // cos (phi) sin (theta), -sin (phi), cos (phi) cos (theta)
596 // (-r) (Vector3D.plusJ) coordinates are :
597 // sin (psi) cos (phi), cos (psi) cos (phi), -sin (phi)
598 // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
599 Vector3D v1 = applyTo(Vector3D.PLUS_K);
600 Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
601 if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
602 throw new CardanEulerSingularityException(true);
603 }
604 return new double[] {
605 FastMath.atan2(v1.getX(), v1.getZ()),
606 -FastMath.asin(v2.getZ()),
607 FastMath.atan2(v2.getX(), v2.getY())
608 };
609
610 } else if (order == RotationOrder.YZX) {
611
612 // r (Vector3D.plusI) coordinates are :
613 // cos (psi) cos (theta), sin (psi), -cos (psi) sin (theta)
614 // (-r) (Vector3D.plusJ) coordinates are :
615 // sin (psi), cos (phi) cos (psi), -sin (phi) cos (psi)
616 // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
617 Vector3D v1 = applyTo(Vector3D.PLUS_I);
618 Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
619 if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
620 throw new CardanEulerSingularityException(true);
621 }
622 return new double[] {
623 FastMath.atan2(-(v1.getZ()), v1.getX()),
624 FastMath.asin(v2.getX()),
625 FastMath.atan2(-(v2.getZ()), v2.getY())
626 };
627
628 } else if (order == RotationOrder.ZXY) {
629
630 // r (Vector3D.plusJ) coordinates are :
631 // -cos (phi) sin (psi), cos (phi) cos (psi), sin (phi)
632 // (-r) (Vector3D.plusK) coordinates are :
633 // -sin (theta) cos (phi), sin (phi), cos (theta) cos (phi)
634 // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
635 Vector3D v1 = applyTo(Vector3D.PLUS_J);
636 Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
637 if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
638 throw new CardanEulerSingularityException(true);
639 }
640 return new double[] {
641 FastMath.atan2(-(v1.getX()), v1.getY()),
642 FastMath.asin(v2.getY()),
643 FastMath.atan2(-(v2.getX()), v2.getZ())
644 };
645
646 } else if (order == RotationOrder.ZYX) {
647
648 // r (Vector3D.plusI) coordinates are :
649 // cos (theta) cos (psi), cos (theta) sin (psi), -sin (theta)
650 // (-r) (Vector3D.plusK) coordinates are :
651 // -sin (theta), sin (phi) cos (theta), cos (phi) cos (theta)
652 // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
653 Vector3D v1 = applyTo(Vector3D.PLUS_I);
654 Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
655 if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
656 throw new CardanEulerSingularityException(true);
657 }
658 return new double[] {
659 FastMath.atan2(v1.getY(), v1.getX()),
660 -FastMath.asin(v2.getX()),
661 FastMath.atan2(v2.getY(), v2.getZ())
662 };
663
664 } else if (order == RotationOrder.XYX) {
665
666 // r (Vector3D.plusI) coordinates are :
667 // cos (theta), sin (phi1) sin (theta), -cos (phi1) sin (theta)
668 // (-r) (Vector3D.plusI) coordinates are :
669 // cos (theta), sin (theta) sin (phi2), sin (theta) cos (phi2)
670 // and we can choose to have theta in the interval [0 ; PI]
671 Vector3D v1 = applyTo(Vector3D.PLUS_I);
672 Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
673 if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
674 throw new CardanEulerSingularityException(false);
675 }
676 return new double[] {
677 FastMath.atan2(v1.getY(), -v1.getZ()),
678 FastMath.acos(v2.getX()),
679 FastMath.atan2(v2.getY(), v2.getZ())
680 };
681
682 } else if (order == RotationOrder.XZX) {
683
684 // r (Vector3D.plusI) coordinates are :
685 // cos (psi), cos (phi1) sin (psi), sin (phi1) sin (psi)
686 // (-r) (Vector3D.plusI) coordinates are :
687 // cos (psi), -sin (psi) cos (phi2), sin (psi) sin (phi2)
688 // and we can choose to have psi in the interval [0 ; PI]
689 Vector3D v1 = applyTo(Vector3D.PLUS_I);
690 Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
691 if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
692 throw new CardanEulerSingularityException(false);
693 }
694 return new double[] {
695 FastMath.atan2(v1.getZ(), v1.getY()),
696 FastMath.acos(v2.getX()),
697 FastMath.atan2(v2.getZ(), -v2.getY())
698 };
699
700 } else if (order == RotationOrder.YXY) {
701
702 // r (Vector3D.plusJ) coordinates are :
703 // sin (theta1) sin (phi), cos (phi), cos (theta1) sin (phi)
704 // (-r) (Vector3D.plusJ) coordinates are :
705 // sin (phi) sin (theta2), cos (phi), -sin (phi) cos (theta2)
706 // and we can choose to have phi in the interval [0 ; PI]
707 Vector3D v1 = applyTo(Vector3D.PLUS_J);
708 Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
709 if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
710 throw new CardanEulerSingularityException(false);
711 }
712 return new double[] {
713 FastMath.atan2(v1.getX(), v1.getZ()),
714 FastMath.acos(v2.getY()),
715 FastMath.atan2(v2.getX(), -v2.getZ())
716 };
717
718 } else if (order == RotationOrder.YZY) {
719
720 // r (Vector3D.plusJ) coordinates are :
721 // -cos (theta1) sin (psi), cos (psi), sin (theta1) sin (psi)
722 // (-r) (Vector3D.plusJ) coordinates are :
723 // sin (psi) cos (theta2), cos (psi), sin (psi) sin (theta2)
724 // and we can choose to have psi in the interval [0 ; PI]
725 Vector3D v1 = applyTo(Vector3D.PLUS_J);
726 Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
727 if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
728 throw new CardanEulerSingularityException(false);
729 }
730 return new double[] {
731 FastMath.atan2(v1.getZ(), -v1.getX()),
732 FastMath.acos(v2.getY()),
733 FastMath.atan2(v2.getZ(), v2.getX())
734 };
735
736 } else if (order == RotationOrder.ZXZ) {
737
738 // r (Vector3D.plusK) coordinates are :
739 // sin (psi1) sin (phi), -cos (psi1) sin (phi), cos (phi)
740 // (-r) (Vector3D.plusK) coordinates are :
741 // sin (phi) sin (psi2), sin (phi) cos (psi2), cos (phi)
742 // and we can choose to have phi in the interval [0 ; PI]
743 Vector3D v1 = applyTo(Vector3D.PLUS_K);
744 Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
745 if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
746 throw new CardanEulerSingularityException(false);
747 }
748 return new double[] {
749 FastMath.atan2(v1.getX(), -v1.getY()),
750 FastMath.acos(v2.getZ()),
751 FastMath.atan2(v2.getX(), v2.getY())
752 };
753
754 } else { // last possibility is ZYZ
755
756 // r (Vector3D.plusK) coordinates are :
757 // cos (psi1) sin (theta), sin (psi1) sin (theta), cos (theta)
758 // (-r) (Vector3D.plusK) coordinates are :
759 // -sin (theta) cos (psi2), sin (theta) sin (psi2), cos (theta)
760 // and we can choose to have theta in the interval [0 ; PI]
761 Vector3D v1 = applyTo(Vector3D.PLUS_K);
762 Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
763 if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
764 throw new CardanEulerSingularityException(false);
765 }
766 return new double[] {
767 FastMath.atan2(v1.getY(), v1.getX()),
768 FastMath.acos(v2.getZ()),
769 FastMath.atan2(v2.getY(), -v2.getX())
770 };
771
772 }
773
774 }
775
776 /** Get the 3X3 matrix corresponding to the instance
777 * @return the matrix corresponding to the instance
778 */
779 public double[][] getMatrix() {
780
781 // products
782 double q0q0 = q0 * q0;
783 double q0q1 = q0 * q1;
784 double q0q2 = q0 * q2;
785 double q0q3 = q0 * q3;
786 double q1q1 = q1 * q1;
787 double q1q2 = q1 * q2;
788 double q1q3 = q1 * q3;
789 double q2q2 = q2 * q2;
790 double q2q3 = q2 * q3;
791 double q3q3 = q3 * q3;
792
793 // create the matrix
794 double[][] m = new double[3][];
795 m[0] = new double[3];
796 m[1] = new double[3];
797 m[2] = new double[3];
798
799 m [0][0] = 2.0 * (q0q0 + q1q1) - 1.0;
800 m [1][0] = 2.0 * (q1q2 - q0q3);
801 m [2][0] = 2.0 * (q1q3 + q0q2);
802
803 m [0][1] = 2.0 * (q1q2 + q0q3);
804 m [1][1] = 2.0 * (q0q0 + q2q2) - 1.0;
805 m [2][1] = 2.0 * (q2q3 - q0q1);
806
807 m [0][2] = 2.0 * (q1q3 - q0q2);
808 m [1][2] = 2.0 * (q2q3 + q0q1);
809 m [2][2] = 2.0 * (q0q0 + q3q3) - 1.0;
810
811 return m;
812
813 }
814
815 /** Apply the rotation to a vector.
816 * @param u vector to apply the rotation to
817 * @return a new vector which is the image of u by the rotation
818 */
819 public Vector3D applyTo(Vector3D u) {
820
821 double x = u.getX();
822 double y = u.getY();
823 double z = u.getZ();
824
825 double s = q1 * x + q2 * y + q3 * z;
826
827 return new Vector3D(2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x,
828 2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y,
829 2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - z);
830
831 }
832
833 /** Apply the rotation to a vector stored in an array.
834 * @param in an array with three items which stores vector to rotate
835 * @param out an array with three items to put result to (it can be the same
836 * array as in)
837 */
838 public void applyTo(final double[] in, final double[] out) {
839
840 final double x = in[0];
841 final double y = in[1];
842 final double z = in[2];
843
844 final double s = q1 * x + q2 * y + q3 * z;
845
846 out[0] = 2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x;
847 out[1] = 2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y;
848 out[2] = 2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - z;
849
850 }
851
852 /** Apply the inverse of the rotation to a vector.
853 * @param u vector to apply the inverse of the rotation to
854 * @return a new vector which such that u is its image by the rotation
855 */
856 public Vector3D applyInverseTo(Vector3D u) {
857
858 double x = u.getX();
859 double y = u.getY();
860 double z = u.getZ();
861
862 double s = q1 * x + q2 * y + q3 * z;
863 double m0 = -q0;
864
865 return new Vector3D(2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x,
866 2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y,
867 2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - z);
868
869 }
870
871 /** Apply the inverse of the rotation to a vector stored in an array.
872 * @param in an array with three items which stores vector to rotate
873 * @param out an array with three items to put result to (it can be the same
874 * array as in)
875 */
876 public void applyInverseTo(final double[] in, final double[] out) {
877
878 final double x = in[0];
879 final double y = in[1];
880 final double z = in[2];
881
882 final double s = q1 * x + q2 * y + q3 * z;
883 final double m0 = -q0;
884
885 out[0] = 2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x;
886 out[1] = 2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y;
887 out[2] = 2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - z;
888
889 }
890
891 /** Apply the instance to another rotation.
892 * Applying the instance to a rotation is computing the composition
893 * in an order compliant with the following rule : let u be any
894 * vector and v its image by r (i.e. r.applyTo(u) = v), let w be the image
895 * of v by the instance (i.e. applyTo(v) = w), then w = comp.applyTo(u),
896 * where comp = applyTo(r).
897 * @param r rotation to apply the rotation to
898 * @return a new rotation which is the composition of r by the instance
899 */
900 public Rotation applyTo(Rotation r) {
901 return new Rotation(r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),
902 r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),
903 r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),
904 r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),
905 false);
906 }
907
908 /** Apply the inverse of the instance to another rotation.
909 * Applying the inverse of the instance to a rotation is computing
910 * the composition in an order compliant with the following rule :
911 * let u be any vector and v its image by r (i.e. r.applyTo(u) = v),
912 * let w be the inverse image of v by the instance
913 * (i.e. applyInverseTo(v) = w), then w = comp.applyTo(u), where
914 * comp = applyInverseTo(r).
915 * @param r rotation to apply the rotation to
916 * @return a new rotation which is the composition of r by the inverse
917 * of the instance
918 */
919 public Rotation applyInverseTo(Rotation r) {
920 return new Rotation(-r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),
921 -r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),
922 -r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),
923 -r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),
924 false);
925 }
926
927 /** Perfect orthogonality on a 3X3 matrix.
928 * @param m initial matrix (not exactly orthogonal)
929 * @param threshold convergence threshold for the iterative
930 * orthogonality correction (convergence is reached when the
931 * difference between two steps of the Frobenius norm of the
932 * correction is below this threshold)
933 * @return an orthogonal matrix close to m
934 * @exception NotARotationMatrixException if the matrix cannot be
935 * orthogonalized with the given threshold after 10 iterations
936 */
937 private double[][] orthogonalizeMatrix(double[][] m, double threshold)
938 throws NotARotationMatrixException {
939 double[] m0 = m[0];
940 double[] m1 = m[1];
941 double[] m2 = m[2];
942 double x00 = m0[0];
943 double x01 = m0[1];
944 double x02 = m0[2];
945 double x10 = m1[0];
946 double x11 = m1[1];
947 double x12 = m1[2];
948 double x20 = m2[0];
949 double x21 = m2[1];
950 double x22 = m2[2];
951 double fn = 0;
952 double fn1;
953
954 double[][] o = new double[3][3];
955 double[] o0 = o[0];
956 double[] o1 = o[1];
957 double[] o2 = o[2];
958
959 // iterative correction: Xn+1 = Xn - 0.5 * (Xn.Mt.Xn - M)
960 int i = 0;
961 while (++i < 11) {
962
963 // Mt.Xn
964 double mx00 = m0[0] * x00 + m1[0] * x10 + m2[0] * x20;
965 double mx10 = m0[1] * x00 + m1[1] * x10 + m2[1] * x20;
966 double mx20 = m0[2] * x00 + m1[2] * x10 + m2[2] * x20;
967 double mx01 = m0[0] * x01 + m1[0] * x11 + m2[0] * x21;
968 double mx11 = m0[1] * x01 + m1[1] * x11 + m2[1] * x21;
969 double mx21 = m0[2] * x01 + m1[2] * x11 + m2[2] * x21;
970 double mx02 = m0[0] * x02 + m1[0] * x12 + m2[0] * x22;
971 double mx12 = m0[1] * x02 + m1[1] * x12 + m2[1] * x22;
972 double mx22 = m0[2] * x02 + m1[2] * x12 + m2[2] * x22;
973
974 // Xn+1
975 o0[0] = x00 - 0.5 * (x00 * mx00 + x01 * mx10 + x02 * mx20 - m0[0]);
976 o0[1] = x01 - 0.5 * (x00 * mx01 + x01 * mx11 + x02 * mx21 - m0[1]);
977 o0[2] = x02 - 0.5 * (x00 * mx02 + x01 * mx12 + x02 * mx22 - m0[2]);
978 o1[0] = x10 - 0.5 * (x10 * mx00 + x11 * mx10 + x12 * mx20 - m1[0]);
979 o1[1] = x11 - 0.5 * (x10 * mx01 + x11 * mx11 + x12 * mx21 - m1[1]);
980 o1[2] = x12 - 0.5 * (x10 * mx02 + x11 * mx12 + x12 * mx22 - m1[2]);
981 o2[0] = x20 - 0.5 * (x20 * mx00 + x21 * mx10 + x22 * mx20 - m2[0]);
982 o2[1] = x21 - 0.5 * (x20 * mx01 + x21 * mx11 + x22 * mx21 - m2[1]);
983 o2[2] = x22 - 0.5 * (x20 * mx02 + x21 * mx12 + x22 * mx22 - m2[2]);
984
985 // correction on each elements
986 double corr00 = o0[0] - m0[0];
987 double corr01 = o0[1] - m0[1];
988 double corr02 = o0[2] - m0[2];
989 double corr10 = o1[0] - m1[0];
990 double corr11 = o1[1] - m1[1];
991 double corr12 = o1[2] - m1[2];
992 double corr20 = o2[0] - m2[0];
993 double corr21 = o2[1] - m2[1];
994 double corr22 = o2[2] - m2[2];
995
996 // Frobenius norm of the correction
997 fn1 = corr00 * corr00 + corr01 * corr01 + corr02 * corr02 +
998 corr10 * corr10 + corr11 * corr11 + corr12 * corr12 +
999 corr20 * corr20 + corr21 * corr21 + corr22 * corr22;
1000
1001 // convergence test
1002 if (FastMath.abs(fn1 - fn) <= threshold) {
1003 return o;
1004 }
1005
1006 // prepare next iteration
1007 x00 = o0[0];
1008 x01 = o0[1];
1009 x02 = o0[2];
1010 x10 = o1[0];
1011 x11 = o1[1];
1012 x12 = o1[2];
1013 x20 = o2[0];
1014 x21 = o2[1];
1015 x22 = o2[2];
1016 fn = fn1;
1017
1018 }
1019
1020 // the algorithm did not converge after 10 iterations
1021 throw new NotARotationMatrixException(
1022 LocalizedFormats.UNABLE_TO_ORTHOGONOLIZE_MATRIX,
1023 i - 1);
1024 }
1025
1026 /** Compute the <i>distance</i> between two rotations.
1027 * <p>The <i>distance</i> is intended here as a way to check if two
1028 * rotations are almost similar (i.e. they transform vectors the same way)
1029 * or very different. It is mathematically defined as the angle of
1030 * the rotation r that prepended to one of the rotations gives the other
1031 * one:</p>
1032 * <pre>
1033 * r<sub>1</sub>(r) = r<sub>2</sub>
1034 * </pre>
1035 * <p>This distance is an angle between 0 and π. Its value is the smallest
1036 * possible upper bound of the angle in radians between r<sub>1</sub>(v)
1037 * and r<sub>2</sub>(v) for all possible vectors v. This upper bound is
1038 * reached for some v. The distance is equal to 0 if and only if the two
1039 * rotations are identical.</p>
1040 * <p>Comparing two rotations should always be done using this value rather
1041 * than for example comparing the components of the quaternions. It is much
1042 * more stable, and has a geometric meaning. Also comparing quaternions
1043 * components is error prone since for example quaternions (0.36, 0.48, -0.48, -0.64)
1044 * and (-0.36, -0.48, 0.48, 0.64) represent exactly the same rotation despite
1045 * their components are different (they are exact opposites).</p>
1046 * @param r1 first rotation
1047 * @param r2 second rotation
1048 * @return <i>distance</i> between r1 and r2
1049 */
1050 public static double distance(Rotation r1, Rotation r2) {
1051 return r1.applyInverseTo(r2).getAngle();
1052 }
1053
1054 }