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1   /*
2    * Licensed to the Apache Software Foundation (ASF) under one or more
3    * contributor license agreements.  See the NOTICE file distributed with
4    * this work for additional information regarding copyright ownership.
5    * The ASF licenses this file to You under the Apache License, Version 2.0
6    * (the "License"); you may not use this file except in compliance with
7    * the License.  You may obtain a copy of the License at
8    *
9    *      http://www.apache.org/licenses/LICENSE-2.0
10   *
11   * Unless required by applicable law or agreed to in writing, software
12   * distributed under the License is distributed on an "AS IS" BASIS,
13   * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
14   * See the License for the specific language governing permissions and
15   * limitations under the License.
16   */
17  
18  package org.apache.commons.math.geometry;
19  
20  import java.io.Serializable;
21  
22  /**
23   * This class implements rotations in a three-dimensional space.
24   *
25   * <p>Rotations can be represented by several different mathematical
26   * entities (matrices, axe and angle, Cardan or Euler angles,
27   * quaternions). This class presents an higher level abstraction, more
28   * user-oriented and hiding this implementation details. Well, for the
29   * curious, we use quaternions for the internal representation. The
30   * user can build a rotation from any of these representations, and
31   * any of these representations can be retrieved from a
32   * <code>Rotation</code> instance (see the various constructors and
33   * getters). In addition, a rotation can also be built implicitely
34   * from a set of vectors and their image.</p>
35   * <p>This implies that this class can be used to convert from one
36   * representation to another one. For example, converting a rotation
37   * matrix into a set of Cardan angles from can be done using the
38   * followong single line of code:</p>
39   * <pre>
40   * double[] angles = new Rotation(matrix, 1.0e-10).getAngles(RotationOrder.XYZ);
41   * </pre>
42   * <p>Focus is oriented on what a rotation <em>do</em> rather than on its
43   * underlying representation. Once it has been built, and regardless of its
44   * internal representation, a rotation is an <em>operator</em> which basically
45   * transforms three dimensional {@link Vector3D vectors} into other three
46   * dimensional {@link Vector3D vectors}. Depending on the application, the
47   * meaning of these vectors may vary and the semantics of the rotation also.</p>
48   * <p>For example in an spacecraft attitude simulation tool, users will often
49   * consider the vectors are fixed (say the Earth direction for example) and the
50   * rotation transforms the coordinates coordinates of this vector in inertial
51   * frame into the coordinates of the same vector in satellite frame. In this
52   * case, the rotation implicitely defines the relation between the two frames.
53   * Another example could be a telescope control application, where the rotation
54   * would transform the sighting direction at rest into the desired observing
55   * direction when the telescope is pointed towards an object of interest. In this
56   * case the rotation transforms the directionf at rest in a topocentric frame
57   * into the sighting direction in the same topocentric frame. In many case, both
58   * approaches will be combined, in our telescope example, we will probably also
59   * need to transform the observing direction in the topocentric frame into the
60   * observing direction in inertial frame taking into account the observatory
61   * location and the Earth rotation.</p>
62   *
63   * <p>These examples show that a rotation is what the user wants it to be, so this
64   * class does not push the user towards one specific definition and hence does not
65   * provide methods like <code>projectVectorIntoDestinationFrame</code> or
66   * <code>computeTransformedDirection</code>. It provides simpler and more generic
67   * methods: {@link #applyTo(Vector3D) applyTo(Vector3D)} and {@link
68   * #applyInverseTo(Vector3D) applyInverseTo(Vector3D)}.</p>
69   *
70   * <p>Since a rotation is basically a vectorial operator, several rotations can be
71   * composed together and the composite operation <code>r = r<sub>1</sub> o
72   * r<sub>2</sub></code> (which means that for each vector <code>u</code>,
73   * <code>r(u) = r<sub>1</sub>(r<sub>2</sub>(u))</code>) is also a rotation. Hence
74   * we can consider that in addition to vectors, a rotation can be applied to other
75   * rotations as well (or to itself). With our previous notations, we would say we
76   * can apply <code>r<sub>1</sub></code> to <code>r<sub>2</sub></code> and the result
77   * we get is <code>r = r<sub>1</sub> o r<sub>2</sub></code>. For this purpose, the
78   * class provides the methods: {@link #applyTo(Rotation) applyTo(Rotation)} and
79   * {@link #applyInverseTo(Rotation) applyInverseTo(Rotation)}.</p>
80   *
81   * <p>Rotations are guaranteed to be immutable objects.</p>
82   *
83   * @version $Revision: 627994 $ $Date: 2008-02-15 03:16:05 -0700 (Fri, 15 Feb 2008) $
84   * @see Vector3D
85   * @see RotationOrder
86   * @since 1.2
87   */
88  
89  public class Rotation implements Serializable {
90  
91    /** Build the identity rotation.
92     */
93    public Rotation() {
94      q0 = 1;
95      q1 = 0;
96      q2 = 0;
97      q3 = 0;
98    }
99  
100   /** Build a rotation from the quaternion coordinates.
101    * <p>A rotation can be built from a <em>normalized</em> quaternion,
102    * i.e. a quaternion for which q<sub>0</sub><sup>2</sup> +
103    * q<sub>1</sub><sup>2</sup> + q<sub>2</sub><sup>2</sup> +
104    * q<sub>3</sub><sup>2</sup> = 1. If the quaternion is not normalized,
105    * the constructor can normalize it in a preprocessing step.</p>
106    * @param q0 scalar part of the quaternion
107    * @param q1 first coordinate of the vectorial part of the quaternion
108    * @param q2 second coordinate of the vectorial part of the quaternion
109    * @param q3 third coordinate of the vectorial part of the quaternion
110    * @param needsNormalization if true, the coordinates are considered
111    * not to be normalized, a normalization preprocessing step is performed
112    * before using them
113    */
114   public Rotation(double q0, double q1, double q2, double q3,
115                   boolean needsNormalization) {
116 
117     if (needsNormalization) {
118       // normalization preprocessing
119       double inv = 1.0 / Math.sqrt(q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3);
120       q0 *= inv;
121       q1 *= inv;
122       q2 *= inv;
123       q3 *= inv;
124     }
125 
126     this.q0 = q0;
127     this.q1 = q1;
128     this.q2 = q2;
129     this.q3 = q3;
130 
131   }
132 
133   /** Build a rotation from an axis and an angle.
134    * <p>We use the convention that angles are oriented according to
135    * the effect of the rotation on vectors around the axis. That means
136    * that if (i, j, k) is a direct frame and if we first provide +k as
137    * the axis and PI/2 as the angle to this constructor, and then
138    * {@link #applyTo(Vector3D) apply} the instance to +i, we will get
139    * +j.</p>
140    * @param axis axis around which to rotate
141    * @param angle rotation angle.
142    * @exception ArithmeticException if the axis norm is zero
143    */
144   public Rotation(Vector3D axis, double angle) {
145 
146     double norm = axis.getNorm();
147     if (norm == 0) {
148       throw new ArithmeticException("zero norm for rotation axis");
149     }
150 
151     double halfAngle = -0.5 * angle;
152     double coeff = Math.sin(halfAngle) / norm;
153 
154     q0 = Math.cos (halfAngle);
155     q1 = coeff * axis.getX();
156     q2 = coeff * axis.getY();
157     q3 = coeff * axis.getZ();
158 
159   }
160 
161   /** Build a rotation from a 3X3 matrix.
162 
163    * <p>Rotation matrices are orthogonal matrices, i.e. unit matrices
164    * (which are matrices for which m.m<sup>T</sup> = I) with real
165    * coefficients. The module of the determinant of unit matrices is
166    * 1, among the orthogonal 3X3 matrices, only the ones having a
167    * positive determinant (+1) are rotation matrices.</p>
168 
169    * <p>When a rotation is defined by a matrix with truncated values
170    * (typically when it is extracted from a technical sheet where only
171    * four to five significant digits are available), the matrix is not
172    * orthogonal anymore. This constructor handles this case
173    * transparently by using a copy of the given matrix and applying a
174    * correction to the copy in order to perfect its orthogonality. If
175    * the Frobenius norm of the correction needed is above the given
176    * threshold, then the matrix is considered to be too far from a
177    * true rotation matrix and an exception is thrown.<p>
178 
179    * @param m rotation matrix
180    * @param threshold convergence threshold for the iterative
181    * orthogonality correction (convergence is reached when the
182    * difference between two steps of the Frobenius norm of the
183    * correction is below this threshold)
184 
185    * @exception NotARotationMatrixException if the matrix is not a 3X3
186    * matrix, or if it cannot be transformed into an orthogonal matrix
187    * with the given threshold, or if the determinant of the resulting
188    * orthogonal matrix is negative
189 
190    */
191   public Rotation(double[][] m, double threshold)
192     throws NotARotationMatrixException {
193 
194     // dimension check
195     if ((m.length != 3) || (m[0].length != 3) ||
196         (m[1].length != 3) || (m[2].length != 3)) {
197       throw new NotARotationMatrixException("a {0}x{1} matrix" +
198                                             " cannot be a rotation matrix",
199                                             new Object[] {
200                                               Integer.toString(m.length),
201                                               Integer.toString(m[0].length)
202                                             });
203     }
204 
205     // compute a "close" orthogonal matrix
206     double[][] ort = orthogonalizeMatrix(m, threshold);
207 
208     // check the sign of the determinant
209     double det = ort[0][0] * (ort[1][1] * ort[2][2] - ort[2][1] * ort[1][2]) -
210                  ort[1][0] * (ort[0][1] * ort[2][2] - ort[2][1] * ort[0][2]) +
211                  ort[2][0] * (ort[0][1] * ort[1][2] - ort[1][1] * ort[0][2]);
212     if (det < 0.0) {
213       throw new NotARotationMatrixException("the closest orthogonal matrix" +
214                                             " has a negative determinant {0}",
215                                             new Object[] {
216                                               Double.toString(det)
217                                             });
218     }
219 
220     // There are different ways to compute the quaternions elements
221     // from the matrix. They all involve computing one element from
222     // the diagonal of the matrix, and computing the three other ones
223     // using a formula involving a division by the first element,
224     // which unfortunately can be zero. Since the norm of the
225     // quaternion is 1, we know at least one element has an absolute
226     // value greater or equal to 0.5, so it is always possible to
227     // select the right formula and avoid division by zero and even
228     // numerical inaccuracy. Checking the elements in turn and using
229     // the first one greater than 0.45 is safe (this leads to a simple
230     // test since qi = 0.45 implies 4 qi^2 - 1 = -0.19)
231     double s = ort[0][0] + ort[1][1] + ort[2][2];
232     if (s > -0.19) {
233       // compute q0 and deduce q1, q2 and q3
234       q0 = 0.5 * Math.sqrt(s + 1.0);
235       double inv = 0.25 / q0;
236       q1 = inv * (ort[1][2] - ort[2][1]);
237       q2 = inv * (ort[2][0] - ort[0][2]);
238       q3 = inv * (ort[0][1] - ort[1][0]);
239     } else {
240       s = ort[0][0] - ort[1][1] - ort[2][2];
241       if (s > -0.19) {
242         // compute q1 and deduce q0, q2 and q3
243         q1 = 0.5 * Math.sqrt(s + 1.0);
244         double inv = 0.25 / q1;
245         q0 = inv * (ort[1][2] - ort[2][1]);
246         q2 = inv * (ort[0][1] + ort[1][0]);
247         q3 = inv * (ort[0][2] + ort[2][0]);
248       } else {
249         s = ort[1][1] - ort[0][0] - ort[2][2];
250         if (s > -0.19) {
251           // compute q2 and deduce q0, q1 and q3
252           q2 = 0.5 * Math.sqrt(s + 1.0);
253           double inv = 0.25 / q2;
254           q0 = inv * (ort[2][0] - ort[0][2]);
255           q1 = inv * (ort[0][1] + ort[1][0]);
256           q3 = inv * (ort[2][1] + ort[1][2]);
257         } else {
258           // compute q3 and deduce q0, q1 and q2
259           s = ort[2][2] - ort[0][0] - ort[1][1];
260           q3 = 0.5 * Math.sqrt(s + 1.0);
261           double inv = 0.25 / q3;
262           q0 = inv * (ort[0][1] - ort[1][0]);
263           q1 = inv * (ort[0][2] + ort[2][0]);
264           q2 = inv * (ort[2][1] + ort[1][2]);
265         }
266       }
267     }
268 
269   }
270 
271   /** Build the rotation that transforms a pair of vector into another pair.
272 
273    * <p>Except for possible scale factors, if the instance were applied to
274    * the pair (u<sub>1</sub>, u<sub>2</sub>) it will produce the pair
275    * (v<sub>1</sub>, v<sub>2</sub>).</p>
276 
277    * <p>If the angular separation between u<sub>1</sub> and u<sub>2</sub> is
278    * not the same as the angular separation between v<sub>1</sub> and
279    * v<sub>2</sub>, then a corrected v'<sub>2</sub> will be used rather than
280    * v<sub>2</sub>, the corrected vector will be in the (v<sub>1</sub>,
281    * v<sub>2</sub>) plane.</p>
282 
283    * @param u1 first vector of the origin pair
284    * @param u2 second vector of the origin pair
285    * @param v1 desired image of u1 by the rotation
286    * @param v2 desired image of u2 by the rotation
287    * @exception IllegalArgumentException if the norm of one of the vectors is zero
288    */
289   public Rotation(Vector3D u1, Vector3D u2, Vector3D v1, Vector3D v2) {
290 
291   // norms computation
292   double u1u1 = Vector3D.dotProduct(u1, u1);
293   double u2u2 = Vector3D.dotProduct(u2, u2);
294   double v1v1 = Vector3D.dotProduct(v1, v1);
295   double v2v2 = Vector3D.dotProduct(v2, v2);
296   if ((u1u1 == 0) || (u2u2 == 0) || (v1v1 == 0) || (v2v2 == 0)) {
297     throw new IllegalArgumentException("zero norm for rotation defining vector");
298   }
299 
300   double u1x = u1.getX();
301   double u1y = u1.getY();
302   double u1z = u1.getZ();
303 
304   double u2x = u2.getX();
305   double u2y = u2.getY();
306   double u2z = u2.getZ();
307 
308   // normalize v1 in order to have (v1'|v1') = (u1|u1)
309   double coeff = Math.sqrt (u1u1 / v1v1);
310   double v1x   = coeff * v1.getX();
311   double v1y   = coeff * v1.getY();
312   double v1z   = coeff * v1.getZ();
313   v1 = new Vector3D(v1x, v1y, v1z);
314 
315   // adjust v2 in order to have (u1|u2) = (v1|v2) and (v2'|v2') = (u2|u2)
316   double u1u2   = Vector3D.dotProduct(u1, u2);
317   double v1v2   = Vector3D.dotProduct(v1, v2);
318   double coeffU = u1u2 / u1u1;
319   double coeffV = v1v2 / u1u1;
320   double beta   = Math.sqrt((u2u2 - u1u2 * coeffU) / (v2v2 - v1v2 * coeffV));
321   double alpha  = coeffU - beta * coeffV;
322   double v2x    = alpha * v1x + beta * v2.getX();
323   double v2y    = alpha * v1y + beta * v2.getY();
324   double v2z    = alpha * v1z + beta * v2.getZ();
325   v2 = new Vector3D(v2x, v2y, v2z);
326 
327   // preliminary computation (we use explicit formulation instead
328   // of relying on the Vector3D class in order to avoid building lots
329   // of temporary objects)
330   Vector3D uRef = u1;
331   Vector3D vRef = v1;
332   double dx1 = v1x - u1.getX();
333   double dy1 = v1y - u1.getY();
334   double dz1 = v1z - u1.getZ();
335   double dx2 = v2x - u2.getX();
336   double dy2 = v2y - u2.getY();
337   double dz2 = v2z - u2.getZ();
338   Vector3D k = new Vector3D(dy1 * dz2 - dz1 * dy2,
339                             dz1 * dx2 - dx1 * dz2,
340                             dx1 * dy2 - dy1 * dx2);
341   double c = k.getX() * (u1y * u2z - u1z * u2y) +
342              k.getY() * (u1z * u2x - u1x * u2z) +
343              k.getZ() * (u1x * u2y - u1y * u2x);
344 
345   if (c == 0) {
346     // the (q1, q2, q3) vector is in the (u1, u2) plane
347     // we try other vectors
348     Vector3D u3 = Vector3D.crossProduct(u1, u2);
349     Vector3D v3 = Vector3D.crossProduct(v1, v2);
350     double u3x  = u3.getX();
351     double u3y  = u3.getY();
352     double u3z  = u3.getZ();
353     double v3x  = v3.getX();
354     double v3y  = v3.getY();
355     double v3z  = v3.getZ();
356 
357     double dx3 = v3x - u3x;
358     double dy3 = v3y - u3y;
359     double dz3 = v3z - u3z;
360     k = new Vector3D(dy1 * dz3 - dz1 * dy3,
361                      dz1 * dx3 - dx1 * dz3,
362                      dx1 * dy3 - dy1 * dx3);
363     c = k.getX() * (u1y * u3z - u1z * u3y) +
364         k.getY() * (u1z * u3x - u1x * u3z) +
365         k.getZ() * (u1x * u3y - u1y * u3x);
366 
367     if (c == 0) {
368       // the (q1, q2, q3) vector is aligned with u1:
369       // we try (u2, u3) and (v2, v3)
370       k = new Vector3D(dy2 * dz3 - dz2 * dy3,
371                        dz2 * dx3 - dx2 * dz3,
372                        dx2 * dy3 - dy2 * dx3);
373       c = k.getX() * (u2y * u3z - u2z * u3y) +
374           k.getY() * (u2z * u3x - u2x * u3z) +
375           k.getZ() * (u2x * u3y - u2y * u3x);
376 
377       if (c == 0) {
378         // the (q1, q2, q3) vector is aligned with everything
379         // this is really the identity rotation
380         q0 = 1.0;
381         q1 = 0.0;
382         q2 = 0.0;
383         q3 = 0.0;
384         return;
385       }
386 
387       // we will have to use u2 and v2 to compute the scalar part
388       uRef = u2;
389       vRef = v2;
390 
391     }
392 
393   }
394 
395   // compute the vectorial part
396   c = Math.sqrt(c);
397   double inv = 1.0 / (c + c);
398   q1 = inv * k.getX();
399   q2 = inv * k.getY();
400   q3 = inv * k.getZ();
401 
402   // compute the scalar part
403    k = new Vector3D(uRef.getY() * q3 - uRef.getZ() * q2,
404                     uRef.getZ() * q1 - uRef.getX() * q3,
405                     uRef.getX() * q2 - uRef.getY() * q1);
406    c = Vector3D.dotProduct(k, k);
407   q0 = Vector3D.dotProduct(vRef, k) / (c + c);
408 
409   }
410 
411   /** Build one of the rotations that transform one vector into another one.
412 
413    * <p>Except for a possible scale factor, if the instance were
414    * applied to the vector u it will produce the vector v. There is an
415    * infinite number of such rotations, this constructor choose the
416    * one with the smallest associated angle (i.e. the one whose axis
417    * is orthogonal to the (u, v) plane). If u and v are colinear, an
418    * arbitrary rotation axis is chosen.</p>
419 
420    * @param u origin vector
421    * @param v desired image of u by the rotation
422    * @exception IllegalArgumentException if the norm of one of the vectors is zero
423    */
424   public Rotation(Vector3D u, Vector3D v) {
425 
426     double normProduct = u.getNorm() * v.getNorm();
427     if (normProduct == 0) {
428       throw new IllegalArgumentException("zero norm for rotation defining vector");
429     }
430 
431     double dot = Vector3D.dotProduct(u, v);
432 
433     if (dot < ((2.0e-15 - 1.0) * normProduct)) {
434       // special case u = -v: we select a PI angle rotation around
435       // an arbitrary vector orthogonal to u
436       Vector3D w = u.orthogonal();
437       q0 = 0.0;
438       q1 = -w.getX();
439       q2 = -w.getY();
440       q3 = -w.getZ();
441     } else {
442       // general case: (u, v) defines a plane, we select
443       // the shortest possible rotation: axis orthogonal to this plane
444       q0 = Math.sqrt(0.5 * (1.0 + dot / normProduct));
445       double coeff = 1.0 / (2.0 * q0 * normProduct);
446       q1 = coeff * (v.getY() * u.getZ() - v.getZ() * u.getY());
447       q2 = coeff * (v.getZ() * u.getX() - v.getX() * u.getZ());
448       q3 = coeff * (v.getX() * u.getY() - v.getY() * u.getX());
449     }
450 
451   }
452 
453   /** Build a rotation from three Cardan or Euler elementary rotations.
454 
455    * <p>Cardan rotations are three successive rotations around the
456    * canonical axes X, Y and Z, each axis beeing used once. There are
457    * 6 such sets of rotations (XYZ, XZY, YXZ, YZX, ZXY and ZYX). Euler
458    * rotations are three successive rotations around the canonical
459    * axes X, Y and Z, the first and last rotations beeing around the
460    * same axis. There are 6 such sets of rotations (XYX, XZX, YXY,
461    * YZY, ZXZ and ZYZ), the most popular one being ZXZ.</p>
462    * <p>Beware that many people routinely use the term Euler angles even
463    * for what really are Cardan angles (this confusion is especially
464    * widespread in the aerospace business where Roll, Pitch and Yaw angles
465    * are often wrongly tagged as Euler angles).</p>
466 
467    * @param order order of rotations to use
468    * @param alpha1 angle of the first elementary rotation
469    * @param alpha2 angle of the second elementary rotation
470    * @param alpha3 angle of the third elementary rotation
471    */
472   public Rotation(RotationOrder order,
473                   double alpha1, double alpha2, double alpha3) {
474     Rotation r1 = new Rotation(order.getA1(), alpha1);
475     Rotation r2 = new Rotation(order.getA2(), alpha2);
476     Rotation r3 = new Rotation(order.getA3(), alpha3);
477     Rotation composed = r1.applyTo(r2.applyTo(r3));
478     q0 = composed.q0;
479     q1 = composed.q1;
480     q2 = composed.q2;
481     q3 = composed.q3;
482   }
483 
484   /** Revert a rotation.
485    * Build a rotation which reverse the effect of another
486    * rotation. This means that if r(u) = v, then r.revert(v) = u. The
487    * instance is not changed.
488    * @return a new rotation whose effect is the reverse of the effect
489    * of the instance
490    */
491   public Rotation revert() {
492     return new Rotation(-q0, q1, q2, q3, false);
493   }
494 
495   /** Get the scalar coordinate of the quaternion.
496    * @return scalar coordinate of the quaternion
497    */
498   public double getQ0() {
499     return q0;
500   }
501 
502   /** Get the first coordinate of the vectorial part of the quaternion.
503    * @return first coordinate of the vectorial part of the quaternion
504    */
505   public double getQ1() {
506     return q1;
507   }
508 
509   /** Get the second coordinate of the vectorial part of the quaternion.
510    * @return second coordinate of the vectorial part of the quaternion
511    */
512   public double getQ2() {
513     return q2;
514   }
515 
516   /** Get the third coordinate of the vectorial part of the quaternion.
517    * @return third coordinate of the vectorial part of the quaternion
518    */
519   public double getQ3() {
520     return q3;
521   }
522 
523   /** Get the normalized axis of the rotation.
524    * @return normalized axis of the rotation
525    */
526   public Vector3D getAxis() {
527     double squaredSine = q1 * q1 + q2 * q2 + q3 * q3;
528     if (squaredSine == 0) {
529       return new Vector3D(1, 0, 0);
530     } else if (q0 < 0) {
531       double inverse = 1 / Math.sqrt(squaredSine);
532       return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
533     }
534     double inverse = -1 / Math.sqrt(squaredSine);
535     return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
536   }
537 
538   /** Get the angle of the rotation.
539    * @return angle of the rotation (between 0 and &pi;)
540    */
541   public double getAngle() {
542     if ((q0 < -0.1) || (q0 > 0.1)) {
543       return 2 * Math.asin(Math.sqrt(q1 * q1 + q2 * q2 + q3 * q3));
544     } else if (q0 < 0) {
545       return 2 * Math.acos(-q0);
546     }
547     return 2 * Math.acos(q0);
548   }
549 
550   /** Get the Cardan or Euler angles corresponding to the instance.
551 
552    * <p>The equations show that each rotation can be defined by two
553    * different values of the Cardan or Euler angles set. For example
554    * if Cardan angles are used, the rotation defined by the angles
555    * a<sub>1</sub>, a<sub>2</sub> and a<sub>3</sub> is the same as
556    * the rotation defined by the angles &pi; + a<sub>1</sub>, &pi;
557    * - a<sub>2</sub> and &pi; + a<sub>3</sub>. This method implements
558    * the following arbitrary choices:</p>
559    * <ul>
560    *   <li>for Cardan angles, the chosen set is the one for which the
561    *   second angle is between -&pi;/2 and &pi;/2 (i.e its cosine is
562    *   positive),</li>
563    *   <li>for Euler angles, the chosen set is the one for which the
564    *   second angle is between 0 and &pi; (i.e its sine is positive).</li>
565    * </ul>
566 
567    * <p>Cardan and Euler angle have a very disappointing drawback: all
568    * of them have singularities. This means that if the instance is
569    * too close to the singularities corresponding to the given
570    * rotation order, it will be impossible to retrieve the angles. For
571    * Cardan angles, this is often called gimbal lock. There is
572    * <em>nothing</em> to do to prevent this, it is an intrinsic problem
573    * with Cardan and Euler representation (but not a problem with the
574    * rotation itself, which is perfectly well defined). For Cardan
575    * angles, singularities occur when the second angle is close to
576    * -&pi;/2 or +&pi;/2, for Euler angle singularities occur when the
577    * second angle is close to 0 or &pi;, this implies that the identity
578    * rotation is always singular for Euler angles!</p>
579 
580    * @param order rotation order to use
581    * @return an array of three angles, in the order specified by the set
582    * @exception CardanEulerSingularityException if the rotation is
583    * singular with respect to the angles set specified
584    */
585   public double[] getAngles(RotationOrder order)
586     throws CardanEulerSingularityException {
587 
588     if (order == RotationOrder.XYZ) {
589 
590       // r (Vector3D.plusK) coordinates are :
591       //  sin (theta), -cos (theta) sin (phi), cos (theta) cos (phi)
592       // (-r) (Vector3D.plusI) coordinates are :
593       // cos (psi) cos (theta), -sin (psi) cos (theta), sin (theta)
594       // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
595       Vector3D v1 = applyTo(Vector3D.plusK);
596       Vector3D v2 = applyInverseTo(Vector3D.plusI);
597       if  ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
598         throw new CardanEulerSingularityException(true);
599       }
600       return new double[] {
601         Math.atan2(-(v1.getY()), v1.getZ()),
602         Math.asin(v2.getZ()),
603         Math.atan2(-(v2.getY()), v2.getX())
604       };
605 
606     } else if (order == RotationOrder.XZY) {
607 
608       // r (Vector3D.plusJ) coordinates are :
609       // -sin (psi), cos (psi) cos (phi), cos (psi) sin (phi)
610       // (-r) (Vector3D.plusI) coordinates are :
611       // cos (theta) cos (psi), -sin (psi), sin (theta) cos (psi)
612       // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
613       Vector3D v1 = applyTo(Vector3D.plusJ);
614       Vector3D v2 = applyInverseTo(Vector3D.plusI);
615       if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
616         throw new CardanEulerSingularityException(true);
617       }
618       return new double[] {
619         Math.atan2(v1.getZ(), v1.getY()),
620        -Math.asin(v2.getY()),
621         Math.atan2(v2.getZ(), v2.getX())
622       };
623 
624     } else if (order == RotationOrder.YXZ) {
625 
626       // r (Vector3D.plusK) coordinates are :
627       //  cos (phi) sin (theta), -sin (phi), cos (phi) cos (theta)
628       // (-r) (Vector3D.plusJ) coordinates are :
629       // sin (psi) cos (phi), cos (psi) cos (phi), -sin (phi)
630       // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
631       Vector3D v1 = applyTo(Vector3D.plusK);
632       Vector3D v2 = applyInverseTo(Vector3D.plusJ);
633       if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
634         throw new CardanEulerSingularityException(true);
635       }
636       return new double[] {
637         Math.atan2(v1.getX(), v1.getZ()),
638        -Math.asin(v2.getZ()),
639         Math.atan2(v2.getX(), v2.getY())
640       };
641 
642     } else if (order == RotationOrder.YZX) {
643 
644       // r (Vector3D.plusI) coordinates are :
645       // cos (psi) cos (theta), sin (psi), -cos (psi) sin (theta)
646       // (-r) (Vector3D.plusJ) coordinates are :
647       // sin (psi), cos (phi) cos (psi), -sin (phi) cos (psi)
648       // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
649       Vector3D v1 = applyTo(Vector3D.plusI);
650       Vector3D v2 = applyInverseTo(Vector3D.plusJ);
651       if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
652         throw new CardanEulerSingularityException(true);
653       }
654       return new double[] {
655         Math.atan2(-(v1.getZ()), v1.getX()),
656         Math.asin(v2.getX()),
657         Math.atan2(-(v2.getZ()), v2.getY())
658       };
659 
660     } else if (order == RotationOrder.ZXY) {
661 
662       // r (Vector3D.plusJ) coordinates are :
663       // -cos (phi) sin (psi), cos (phi) cos (psi), sin (phi)
664       // (-r) (Vector3D.plusK) coordinates are :
665       // -sin (theta) cos (phi), sin (phi), cos (theta) cos (phi)
666       // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
667       Vector3D v1 = applyTo(Vector3D.plusJ);
668       Vector3D v2 = applyInverseTo(Vector3D.plusK);
669       if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
670         throw new CardanEulerSingularityException(true);
671       }
672       return new double[] {
673         Math.atan2(-(v1.getX()), v1.getY()),
674         Math.asin(v2.getY()),
675         Math.atan2(-(v2.getX()), v2.getZ())
676       };
677 
678     } else if (order == RotationOrder.ZYX) {
679 
680       // r (Vector3D.plusI) coordinates are :
681       //  cos (theta) cos (psi), cos (theta) sin (psi), -sin (theta)
682       // (-r) (Vector3D.plusK) coordinates are :
683       // -sin (theta), sin (phi) cos (theta), cos (phi) cos (theta)
684       // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
685       Vector3D v1 = applyTo(Vector3D.plusI);
686       Vector3D v2 = applyInverseTo(Vector3D.plusK);
687       if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
688         throw new CardanEulerSingularityException(true);
689       }
690       return new double[] {
691         Math.atan2(v1.getY(), v1.getX()),
692        -Math.asin(v2.getX()),
693         Math.atan2(v2.getY(), v2.getZ())
694       };
695 
696     } else if (order == RotationOrder.XYX) {
697 
698       // r (Vector3D.plusI) coordinates are :
699       //  cos (theta), sin (phi1) sin (theta), -cos (phi1) sin (theta)
700       // (-r) (Vector3D.plusI) coordinates are :
701       // cos (theta), sin (theta) sin (phi2), sin (theta) cos (phi2)
702       // and we can choose to have theta in the interval [0 ; PI]
703       Vector3D v1 = applyTo(Vector3D.plusI);
704       Vector3D v2 = applyInverseTo(Vector3D.plusI);
705       if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
706         throw new CardanEulerSingularityException(false);
707       }
708       return new double[] {
709         Math.atan2(v1.getY(), -v1.getZ()),
710         Math.acos(v2.getX()),
711         Math.atan2(v2.getY(), v2.getZ())
712       };
713 
714     } else if (order == RotationOrder.XZX) {
715 
716       // r (Vector3D.plusI) coordinates are :
717       //  cos (psi), cos (phi1) sin (psi), sin (phi1) sin (psi)
718       // (-r) (Vector3D.plusI) coordinates are :
719       // cos (psi), -sin (psi) cos (phi2), sin (psi) sin (phi2)
720       // and we can choose to have psi in the interval [0 ; PI]
721       Vector3D v1 = applyTo(Vector3D.plusI);
722       Vector3D v2 = applyInverseTo(Vector3D.plusI);
723       if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
724         throw new CardanEulerSingularityException(false);
725       }
726       return new double[] {
727         Math.atan2(v1.getZ(), v1.getY()),
728         Math.acos(v2.getX()),
729         Math.atan2(v2.getZ(), -v2.getY())
730       };
731 
732     } else if (order == RotationOrder.YXY) {
733 
734       // r (Vector3D.plusJ) coordinates are :
735       //  sin (theta1) sin (phi), cos (phi), cos (theta1) sin (phi)
736       // (-r) (Vector3D.plusJ) coordinates are :
737       // sin (phi) sin (theta2), cos (phi), -sin (phi) cos (theta2)
738       // and we can choose to have phi in the interval [0 ; PI]
739       Vector3D v1 = applyTo(Vector3D.plusJ);
740       Vector3D v2 = applyInverseTo(Vector3D.plusJ);
741       if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
742         throw new CardanEulerSingularityException(false);
743       }
744       return new double[] {
745         Math.atan2(v1.getX(), v1.getZ()),
746         Math.acos(v2.getY()),
747         Math.atan2(v2.getX(), -v2.getZ())
748       };
749 
750     } else if (order == RotationOrder.YZY) {
751 
752       // r (Vector3D.plusJ) coordinates are :
753       //  -cos (theta1) sin (psi), cos (psi), sin (theta1) sin (psi)
754       // (-r) (Vector3D.plusJ) coordinates are :
755       // sin (psi) cos (theta2), cos (psi), sin (psi) sin (theta2)
756       // and we can choose to have psi in the interval [0 ; PI]
757       Vector3D v1 = applyTo(Vector3D.plusJ);
758       Vector3D v2 = applyInverseTo(Vector3D.plusJ);
759       if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
760         throw new CardanEulerSingularityException(false);
761       }
762       return new double[] {
763         Math.atan2(v1.getZ(), -v1.getX()),
764         Math.acos(v2.getY()),
765         Math.atan2(v2.getZ(), v2.getX())
766       };
767 
768     } else if (order == RotationOrder.ZXZ) {
769 
770       // r (Vector3D.plusK) coordinates are :
771       //  sin (psi1) sin (phi), -cos (psi1) sin (phi), cos (phi)
772       // (-r) (Vector3D.plusK) coordinates are :
773       // sin (phi) sin (psi2), sin (phi) cos (psi2), cos (phi)
774       // and we can choose to have phi in the interval [0 ; PI]
775       Vector3D v1 = applyTo(Vector3D.plusK);
776       Vector3D v2 = applyInverseTo(Vector3D.plusK);
777       if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
778         throw new CardanEulerSingularityException(false);
779       }
780       return new double[] {
781         Math.atan2(v1.getX(), -v1.getY()),
782         Math.acos(v2.getZ()),
783         Math.atan2(v2.getX(), v2.getY())
784       };
785 
786     } else { // last possibility is ZYZ
787 
788       // r (Vector3D.plusK) coordinates are :
789       //  cos (psi1) sin (theta), sin (psi1) sin (theta), cos (theta)
790       // (-r) (Vector3D.plusK) coordinates are :
791       // -sin (theta) cos (psi2), sin (theta) sin (psi2), cos (theta)
792       // and we can choose to have theta in the interval [0 ; PI]
793       Vector3D v1 = applyTo(Vector3D.plusK);
794       Vector3D v2 = applyInverseTo(Vector3D.plusK);
795       if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
796         throw new CardanEulerSingularityException(false);
797       }
798       return new double[] {
799         Math.atan2(v1.getY(), v1.getX()),
800         Math.acos(v2.getZ()),
801         Math.atan2(v2.getY(), -v2.getX())
802       };
803 
804     }
805 
806   }
807 
808   /** Get the 3X3 matrix corresponding to the instance
809    * @return the matrix corresponding to the instance
810    */
811   public double[][] getMatrix() {
812 
813     // products
814     double q0q0  = q0 * q0;
815     double q0q1  = q0 * q1;
816     double q0q2  = q0 * q2;
817     double q0q3  = q0 * q3;
818     double q1q1  = q1 * q1;
819     double q1q2  = q1 * q2;
820     double q1q3  = q1 * q3;
821     double q2q2  = q2 * q2;
822     double q2q3  = q2 * q3;
823     double q3q3  = q3 * q3;
824 
825     // create the matrix
826     double[][] m = new double[3][];
827     m[0] = new double[3];
828     m[1] = new double[3];
829     m[2] = new double[3];
830 
831     m [0][0] = 2.0 * (q0q0 + q1q1) - 1.0;
832     m [1][0] = 2.0 * (q1q2 - q0q3);
833     m [2][0] = 2.0 * (q1q3 + q0q2);
834 
835     m [0][1] = 2.0 * (q1q2 + q0q3);
836     m [1][1] = 2.0 * (q0q0 + q2q2) - 1.0;
837     m [2][1] = 2.0 * (q2q3 - q0q1);
838 
839     m [0][2] = 2.0 * (q1q3 - q0q2);
840     m [1][2] = 2.0 * (q2q3 + q0q1);
841     m [2][2] = 2.0 * (q0q0 + q3q3) - 1.0;
842 
843     return m;
844 
845   }
846 
847   /** Apply the rotation to a vector.
848    * @param u vector to apply the rotation to
849    * @return a new vector which is the image of u by the rotation
850    */
851   public Vector3D applyTo(Vector3D u) {
852 
853     double x = u.getX();
854     double y = u.getY();
855     double z = u.getZ();
856 
857     double s = q1 * x + q2 * y + q3 * z;
858 
859     return new Vector3D(2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x,
860                         2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y,
861                         2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - z);
862 
863   }
864 
865   /** Apply the inverse of the rotation to a vector.
866    * @param u vector to apply the inverse of the rotation to
867    * @return a new vector which such that u is its image by the rotation
868    */
869   public Vector3D applyInverseTo(Vector3D u) {
870 
871     double x = u.getX();
872     double y = u.getY();
873     double z = u.getZ();
874 
875     double s = q1 * x + q2 * y + q3 * z;
876     double m0 = -q0;
877 
878     return new Vector3D(2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x,
879                         2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y,
880                         2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - z);
881 
882   }
883 
884   /** Apply the instance to another rotation.
885    * Applying the instance to a rotation is computing the composition
886    * in an order compliant with the following rule : let u be any
887    * vector and v its image by r (i.e. r.applyTo(u) = v), let w be the image
888    * of v by the instance (i.e. applyTo(v) = w), then w = comp.applyTo(u),
889    * where comp = applyTo(r).
890    * @param r rotation to apply the rotation to
891    * @return a new rotation which is the composition of r by the instance
892    */
893   public Rotation applyTo(Rotation r) {
894     return new Rotation(r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),
895                         r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),
896                         r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),
897                         r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),
898                         false);
899   }
900 
901   /** Apply the inverse of the instance to another rotation.
902    * Applying the inverse of the instance to a rotation is computing
903    * the composition in an order compliant with the following rule :
904    * let u be any vector and v its image by r (i.e. r.applyTo(u) = v),
905    * let w be the inverse image of v by the instance
906    * (i.e. applyInverseTo(v) = w), then w = comp.applyTo(u), where
907    * comp = applyInverseTo(r).
908    * @param r rotation to apply the rotation to
909    * @return a new rotation which is the composition of r by the inverse
910    * of the instance
911    */
912   public Rotation applyInverseTo(Rotation r) {
913     return new Rotation(-r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),
914                         -r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),
915                         -r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),
916                         -r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),
917                         false);
918   }
919 
920   /** Perfect orthogonality on a 3X3 matrix.
921    * @param m initial matrix (not exactly orthogonal)
922    * @param threshold convergence threshold for the iterative
923    * orthogonality correction (convergence is reached when the
924    * difference between two steps of the Frobenius norm of the
925    * correction is below this threshold)
926    * @return an orthogonal matrix close to m
927    * @exception NotARotationMatrixException if the matrix cannot be
928    * orthogonalized with the given threshold after 10 iterations
929    */
930   private double[][] orthogonalizeMatrix(double[][] m, double threshold)
931     throws NotARotationMatrixException {
932     double[] m0 = m[0];
933     double[] m1 = m[1];
934     double[] m2 = m[2];
935     double x00 = m0[0];
936     double x01 = m0[1];
937     double x02 = m0[2];
938     double x10 = m1[0];
939     double x11 = m1[1];
940     double x12 = m1[2];
941     double x20 = m2[0];
942     double x21 = m2[1];
943     double x22 = m2[2];
944     double fn = 0;
945     double fn1;
946 
947     double[][] o = new double[3][3];
948     double[] o0 = o[0];
949     double[] o1 = o[1];
950     double[] o2 = o[2];
951 
952     // iterative correction: Xn+1 = Xn - 0.5 * (Xn.Mt.Xn - M)
953     int i = 0;
954     while (++i < 11) {
955 
956       // Mt.Xn
957       double mx00 = m0[0] * x00 + m1[0] * x10 + m2[0] * x20;
958       double mx10 = m0[1] * x00 + m1[1] * x10 + m2[1] * x20;
959       double mx20 = m0[2] * x00 + m1[2] * x10 + m2[2] * x20;
960       double mx01 = m0[0] * x01 + m1[0] * x11 + m2[0] * x21;
961       double mx11 = m0[1] * x01 + m1[1] * x11 + m2[1] * x21;
962       double mx21 = m0[2] * x01 + m1[2] * x11 + m2[2] * x21;
963       double mx02 = m0[0] * x02 + m1[0] * x12 + m2[0] * x22;
964       double mx12 = m0[1] * x02 + m1[1] * x12 + m2[1] * x22;
965       double mx22 = m0[2] * x02 + m1[2] * x12 + m2[2] * x22;
966 
967       // Xn+1
968       o0[0] = x00 - 0.5 * (x00 * mx00 + x01 * mx10 + x02 * mx20 - m0[0]);
969       o0[1] = x01 - 0.5 * (x00 * mx01 + x01 * mx11 + x02 * mx21 - m0[1]);
970       o0[2] = x02 - 0.5 * (x00 * mx02 + x01 * mx12 + x02 * mx22 - m0[2]);
971       o1[0] = x10 - 0.5 * (x10 * mx00 + x11 * mx10 + x12 * mx20 - m1[0]);
972       o1[1] = x11 - 0.5 * (x10 * mx01 + x11 * mx11 + x12 * mx21 - m1[1]);
973       o1[2] = x12 - 0.5 * (x10 * mx02 + x11 * mx12 + x12 * mx22 - m1[2]);
974       o2[0] = x20 - 0.5 * (x20 * mx00 + x21 * mx10 + x22 * mx20 - m2[0]);
975       o2[1] = x21 - 0.5 * (x20 * mx01 + x21 * mx11 + x22 * mx21 - m2[1]);
976       o2[2] = x22 - 0.5 * (x20 * mx02 + x21 * mx12 + x22 * mx22 - m2[2]);
977 
978       // correction on each elements
979       double corr00 = o0[0] - m0[0];
980       double corr01 = o0[1] - m0[1];
981       double corr02 = o0[2] - m0[2];
982       double corr10 = o1[0] - m1[0];
983       double corr11 = o1[1] - m1[1];
984       double corr12 = o1[2] - m1[2];
985       double corr20 = o2[0] - m2[0];
986       double corr21 = o2[1] - m2[1];
987       double corr22 = o2[2] - m2[2];
988 
989       // Frobenius norm of the correction
990       fn1 = corr00 * corr00 + corr01 * corr01 + corr02 * corr02 +
991             corr10 * corr10 + corr11 * corr11 + corr12 * corr12 +
992             corr20 * corr20 + corr21 * corr21 + corr22 * corr22;
993 
994       // convergence test
995       if (Math.abs(fn1 - fn) <= threshold)
996         return o;
997 
998       // prepare next iteration
999       x00 = o0[0];
1000       x01 = o0[1];
1001       x02 = o0[2];
1002       x10 = o1[0];
1003       x11 = o1[1];
1004       x12 = o1[2];
1005       x20 = o2[0];
1006       x21 = o2[1];
1007       x22 = o2[2];
1008       fn  = fn1;
1009 
1010     }
1011 
1012     // the algorithm did not converge after 10 iterations
1013     throw new NotARotationMatrixException("unable to orthogonalize matrix" +
1014                                           " in {0} iterations",
1015                                           new Object[] {
1016                                             Integer.toString(i - 1)
1017                                           });
1018   }
1019 
1020   /** Scalar coordinate of the quaternion. */
1021   private final double q0;
1022 
1023   /** First coordinate of the vectorial part of the quaternion. */
1024   private final double q1;
1025 
1026   /** Second coordinate of the vectorial part of the quaternion. */
1027   private final double q2;
1028 
1029   /** Third coordinate of the vectorial part of the quaternion. */
1030   private final double q3;
1031 
1032   /** Serializable version identifier */
1033   private static final long serialVersionUID = 8225864499430109352L;
1034 
1035 }