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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    *      https://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  /*
19   * This is not the original file distributed by the Apache Software Foundation
20   * It has been modified by the Hipparchus project
21   */
22  
23  package org.hipparchus.geometry.euclidean.threed;
24  
25  import java.io.Serializable;
26  import java.util.function.DoubleSupplier;
27  
28  import org.hipparchus.exception.MathIllegalArgumentException;
29  import org.hipparchus.exception.MathIllegalStateException;
30  import org.hipparchus.exception.MathRuntimeException;
31  import org.hipparchus.geometry.LocalizedGeometryFormats;
32  import org.hipparchus.util.FastMath;
33  import org.hipparchus.util.MathArrays;
34  import org.hipparchus.util.SinCos;
35  
36  /**
37   * This class implements rotations in a three-dimensional space.
38   *
39   * <p>Rotations can be represented by several different mathematical
40   * entities (matrices, axe and angle, Cardan or Euler angles,
41   * quaternions). This class presents an higher level abstraction, more
42   * user-oriented and hiding this implementation details. Well, for the
43   * curious, we use quaternions for the internal representation. The
44   * user can build a rotation from any of these representations, and
45   * any of these representations can be retrieved from a
46   * <code>Rotation</code> instance (see the various constructors and
47   * getters). In addition, a rotation can also be built implicitly
48   * from a set of vectors and their image.</p>
49   * <p>This implies that this class can be used to convert from one
50   * representation to another one. For example, converting a rotation
51   * matrix into a set of Cardan angles from can be done using the
52   * following single line of code:</p>
53   * <pre>
54   * double[] angles = new Rotation(matrix, 1.0e-10).getAngles(RotationOrder.XYZ);
55   * </pre>
56   * <p>Focus is oriented on what a rotation <em>do</em> rather than on its
57   * underlying representation. Once it has been built, and regardless of its
58   * internal representation, a rotation is an <em>operator</em> which basically
59   * transforms three dimensional {@link Vector3D vectors} into other three
60   * dimensional {@link Vector3D vectors}. Depending on the application, the
61   * meaning of these vectors may vary and the semantics of the rotation also.</p>
62   * <p>For example in an spacecraft attitude simulation tool, users will often
63   * consider the vectors are fixed (say the Earth direction for example) and the
64   * frames change. The rotation transforms the coordinates of the vector in inertial
65   * frame into the coordinates of the same vector in satellite frame. In this
66   * case, the rotation implicitly defines the relation between the two frames.</p>
67   * <p>Another example could be a telescope control application, where the rotation
68   * would transform the sighting direction at rest into the desired observing
69   * direction when the telescope is pointed towards an object of interest. In this
70   * case the rotation transforms the direction at rest in a topocentric frame
71   * into the sighting direction in the same topocentric frame. This implies in this
72   * case the frame is fixed and the vector moves.</p>
73   * <p>In many case, both approaches will be combined. In our telescope example,
74   * we will probably also need to transform the observing direction in the topocentric
75   * frame into the observing direction in inertial frame taking into account the observatory
76   * location and the Earth rotation, which would essentially be an application of the
77   * first approach.</p>
78   *
79   * <p>These examples show that a rotation is what the user wants it to be. This
80   * class does not push the user towards one specific definition and hence does not
81   * provide methods like <code>projectVectorIntoDestinationFrame</code> or
82   * <code>computeTransformedDirection</code>. It provides simpler and more generic
83   * methods: {@link #applyTo(Vector3D) applyTo(Vector3D)} and {@link
84   * #applyInverseTo(Vector3D) applyInverseTo(Vector3D)}.</p>
85   *
86   * <p>Since a rotation is basically a vectorial operator, several rotations can be
87   * composed together and the composite operation <code>r = r<sub>1</sub> o
88   * r<sub>2</sub></code> (which means that for each vector <code>u</code>,
89   * <code>r(u) = r<sub>1</sub>(r<sub>2</sub>(u))</code>) is also a rotation. Hence
90   * we can consider that in addition to vectors, a rotation can be applied to other
91   * rotations as well (or to itself). With our previous notations, we would say we
92   * can apply <code>r<sub>1</sub></code> to <code>r<sub>2</sub></code> and the result
93   * we get is <code>r = r<sub>1</sub> o r<sub>2</sub></code>. For this purpose, the
94   * class provides the methods: {@link #applyTo(Rotation) applyTo(Rotation)} and
95   * {@link #applyInverseTo(Rotation) applyInverseTo(Rotation)}.</p>
96   *
97   * <p>Rotations are guaranteed to be immutable objects.</p>
98   *
99   * @see Vector3D
100  * @see RotationOrder
101  */
102 
103 public class Rotation implements Serializable {
104 
105   /** Identity rotation. */
106   public static final Rotation IDENTITY = new Rotation(1.0, 0.0, 0.0, 0.0, false);
107 
108   /** Switch to safe computation of asin/acos.
109    * @since 3.0
110    */
111   private static final double SAFE_SWITCH = 0.999;
112 
113   /** Serializable version identifier */
114   private static final long serialVersionUID = -2153622329907944313L;
115 
116   /** Scalar coordinate of the quaternion. */
117   private final double q0;
118 
119   /** First coordinate of the vectorial part of the quaternion. */
120   private final double q1;
121 
122   /** Second coordinate of the vectorial part of the quaternion. */
123   private final double q2;
124 
125   /** Third coordinate of the vectorial part of the quaternion. */
126   private final double q3;
127 
128   /** Build a rotation from the quaternion coordinates.
129    * <p>A rotation can be built from a <em>normalized</em> quaternion,
130    * i.e. a quaternion for which q<sub>0</sub><sup>2</sup> +
131    * q<sub>1</sub><sup>2</sup> + q<sub>2</sub><sup>2</sup> +
132    * q<sub>3</sub><sup>2</sup> = 1. If the quaternion is not normalized,
133    * the constructor can normalize it in a preprocessing step.</p>
134    * <p>Note that some conventions put the scalar part of the quaternion
135    * as the 4<sup>th</sup> component and the vector part as the first three
136    * components. This is <em>not</em> our convention. We put the scalar part
137    * as the first component.</p>
138    * @param q0 scalar part of the quaternion
139    * @param q1 first coordinate of the vectorial part of the quaternion
140    * @param q2 second coordinate of the vectorial part of the quaternion
141    * @param q3 third coordinate of the vectorial part of the quaternion
142    * @param needsNormalization if true, the coordinates are considered
143    * not to be normalized, a normalization preprocessing step is performed
144    * before using them
145    */
146   public Rotation(double q0, double q1, double q2, double q3,
147                   boolean needsNormalization) {
148 
149     if (needsNormalization) {
150       // normalization preprocessing
151       double inv = 1.0 / FastMath.sqrt(q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3);
152       q0 *= inv;
153       q1 *= inv;
154       q2 *= inv;
155       q3 *= inv;
156     }
157 
158     this.q0 = q0;
159     this.q1 = q1;
160     this.q2 = q2;
161     this.q3 = q3;
162 
163   }
164 
165   /** Build a rotation from an axis and an angle.
166    * @param axis axis around which to rotate
167    * @param angle rotation angle
168    * @param convention convention to use for the semantics of the angle
169    * @exception MathIllegalArgumentException if the axis norm is zero
170    */
171   public Rotation(final Vector3D axis, final double angle, final RotationConvention convention)
172       throws MathIllegalArgumentException {
173 
174     double norm = axis.getNorm();
175     if (norm == 0) {
176       throw new MathIllegalArgumentException(LocalizedGeometryFormats.ZERO_NORM_FOR_ROTATION_AXIS);
177     }
178 
179     double halfAngle = convention == RotationConvention.VECTOR_OPERATOR ? -0.5 * angle : +0.5 * angle;
180     SinCos sinCos = FastMath.sinCos(halfAngle);
181     double coeff = sinCos.sin() / norm;
182 
183     q0 = sinCos.cos();
184     q1 = coeff * axis.getX();
185     q2 = coeff * axis.getY();
186     q3 = coeff * axis.getZ();
187 
188   }
189 
190   /** Build a rotation from a 3X3 matrix.
191 
192    * <p>Rotation matrices are orthogonal matrices, i.e. unit matrices
193    * (which are matrices for which m.m<sup>T</sup> = I) with real
194    * coefficients. The module of the determinant of unit matrices is
195    * 1, among the orthogonal 3X3 matrices, only the ones having a
196    * positive determinant (+1) are rotation matrices.</p>
197 
198    * <p>When a rotation is defined by a matrix with truncated values
199    * (typically when it is extracted from a technical sheet where only
200    * four to five significant digits are available), the matrix is not
201    * orthogonal anymore. This constructor handles this case
202    * transparently by using a copy of the given matrix and applying a
203    * correction to the copy in order to perfect its orthogonality. If
204    * the Frobenius norm of the correction needed is above the given
205    * threshold, then the matrix is considered to be too far from a
206    * true rotation matrix and an exception is thrown.</p>
207 
208    * @param m rotation matrix
209    * @param threshold convergence threshold for the iterative
210    * orthogonality correction (convergence is reached when the
211    * difference between two steps of the Frobenius norm of the
212    * correction is below this threshold)
213 
214    * @exception MathIllegalArgumentException if the matrix is not a 3X3
215    * matrix, or if it cannot be transformed into an orthogonal matrix
216    * with the given threshold, or if the determinant of the resulting
217    * orthogonal matrix is negative
218 
219    */
220   public Rotation(double[][] m, double threshold)
221     throws MathIllegalArgumentException {
222 
223     // dimension check
224     if ((m.length != 3) || (m[0].length != 3) ||
225         (m[1].length != 3) || (m[2].length != 3)) {
226       throw new MathIllegalArgumentException(LocalizedGeometryFormats.ROTATION_MATRIX_DIMENSIONS,
227                                              m.length, m[0].length);
228     }
229 
230     // compute a "close" orthogonal matrix
231     double[][] ort = orthogonalizeMatrix(m, threshold);
232 
233     // check the sign of the determinant
234     double det = ort[0][0] * (ort[1][1] * ort[2][2] - ort[2][1] * ort[1][2]) -
235                  ort[1][0] * (ort[0][1] * ort[2][2] - ort[2][1] * ort[0][2]) +
236                  ort[2][0] * (ort[0][1] * ort[1][2] - ort[1][1] * ort[0][2]);
237     if (det < 0.0) {
238       throw new MathIllegalArgumentException(LocalizedGeometryFormats.CLOSEST_ORTHOGONAL_MATRIX_HAS_NEGATIVE_DETERMINANT,
239                                              det);
240     }
241 
242     double[] quat = mat2quat(ort);
243     q0 = quat[0];
244     q1 = quat[1];
245     q2 = quat[2];
246     q3 = quat[3];
247 
248   }
249 
250   /** Build the rotation that transforms a pair of vectors into another pair.
251 
252    * <p>Except for possible scale factors, if the instance were applied to
253    * the pair (u<sub>1</sub>, u<sub>2</sub>) it will produce the pair
254    * (v<sub>1</sub>, v<sub>2</sub>).</p>
255 
256    * <p>If the angular separation between u<sub>1</sub> and u<sub>2</sub> is
257    * not the same as the angular separation between v<sub>1</sub> and
258    * v<sub>2</sub>, then a corrected v'<sub>2</sub> will be used rather than
259    * v<sub>2</sub>, the corrected vector will be in the (±v<sub>1</sub>,
260    * +v<sub>2</sub>) half-plane.</p>
261    * @param u1 first vector of the origin pair
262    * @param u2 second vector of the origin pair
263    * @param v1 desired image of u1 by the rotation
264    * @param v2 desired image of u2 by the rotation
265    * @exception MathRuntimeException if the norm of one of the vectors is zero,
266    * or if one of the pair is degenerated (i.e. the vectors of the pair are collinear)
267    */
268   public Rotation(Vector3D u1, Vector3D u2, Vector3D v1, Vector3D v2)
269       throws MathRuntimeException {
270 
271       // build orthonormalized base from u1, u2
272       // this fails when vectors are null or collinear, which is forbidden to define a rotation
273       final Vector3D u3 = u1.crossProduct(u2).normalize();
274       u2 = u3.crossProduct(u1).normalize();
275       u1 = u1.normalize();
276 
277       // build an orthonormalized base from v1, v2
278       // this fails when vectors are null or collinear, which is forbidden to define a rotation
279       final Vector3D v3 = v1.crossProduct(v2).normalize();
280       v2 = v3.crossProduct(v1).normalize();
281       v1 = v1.normalize();
282 
283       // buid a matrix transforming the first base into the second one
284       final double[][] m = {
285           {
286               MathArrays.linearCombination(u1.getX(), v1.getX(), u2.getX(), v2.getX(), u3.getX(), v3.getX()),
287               MathArrays.linearCombination(u1.getY(), v1.getX(), u2.getY(), v2.getX(), u3.getY(), v3.getX()),
288               MathArrays.linearCombination(u1.getZ(), v1.getX(), u2.getZ(), v2.getX(), u3.getZ(), v3.getX())
289           },
290           {
291               MathArrays.linearCombination(u1.getX(), v1.getY(), u2.getX(), v2.getY(), u3.getX(), v3.getY()),
292               MathArrays.linearCombination(u1.getY(), v1.getY(), u2.getY(), v2.getY(), u3.getY(), v3.getY()),
293               MathArrays.linearCombination(u1.getZ(), v1.getY(), u2.getZ(), v2.getY(), u3.getZ(), v3.getY())
294           },
295           {
296               MathArrays.linearCombination(u1.getX(), v1.getZ(), u2.getX(), v2.getZ(), u3.getX(), v3.getZ()),
297               MathArrays.linearCombination(u1.getY(), v1.getZ(), u2.getY(), v2.getZ(), u3.getY(), v3.getZ()),
298               MathArrays.linearCombination(u1.getZ(), v1.getZ(), u2.getZ(), v2.getZ(), u3.getZ(), v3.getZ())
299           }
300       };
301 
302       double[] quat = mat2quat(m);
303       q0 = quat[0];
304       q1 = quat[1];
305       q2 = quat[2];
306       q3 = quat[3];
307 
308   }
309 
310   /** Build one of the rotations that transform one vector into another one.
311 
312    * <p>Except for a possible scale factor, if the instance were
313    * applied to the vector u it will produce the vector v. There is an
314    * infinite number of such rotations, this constructor choose the
315    * one with the smallest associated angle (i.e. the one whose axis
316    * is orthogonal to the (u, v) plane). If u and v are collinear, an
317    * arbitrary rotation axis is chosen.</p>
318 
319    * @param u origin vector
320    * @param v desired image of u by the rotation
321    * @exception MathRuntimeException if the norm of one of the vectors is zero
322    */
323   public Rotation(Vector3D u, Vector3D v) throws MathRuntimeException {
324 
325     double normProduct = u.getNorm() * v.getNorm();
326     if (normProduct == 0) {
327         throw new MathRuntimeException(LocalizedGeometryFormats.ZERO_NORM_FOR_ROTATION_DEFINING_VECTOR);
328     }
329 
330     double dot = u.dotProduct(v);
331 
332     if (dot < ((2.0e-15 - 1.0) * normProduct)) {
333       // special case u = -v: we select a PI angle rotation around
334       // an arbitrary vector orthogonal to u
335       Vector3D w = u.orthogonal();
336       q0 = 0.0;
337       q1 = -w.getX();
338       q2 = -w.getY();
339       q3 = -w.getZ();
340     } else {
341       // general case: (u, v) defines a plane, we select
342       // the shortest possible rotation: axis orthogonal to this plane
343       q0 = FastMath.sqrt(0.5 * (1.0 + dot / normProduct));
344       double coeff = 1.0 / (2.0 * q0 * normProduct);
345       Vector3D q = v.crossProduct(u);
346       q1 = coeff * q.getX();
347       q2 = coeff * q.getY();
348       q3 = coeff * q.getZ();
349     }
350 
351   }
352 
353   /** Build a rotation from three Cardan or Euler elementary rotations.
354 
355    * <p>Cardan rotations are three successive rotations around the
356    * canonical axes X, Y and Z, each axis being used once. There are
357    * 6 such sets of rotations (XYZ, XZY, YXZ, YZX, ZXY and ZYX). Euler
358    * rotations are three successive rotations around the canonical
359    * axes X, Y and Z, the first and last rotations being around the
360    * same axis. There are 6 such sets of rotations (XYX, XZX, YXY,
361    * YZY, ZXZ and ZYZ), the most popular one being ZXZ.</p>
362    * <p>Beware that many people routinely use the term Euler angles even
363    * for what really are Cardan angles (this confusion is especially
364    * widespread in the aerospace business where Roll, Pitch and Yaw angles
365    * are often wrongly tagged as Euler angles).</p>
366 
367    * @param order order of rotations to compose, from left to right
368    * (i.e. we will use {@code r1.compose(r2.compose(r3, convention), convention)})
369    * @param convention convention to use for the semantics of the angle
370    * @param alpha1 angle of the first elementary rotation
371    * @param alpha2 angle of the second elementary rotation
372    * @param alpha3 angle of the third elementary rotation
373    */
374   public Rotation(RotationOrder order, RotationConvention convention,
375                   double alpha1, double alpha2, double alpha3) {
376       Rotation r1 = new Rotation(order.getA1(), alpha1, convention);
377       Rotation r2 = new Rotation(order.getA2(), alpha2, convention);
378       Rotation r3 = new Rotation(order.getA3(), alpha3, convention);
379       Rotation composed = r1.compose(r2.compose(r3, convention), convention);
380       q0 = composed.q0;
381       q1 = composed.q1;
382       q2 = composed.q2;
383       q3 = composed.q3;
384   }
385 
386   /** Convert an orthogonal rotation matrix to a quaternion.
387    * @param ort orthogonal rotation matrix
388    * @return quaternion corresponding to the matrix
389    */
390   private static double[] mat2quat(final double[][] ort) {
391 
392       final double[] quat = new double[4];
393 
394       // There are different ways to compute the quaternions elements
395       // from the matrix. They all involve computing one element from
396       // the diagonal of the matrix, and computing the three other ones
397       // using a formula involving a division by the first element,
398       // which unfortunately can be zero. Since the norm of the
399       // quaternion is 1, we know at least one element has an absolute
400       // value greater or equal to 0.5, so it is always possible to
401       // select the right formula and avoid division by zero and even
402       // numerical inaccuracy. Checking the elements in turn and using
403       // the first one greater than 0.45 is safe (this leads to a simple
404       // test since qi = 0.45 implies 4 qi^2 - 1 = -0.19)
405       double s = ort[0][0] + ort[1][1] + ort[2][2];
406       if (s > -0.19) {
407           // compute q0 and deduce q1, q2 and q3
408           quat[0] = 0.5 * FastMath.sqrt(s + 1.0);
409           double inv = 0.25 / quat[0];
410           quat[1] = inv * (ort[1][2] - ort[2][1]);
411           quat[2] = inv * (ort[2][0] - ort[0][2]);
412           quat[3] = inv * (ort[0][1] - ort[1][0]);
413       } else {
414           s = ort[0][0] - ort[1][1] - ort[2][2];
415           if (s > -0.19) {
416               // compute q1 and deduce q0, q2 and q3
417               quat[1] = 0.5 * FastMath.sqrt(s + 1.0);
418               double inv = 0.25 / quat[1];
419               quat[0] = inv * (ort[1][2] - ort[2][1]);
420               quat[2] = inv * (ort[0][1] + ort[1][0]);
421               quat[3] = inv * (ort[0][2] + ort[2][0]);
422           } else {
423               s = ort[1][1] - ort[0][0] - ort[2][2];
424               if (s > -0.19) {
425                   // compute q2 and deduce q0, q1 and q3
426                   quat[2] = 0.5 * FastMath.sqrt(s + 1.0);
427                   double inv = 0.25 / quat[2];
428                   quat[0] = inv * (ort[2][0] - ort[0][2]);
429                   quat[1] = inv * (ort[0][1] + ort[1][0]);
430                   quat[3] = inv * (ort[2][1] + ort[1][2]);
431               } else {
432                   // compute q3 and deduce q0, q1 and q2
433                   s = ort[2][2] - ort[0][0] - ort[1][1];
434                   quat[3] = 0.5 * FastMath.sqrt(s + 1.0);
435                   double inv = 0.25 / quat[3];
436                   quat[0] = inv * (ort[0][1] - ort[1][0]);
437                   quat[1] = inv * (ort[0][2] + ort[2][0]);
438                   quat[2] = inv * (ort[2][1] + ort[1][2]);
439               }
440           }
441       }
442 
443       return quat;
444 
445   }
446 
447   /** Revert a rotation.
448    * Build a rotation which reverse the effect of another
449    * rotation. This means that if r(u) = v, then r.revert(v) = u. The
450    * instance is not changed.
451    * @return a new rotation whose effect is the reverse of the effect
452    * of the instance
453    */
454   public Rotation revert() {
455     return new Rotation(-q0, q1, q2, q3, false);
456   }
457 
458   /** Get the scalar coordinate of the quaternion.
459    * @return scalar coordinate of the quaternion
460    */
461   public double getQ0() {
462     return q0;
463   }
464 
465   /** Get the first coordinate of the vectorial part of the quaternion.
466    * @return first coordinate of the vectorial part of the quaternion
467    */
468   public double getQ1() {
469     return q1;
470   }
471 
472   /** Get the second coordinate of the vectorial part of the quaternion.
473    * @return second coordinate of the vectorial part of the quaternion
474    */
475   public double getQ2() {
476     return q2;
477   }
478 
479   /** Get the third coordinate of the vectorial part of the quaternion.
480    * @return third coordinate of the vectorial part of the quaternion
481    */
482   public double getQ3() {
483     return q3;
484   }
485 
486   /** Get the normalized axis of the rotation.
487    * <p>
488    * Note that as {@link #getAngle()} always returns an angle
489    * between 0 and &pi;, changing the convention changes the
490    * direction of the axis, not the sign of the angle.
491    * </p>
492    * @param convention convention to use for the semantics of the angle
493    * @return normalized axis of the rotation
494    * @see #Rotation(Vector3D, double, RotationConvention)
495    */
496   public Vector3D getAxis(final RotationConvention convention) {
497     final double squaredSine = q1 * q1 + q2 * q2 + q3 * q3;
498     if (squaredSine == 0) {
499       return convention == RotationConvention.VECTOR_OPERATOR ? Vector3D.PLUS_I : Vector3D.MINUS_I;
500     } else {
501         final double sgn = convention == RotationConvention.VECTOR_OPERATOR ? +1 : -1;
502         if (q0 < 0) {
503             final double inverse = sgn / FastMath.sqrt(squaredSine);
504             return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
505         }
506         final double inverse = -sgn / FastMath.sqrt(squaredSine);
507         return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
508     }
509   }
510 
511   /** Get the angle of the rotation.
512    * @return angle of the rotation (between 0 and &pi;)
513    * @see #Rotation(Vector3D, double, RotationConvention)
514    */
515   public double getAngle() {
516     if ((q0 < -0.1) || (q0 > 0.1)) {
517       return 2 * FastMath.asin(FastMath.sqrt(q1 * q1 + q2 * q2 + q3 * q3));
518     } else if (q0 < 0) {
519       return 2 * FastMath.acos(-q0);
520     }
521     return 2 * FastMath.acos(q0);
522   }
523 
524   /** Get the Cardan or Euler angles corresponding to the instance.
525 
526    * <p>The equations show that each rotation can be defined by two
527    * different values of the Cardan or Euler angles set. For example
528    * if Cardan angles are used, the rotation defined by the angles
529    * a<sub>1</sub>, a<sub>2</sub> and a<sub>3</sub> is the same as
530    * the rotation defined by the angles &pi; + a<sub>1</sub>, &pi;
531    * - a<sub>2</sub> and &pi; + a<sub>3</sub>. This method implements
532    * the following arbitrary choices:</p>
533    * <ul>
534    *   <li>for Cardan angles, the chosen set is the one for which the
535    *   second angle is between -&pi;/2 and &pi;/2 (i.e its cosine is
536    *   positive),</li>
537    *   <li>for Euler angles, the chosen set is the one for which the
538    *   second angle is between 0 and &pi; (i.e its sine is positive).</li>
539    * </ul>
540 
541    * <p>Cardan and Euler angle have a very disappointing drawback: all
542    * of them have singularities. This means that if the instance is
543    * too close to the singularities corresponding to the given
544    * rotation order, it will be impossible to retrieve the angles. For
545    * Cardan angles, this is often called gimbal lock. There is
546    * <em>nothing</em> to do to prevent this, it is an intrinsic problem
547    * with Cardan and Euler representation (but not a problem with the
548    * rotation itself, which is perfectly well defined). For Cardan
549    * angles, singularities occur when the second angle is close to
550    * -&pi;/2 or +&pi;/2, for Euler angle singularities occur when the
551    * second angle is close to 0 or &pi;, this implies that the identity
552    * rotation is always singular for Euler angles!</p>
553 
554    * @param order rotation order to use
555    * @param convention convention to use for the semantics of the angle
556    * @return an array of three angles, in the order specified by the set
557    * @exception MathIllegalStateException if the rotation is
558    * singular with respect to the angles set specified
559    */
560   public double[] getAngles(RotationOrder order, RotationConvention convention)
561       throws MathIllegalStateException {
562 
563       if (convention == RotationConvention.VECTOR_OPERATOR) {
564           if (order == RotationOrder.XYZ) {
565 
566               // r (Vector3D.plusK) coordinates are :
567               //  sin (theta), -cos (theta) sin (phi), cos (theta) cos (phi)
568               // (-r) (Vector3D.plusI) coordinates are :
569               // cos (psi) cos (theta), -sin (psi) cos (theta), sin (theta)
570               // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
571               Vector3D v1 = applyTo(Vector3D.PLUS_K);
572               Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
573               return new double[] {
574                   FastMath.atan2(-(v1.getY()), v1.getZ()),
575                   safeAsin(v2::getZ, v2::getX, v2::getY),
576                   FastMath.atan2(-(v2.getY()), v2.getX())
577               };
578 
579           } else if (order == RotationOrder.XZY) {
580 
581               // r (Vector3D.plusJ) coordinates are :
582               // -sin (psi), cos (psi) cos (phi), cos (psi) sin (phi)
583               // (-r) (Vector3D.plusI) coordinates are :
584               // cos (theta) cos (psi), -sin (psi), sin (theta) cos (psi)
585               // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
586               Vector3D v1 = applyTo(Vector3D.PLUS_J);
587               Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
588               return new double[] {
589                   FastMath.atan2(v1.getZ(), v1.getY()),
590                  -safeAsin(v2::getY, v2::getZ, v2::getX),
591                   FastMath.atan2(v2.getZ(), v2.getX())
592               };
593 
594           } else if (order == RotationOrder.YXZ) {
595 
596               // r (Vector3D.plusK) coordinates are :
597               //  cos (phi) sin (theta), -sin (phi), cos (phi) cos (theta)
598               // (-r) (Vector3D.plusJ) coordinates are :
599               // sin (psi) cos (phi), cos (psi) cos (phi), -sin (phi)
600               // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
601               Vector3D v1 = applyTo(Vector3D.PLUS_K);
602               Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
603               return new double[] {
604                   FastMath.atan2(v1.getX(), v1.getZ()),
605                  -safeAsin(v2::getZ, v2::getX, v2::getY),
606                   FastMath.atan2(v2.getX(), v2.getY())
607               };
608 
609           } else if (order == RotationOrder.YZX) {
610 
611               // r (Vector3D.plusI) coordinates are :
612               // cos (psi) cos (theta), sin (psi), -cos (psi) sin (theta)
613               // (-r) (Vector3D.plusJ) coordinates are :
614               // sin (psi), cos (phi) cos (psi), -sin (phi) cos (psi)
615               // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
616               Vector3D v1 = applyTo(Vector3D.PLUS_I);
617               Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
618               return new double[] {
619                   FastMath.atan2(-(v1.getZ()), v1.getX()),
620                   safeAsin(v2::getX, v2::getY, v2::getZ),
621                   FastMath.atan2(-(v2.getZ()), v2.getY())
622               };
623 
624           } else if (order == RotationOrder.ZXY) {
625 
626               // r (Vector3D.plusJ) coordinates are :
627               // -cos (phi) sin (psi), cos (phi) cos (psi), sin (phi)
628               // (-r) (Vector3D.plusK) coordinates are :
629               // -sin (theta) cos (phi), sin (phi), cos (theta) cos (phi)
630               // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
631               Vector3D v1 = applyTo(Vector3D.PLUS_J);
632               Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
633               return new double[] {
634                   FastMath.atan2(-(v1.getX()), v1.getY()),
635                   safeAsin(v2::getY, v2::getZ, v2::getX),
636                   FastMath.atan2(-(v2.getX()), v2.getZ())
637               };
638 
639           } else if (order == RotationOrder.ZYX) {
640 
641               // r (Vector3D.plusI) coordinates are :
642               //  cos (theta) cos (psi), cos (theta) sin (psi), -sin (theta)
643               // (-r) (Vector3D.plusK) coordinates are :
644               // -sin (theta), sin (phi) cos (theta), cos (phi) cos (theta)
645               // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
646               Vector3D v1 = applyTo(Vector3D.PLUS_I);
647               Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
648               return new double[] {
649                   FastMath.atan2(v1.getY(), v1.getX()),
650                  -safeAsin(v2::getX, v2::getY, v2::getZ),
651                   FastMath.atan2(v2.getY(), v2.getZ())
652               };
653 
654           } else if (order == RotationOrder.XYX) {
655 
656               // r (Vector3D.plusI) coordinates are :
657               //  cos (theta), sin (phi1) sin (theta), -cos (phi1) sin (theta)
658               // (-r) (Vector3D.plusI) coordinates are :
659               // cos (theta), sin (theta) sin (phi2), sin (theta) cos (phi2)
660               // and we can choose to have theta in the interval [0 ; PI]
661               Vector3D v1 = applyTo(Vector3D.PLUS_I);
662               Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
663               return new double[] {
664                   FastMath.atan2(v1.getY(), -v1.getZ()),
665                   safeAcos(v2::getX, v2::getY, v2::getZ),
666                   FastMath.atan2(v2.getY(), v2.getZ())
667               };
668 
669           } else if (order == RotationOrder.XZX) {
670 
671               // r (Vector3D.plusI) coordinates are :
672               //  cos (psi), cos (phi1) sin (psi), sin (phi1) sin (psi)
673               // (-r) (Vector3D.plusI) coordinates are :
674               // cos (psi), -sin (psi) cos (phi2), sin (psi) sin (phi2)
675               // and we can choose to have psi in the interval [0 ; PI]
676               Vector3D v1 = applyTo(Vector3D.PLUS_I);
677               Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
678               return new double[] {
679                   FastMath.atan2(v1.getZ(), v1.getY()),
680                   safeAcos(v2::getX, v2::getY, v2::getZ),
681                   FastMath.atan2(v2.getZ(), -v2.getY())
682               };
683 
684           } else if (order == RotationOrder.YXY) {
685 
686               // r (Vector3D.plusJ) coordinates are :
687               //  sin (theta1) sin (phi), cos (phi), cos (theta1) sin (phi)
688               // (-r) (Vector3D.plusJ) coordinates are :
689               // sin (phi) sin (theta2), cos (phi), -sin (phi) cos (theta2)
690               // and we can choose to have phi in the interval [0 ; PI]
691               Vector3D v1 = applyTo(Vector3D.PLUS_J);
692               Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
693               return new double[] {
694                   FastMath.atan2(v1.getX(), v1.getZ()),
695                   safeAcos(v2::getY, v2::getZ, v2::getX),
696                   FastMath.atan2(v2.getX(), -v2.getZ())
697               };
698 
699           } else if (order == RotationOrder.YZY) {
700 
701               // r (Vector3D.plusJ) coordinates are :
702               //  -cos (theta1) sin (psi), cos (psi), sin (theta1) sin (psi)
703               // (-r) (Vector3D.plusJ) coordinates are :
704               // sin (psi) cos (theta2), cos (psi), sin (psi) sin (theta2)
705               // and we can choose to have psi in the interval [0 ; PI]
706               Vector3D v1 = applyTo(Vector3D.PLUS_J);
707               Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
708               return new double[] {
709                   FastMath.atan2(v1.getZ(), -v1.getX()),
710                   safeAcos(v2::getY, v2::getZ, v2::getX),
711                   FastMath.atan2(v2.getZ(), v2.getX())
712               };
713 
714           } else if (order == RotationOrder.ZXZ) {
715 
716               // r (Vector3D.plusK) coordinates are :
717               //  sin (psi1) sin (phi), -cos (psi1) sin (phi), cos (phi)
718               // (-r) (Vector3D.plusK) coordinates are :
719               // sin (phi) sin (psi2), sin (phi) cos (psi2), cos (phi)
720               // and we can choose to have phi in the interval [0 ; PI]
721               Vector3D v1 = applyTo(Vector3D.PLUS_K);
722               Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
723               return new double[] {
724                   FastMath.atan2(v1.getX(), -v1.getY()),
725                   safeAcos(v2::getZ, v2::getX, v2::getY),
726                   FastMath.atan2(v2.getX(), v2.getY())
727               };
728 
729           } else { // last possibility is ZYZ
730 
731               // r (Vector3D.plusK) coordinates are :
732               //  cos (psi1) sin (theta), sin (psi1) sin (theta), cos (theta)
733               // (-r) (Vector3D.plusK) coordinates are :
734               // -sin (theta) cos (psi2), sin (theta) sin (psi2), cos (theta)
735               // and we can choose to have theta in the interval [0 ; PI]
736               Vector3D v1 = applyTo(Vector3D.PLUS_K);
737               Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
738               return new double[] {
739                   FastMath.atan2(v1.getY(), v1.getX()),
740                   safeAcos(v2::getZ, v2::getX, v2::getY),
741                   FastMath.atan2(v2.getY(), -v2.getX())
742               };
743 
744           }
745       } else {
746           if (order == RotationOrder.XYZ) {
747 
748               // r (Vector3D.plusI) coordinates are :
749               //  cos (theta) cos (psi), -cos (theta) sin (psi), sin (theta)
750               // (-r) (Vector3D.plusK) coordinates are :
751               // sin (theta), -sin (phi) cos (theta), cos (phi) cos (theta)
752               // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
753               Vector3D v1 = applyTo(Vector3D.PLUS_I);
754               Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
755               return new double[] {
756                   FastMath.atan2(-v2.getY(), v2.getZ()),
757                   safeAsin(v2::getX, v2::getY, v2::getZ),
758                   FastMath.atan2(-v1.getY(), v1.getX())
759               };
760 
761           } else if (order == RotationOrder.XZY) {
762 
763               // r (Vector3D.plusI) coordinates are :
764               // cos (psi) cos (theta), -sin (psi), cos (psi) sin (theta)
765               // (-r) (Vector3D.plusJ) coordinates are :
766               // -sin (psi), cos (phi) cos (psi), sin (phi) cos (psi)
767               // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
768               Vector3D v1 = applyTo(Vector3D.PLUS_I);
769               Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
770               return new double[] {
771                   FastMath.atan2(v2.getZ(), v2.getY()),
772                  -safeAsin(v2::getX, v2::getY, v2::getZ),
773                   FastMath.atan2(v1.getZ(), v1.getX())
774               };
775 
776           } else if (order == RotationOrder.YXZ) {
777 
778               // r (Vector3D.plusJ) coordinates are :
779               // cos (phi) sin (psi), cos (phi) cos (psi), -sin (phi)
780               // (-r) (Vector3D.plusK) coordinates are :
781               // sin (theta) cos (phi), -sin (phi), cos (theta) cos (phi)
782               // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
783               Vector3D v1 = applyTo(Vector3D.PLUS_J);
784               Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
785               return new double[] {
786                   FastMath.atan2(v2.getX(), v2.getZ()),
787                  -safeAsin(v2::getY, v2::getZ, v2::getX),
788                   FastMath.atan2(v1.getX(), v1.getY())
789               };
790 
791           } else if (order == RotationOrder.YZX) {
792 
793               // r (Vector3D.plusJ) coordinates are :
794               // sin (psi), cos (psi) cos (phi), -cos (psi) sin (phi)
795               // (-r) (Vector3D.plusI) coordinates are :
796               // cos (theta) cos (psi), sin (psi), -sin (theta) cos (psi)
797               // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
798               Vector3D v1 = applyTo(Vector3D.PLUS_J);
799               Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
800               return new double[] {
801                   FastMath.atan2(-v2.getZ(), v2.getX()),
802                   safeAsin(v2::getY, v2::getZ, v2::getX),
803                   FastMath.atan2(-v1.getZ(), v1.getY())
804               };
805 
806           } else if (order == RotationOrder.ZXY) {
807 
808               // r (Vector3D.plusK) coordinates are :
809               //  -cos (phi) sin (theta), sin (phi), cos (phi) cos (theta)
810               // (-r) (Vector3D.plusJ) coordinates are :
811               // -sin (psi) cos (phi), cos (psi) cos (phi), sin (phi)
812               // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
813               Vector3D v1 = applyTo(Vector3D.PLUS_K);
814               Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
815               return new double[] {
816                   FastMath.atan2(-v2.getX(), v2.getY()),
817                   safeAsin(v2::getZ, v2::getX, v2::getY),
818                   FastMath.atan2(-v1.getX(), v1.getZ())
819               };
820 
821           } else if (order == RotationOrder.ZYX) {
822 
823               // r (Vector3D.plusK) coordinates are :
824               //  -sin (theta), cos (theta) sin (phi), cos (theta) cos (phi)
825               // (-r) (Vector3D.plusI) coordinates are :
826               // cos (psi) cos (theta), sin (psi) cos (theta), -sin (theta)
827               // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
828               Vector3D v1 = applyTo(Vector3D.PLUS_K);
829               Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
830               return new double[] {
831                   FastMath.atan2(v2.getY(), v2.getX()),
832                  -safeAsin(v2::getZ, v2::getX, v2::getY),
833                   FastMath.atan2(v1.getY(), v1.getZ())
834               };
835 
836           } else if (order == RotationOrder.XYX) {
837 
838               // r (Vector3D.plusI) coordinates are :
839               //  cos (theta), sin (phi2) sin (theta), cos (phi2) sin (theta)
840               // (-r) (Vector3D.plusI) coordinates are :
841               // cos (theta), sin (theta) sin (phi1), -sin (theta) cos (phi1)
842               // and we can choose to have theta in the interval [0 ; PI]
843               Vector3D v1 = applyTo(Vector3D.PLUS_I);
844               Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
845               return new double[] {
846                   FastMath.atan2(v2.getY(), -v2.getZ()),
847                   safeAcos(v2::getX, v2::getY, v2::getZ),
848                   FastMath.atan2(v1.getY(), v1.getZ())
849               };
850 
851           } else if (order == RotationOrder.XZX) {
852 
853               // r (Vector3D.plusI) coordinates are :
854               //  cos (psi), -cos (phi2) sin (psi), sin (phi2) sin (psi)
855               // (-r) (Vector3D.plusI) coordinates are :
856               // cos (psi), sin (psi) cos (phi1), sin (psi) sin (phi1)
857               // and we can choose to have psi in the interval [0 ; PI]
858               Vector3D v1 = applyTo(Vector3D.PLUS_I);
859               Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
860               return new double[] {
861                   FastMath.atan2(v2.getZ(), v2.getY()),
862                   safeAcos(v2::getX, v2::getY, v2::getZ),
863                   FastMath.atan2(v1.getZ(), -v1.getY())
864               };
865 
866           } else if (order == RotationOrder.YXY) {
867 
868               // r (Vector3D.plusJ) coordinates are :
869               // sin (phi) sin (theta2), cos (phi), -sin (phi) cos (theta2)
870               // (-r) (Vector3D.plusJ) coordinates are :
871               //  sin (theta1) sin (phi), cos (phi), cos (theta1) sin (phi)
872               // and we can choose to have phi in the interval [0 ; PI]
873               Vector3D v1 = applyTo(Vector3D.PLUS_J);
874               Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
875               return new double[] {
876                   FastMath.atan2(v2.getX(), v2.getZ()),
877                   safeAcos(v2::getY, v2::getZ, v2::getX),
878                   FastMath.atan2(v1.getX(), -v1.getZ())
879               };
880 
881           } else if (order == RotationOrder.YZY) {
882 
883               // r (Vector3D.plusJ) coordinates are :
884               // sin (psi) cos (theta2), cos (psi), sin (psi) sin (theta2)
885               // (-r) (Vector3D.plusJ) coordinates are :
886               //  -cos (theta1) sin (psi), cos (psi), sin (theta1) sin (psi)
887               // and we can choose to have psi in the interval [0 ; PI]
888               Vector3D v1 = applyTo(Vector3D.PLUS_J);
889               Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
890               return new double[] {
891                   FastMath.atan2(v2.getZ(), -v2.getX()),
892                   safeAcos(v2::getY, v2::getZ, v2::getX),
893                   FastMath.atan2(v1.getZ(), v1.getX())
894               };
895 
896           } else if (order == RotationOrder.ZXZ) {
897 
898               // r (Vector3D.plusK) coordinates are :
899               // sin (phi) sin (psi2), sin (phi) cos (psi2), cos (phi)
900               // (-r) (Vector3D.plusK) coordinates are :
901               //  sin (psi1) sin (phi), -cos (psi1) sin (phi), cos (phi)
902               // and we can choose to have phi in the interval [0 ; PI]
903               Vector3D v1 = applyTo(Vector3D.PLUS_K);
904               Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
905               return new double[] {
906                   FastMath.atan2(v2.getX(), -v2.getY()),
907                   safeAcos(v2::getZ, v2::getX, v2::getY),
908                   FastMath.atan2(v1.getX(), v1.getY())
909               };
910 
911           } else { // last possibility is ZYZ
912 
913               // r (Vector3D.plusK) coordinates are :
914               // -sin (theta) cos (psi2), sin (theta) sin (psi2), cos (theta)
915               // (-r) (Vector3D.plusK) coordinates are :
916               //  cos (psi1) sin (theta), sin (psi1) sin (theta), cos (theta)
917               // and we can choose to have theta in the interval [0 ; PI]
918               Vector3D v1 = applyTo(Vector3D.PLUS_K);
919               Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
920               return new double[] {
921                   FastMath.atan2(v2.getY(), v2.getX()),
922                   safeAcos(v2::getZ, v2::getX, v2::getY),
923                   FastMath.atan2(v1.getY(), -v1.getX())
924               };
925 
926           }
927       }
928 
929   }
930 
931   /** Safe computation of acos(some vector coordinate) working around singularities.
932    * @param cosGetter getter for the cosine coordinate
933    * @param sin1Getter getter for one of the sine coordinates
934    * @param sin2Getter getter for the other sine coordinate
935    * @return acos of the coordinate
936    * @since 3.0
937    */
938   private double safeAcos(final DoubleSupplier cosGetter,
939                           final DoubleSupplier sin1Getter,
940                           final DoubleSupplier sin2Getter) {
941       final double cos = cosGetter.getAsDouble();
942       if (cos < -SAFE_SWITCH) {
943           final double s1 = sin1Getter.getAsDouble();
944           final double s2 = sin2Getter.getAsDouble();
945           return FastMath.PI - FastMath.asin(FastMath.sqrt(s1 * s1 + s2 * s2));
946       } else if (cos > SAFE_SWITCH) {
947           final double s1 = sin1Getter.getAsDouble();
948           final double s2 = sin2Getter.getAsDouble();
949           return FastMath.asin(FastMath.sqrt(s1 * s1 + s2 * s2));
950       } else {
951           return FastMath.acos(cos);
952       }
953   }
954 
955   /** Safe computation of asin(some vector coordinate) working around singularities.
956    * @param sinGetter getter for the sine coordinate
957    * @param cos1Getter getter for one of the cosine coordinates
958    * @param cos2Getter getter for the other cosine coordinate
959    * @return acos of the coordinate
960    * @since 3.0
961    */
962   private double safeAsin(final DoubleSupplier sinGetter,
963                           final DoubleSupplier cos1Getter,
964                           final DoubleSupplier cos2Getter) {
965       final double sin = sinGetter.getAsDouble();
966       if (sin < -SAFE_SWITCH) {
967           final double c1 = cos1Getter.getAsDouble();
968           final double c2 = cos2Getter.getAsDouble();
969           return -FastMath.acos(FastMath.sqrt(c1 * c1 + c2 * c2));
970       } else if (sin > SAFE_SWITCH) {
971           final double c1 = cos1Getter.getAsDouble();
972           final double c2 = cos2Getter.getAsDouble();
973           return FastMath.acos(FastMath.sqrt(c1 * c1 + c2 * c2));
974       } else {
975           return FastMath.asin(sin);
976       }
977   }
978 
979   /** Get the 3X3 matrix corresponding to the instance
980    * @return the matrix corresponding to the instance
981    */
982   public double[][] getMatrix() {
983 
984     // products
985     double q0q0  = q0 * q0;
986     double q0q1  = q0 * q1;
987     double q0q2  = q0 * q2;
988     double q0q3  = q0 * q3;
989     double q1q1  = q1 * q1;
990     double q1q2  = q1 * q2;
991     double q1q3  = q1 * q3;
992     double q2q2  = q2 * q2;
993     double q2q3  = q2 * q3;
994     double q3q3  = q3 * q3;
995 
996     // create the matrix
997     double[][] m = new double[3][];
998     m[0] = new double[3];
999     m[1] = new double[3];
1000     m[2] = new double[3];
1001 
1002     m [0][0] = 2.0 * (q0q0 + q1q1) - 1.0;
1003     m [1][0] = 2.0 * (q1q2 - q0q3);
1004     m [2][0] = 2.0 * (q1q3 + q0q2);
1005 
1006     m [0][1] = 2.0 * (q1q2 + q0q3);
1007     m [1][1] = 2.0 * (q0q0 + q2q2) - 1.0;
1008     m [2][1] = 2.0 * (q2q3 - q0q1);
1009 
1010     m [0][2] = 2.0 * (q1q3 - q0q2);
1011     m [1][2] = 2.0 * (q2q3 + q0q1);
1012     m [2][2] = 2.0 * (q0q0 + q3q3) - 1.0;
1013 
1014     return m;
1015 
1016   }
1017 
1018   /** Apply the rotation to a vector.
1019    * @param u vector to apply the rotation to
1020    * @return a new vector which is the image of u by the rotation
1021    */
1022   public Vector3D applyTo(Vector3D u) {
1023 
1024     double x = u.getX();
1025     double y = u.getY();
1026     double z = u.getZ();
1027 
1028     double s = q1 * x + q2 * y + q3 * z;
1029 
1030     return new Vector3D(2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x,
1031                         2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y,
1032                         2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - z);
1033 
1034   }
1035 
1036   /** Apply the rotation to a vector stored in an array.
1037    * @param in an array with three items which stores vector to rotate
1038    * @param out an array with three items to put result to (it can be the same
1039    * array as in)
1040    */
1041   public void applyTo(final double[] in, final double[] out) {
1042 
1043       final double x = in[0];
1044       final double y = in[1];
1045       final double z = in[2];
1046 
1047       final double s = q1 * x + q2 * y + q3 * z;
1048 
1049       out[0] = 2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x;
1050       out[1] = 2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y;
1051       out[2] = 2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - z;
1052 
1053   }
1054 
1055   /** Apply the inverse of the rotation to a vector.
1056    * @param u vector to apply the inverse of the rotation to
1057    * @return a new vector which such that u is its image by the rotation
1058    */
1059   public Vector3D applyInverseTo(Vector3D u) {
1060 
1061     double x = u.getX();
1062     double y = u.getY();
1063     double z = u.getZ();
1064 
1065     double s = q1 * x + q2 * y + q3 * z;
1066     double m0 = -q0;
1067 
1068     return new Vector3D(2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x,
1069                         2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y,
1070                         2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - z);
1071 
1072   }
1073 
1074   /** Apply the inverse of the rotation to a vector stored in an array.
1075    * @param in an array with three items which stores vector to rotate
1076    * @param out an array with three items to put result to (it can be the same
1077    * array as in)
1078    */
1079   public void applyInverseTo(final double[] in, final double[] out) {
1080 
1081       final double x = in[0];
1082       final double y = in[1];
1083       final double z = in[2];
1084 
1085       final double s = q1 * x + q2 * y + q3 * z;
1086       final double m0 = -q0;
1087 
1088       out[0] = 2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x;
1089       out[1] = 2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y;
1090       out[2] = 2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - z;
1091 
1092   }
1093 
1094   /** Apply the instance to another rotation.
1095    * <p>
1096    * Calling this method is equivalent to call
1097    * {@link #compose(Rotation, RotationConvention)
1098    * compose(r, RotationConvention.VECTOR_OPERATOR)}.
1099    * </p>
1100    * @param r rotation to apply the rotation to
1101    * @return a new rotation which is the composition of r by the instance
1102    */
1103   public Rotation applyTo(Rotation r) {
1104     return compose(r, RotationConvention.VECTOR_OPERATOR);
1105   }
1106 
1107   /** Compose the instance with another rotation.
1108    * <p>
1109    * If the semantics of the rotations composition corresponds to a
1110    * {@link RotationConvention#VECTOR_OPERATOR vector operator} convention,
1111    * applying the instance to a rotation is computing the composition
1112    * in an order compliant with the following rule : let {@code u} be any
1113    * vector and {@code v} its image by {@code r1} (i.e.
1114    * {@code r1.applyTo(u) = v}). Let {@code w} be the image of {@code v} by
1115    * rotation {@code r2} (i.e. {@code r2.applyTo(v) = w}). Then
1116    * {@code w = comp.applyTo(u)}, where
1117    * {@code comp = r2.compose(r1, RotationConvention.VECTOR_OPERATOR)}.
1118    * </p>
1119    * <p>
1120    * If the semantics of the rotations composition corresponds to a
1121    * {@link RotationConvention#FRAME_TRANSFORM frame transform} convention,
1122    * the application order will be reversed. So keeping the exact same
1123    * meaning of all {@code r1}, {@code r2}, {@code u}, {@code v}, {@code w}
1124    * and  {@code comp} as above, {@code comp} could also be computed as
1125    * {@code comp = r1.compose(r2, RotationConvention.FRAME_TRANSFORM)}.
1126    * </p>
1127    * @param r rotation to apply the rotation to
1128    * @param convention convention to use for the semantics of the angle
1129    * @return a new rotation which is the composition of r by the instance
1130    */
1131   public Rotation compose(final Rotation r, final RotationConvention convention) {
1132     return convention == RotationConvention.VECTOR_OPERATOR ?
1133            composeInternal(r) : r.composeInternal(this);
1134   }
1135 
1136   /** Compose the instance with another rotation using vector operator convention.
1137    * @param r rotation to apply the rotation to
1138    * @return a new rotation which is the composition of r by the instance
1139    * using vector operator convention
1140    */
1141   private Rotation composeInternal(final Rotation r) {
1142     return new Rotation(r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),
1143                         r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),
1144                         r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),
1145                         r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),
1146                         false);
1147   }
1148 
1149   /** Apply the inverse of the instance to another rotation.
1150    * <p>
1151    * Calling this method is equivalent to call
1152    * {@link #composeInverse(Rotation, RotationConvention)
1153    * composeInverse(r, RotationConvention.VECTOR_OPERATOR)}.
1154    * </p>
1155    * @param r rotation to apply the rotation to
1156    * @return a new rotation which is the composition of r by the inverse
1157    * of the instance
1158    */
1159   public Rotation applyInverseTo(Rotation r) {
1160     return composeInverse(r, RotationConvention.VECTOR_OPERATOR);
1161   }
1162 
1163   /** Compose the inverse of the instance with another rotation.
1164    * <p>
1165    * If the semantics of the rotations composition corresponds to a
1166    * {@link RotationConvention#VECTOR_OPERATOR vector operator} convention,
1167    * applying the inverse of the instance to a rotation is computing
1168    * the composition in an order compliant with the following rule :
1169    * let {@code u} be any vector and {@code v} its image by {@code r1}
1170    * (i.e. {@code r1.applyTo(u) = v}). Let {@code w} be the inverse image
1171    * of {@code v} by {@code r2} (i.e. {@code r2.applyInverseTo(v) = w}).
1172    * Then {@code w = comp.applyTo(u)}, where
1173    * {@code comp = r2.composeInverse(r1)}.
1174    * </p>
1175    * <p>
1176    * If the semantics of the rotations composition corresponds to a
1177    * {@link RotationConvention#FRAME_TRANSFORM frame transform} convention,
1178    * the application order will be reversed, which means it is the
1179    * <em>innermost</em> rotation that will be reversed. So keeping the exact same
1180    * meaning of all {@code r1}, {@code r2}, {@code u}, {@code v}, {@code w}
1181    * and  {@code comp} as above, {@code comp} could also be computed as
1182    * {@code comp = r1.revert().composeInverse(r2.revert(), RotationConvention.FRAME_TRANSFORM)}.
1183    * </p>
1184    * @param r rotation to apply the rotation to
1185    * @param convention convention to use for the semantics of the angle
1186    * @return a new rotation which is the composition of r by the inverse
1187    * of the instance
1188    */
1189   public Rotation composeInverse(final Rotation r, final RotationConvention convention) {
1190     return convention == RotationConvention.VECTOR_OPERATOR ?
1191            composeInverseInternal(r) : r.composeInternal(revert());
1192   }
1193 
1194   /** Compose the inverse of the instance with another rotation
1195    * using vector operator convention.
1196    * @param r rotation to apply the rotation to
1197    * @return a new rotation which is the composition of r by the inverse
1198    * of the instance using vector operator convention
1199    */
1200   private Rotation composeInverseInternal(Rotation r) {
1201     return new Rotation(-r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),
1202                         -r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),
1203                         -r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),
1204                         -r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),
1205                         false);
1206   }
1207 
1208   /** Perfect orthogonality on a 3X3 matrix.
1209    * @param m initial matrix (not exactly orthogonal)
1210    * @param threshold convergence threshold for the iterative
1211    * orthogonality correction (convergence is reached when the
1212    * difference between two steps of the Frobenius norm of the
1213    * correction is below this threshold)
1214    * @return an orthogonal matrix close to m
1215    * @exception MathIllegalArgumentException if the matrix cannot be
1216    * orthogonalized with the given threshold after 10 iterations
1217    */
1218   private double[][] orthogonalizeMatrix(double[][] m, double threshold)
1219     throws MathIllegalArgumentException {
1220     double[] m0 = m[0];
1221     double[] m1 = m[1];
1222     double[] m2 = m[2];
1223     double x00 = m0[0];
1224     double x01 = m0[1];
1225     double x02 = m0[2];
1226     double x10 = m1[0];
1227     double x11 = m1[1];
1228     double x12 = m1[2];
1229     double x20 = m2[0];
1230     double x21 = m2[1];
1231     double x22 = m2[2];
1232     double fn = 0;
1233     double fn1;
1234 
1235     double[][] o = new double[3][3];
1236     double[] o0 = o[0];
1237     double[] o1 = o[1];
1238     double[] o2 = o[2];
1239 
1240     // iterative correction: Xn+1 = Xn - 0.5 * (Xn.Mt.Xn - M)
1241     int i;
1242     for (i = 0; i < 11; ++i) {
1243 
1244       // Mt.Xn
1245       double mx00 = m0[0] * x00 + m1[0] * x10 + m2[0] * x20;
1246       double mx10 = m0[1] * x00 + m1[1] * x10 + m2[1] * x20;
1247       double mx20 = m0[2] * x00 + m1[2] * x10 + m2[2] * x20;
1248       double mx01 = m0[0] * x01 + m1[0] * x11 + m2[0] * x21;
1249       double mx11 = m0[1] * x01 + m1[1] * x11 + m2[1] * x21;
1250       double mx21 = m0[2] * x01 + m1[2] * x11 + m2[2] * x21;
1251       double mx02 = m0[0] * x02 + m1[0] * x12 + m2[0] * x22;
1252       double mx12 = m0[1] * x02 + m1[1] * x12 + m2[1] * x22;
1253       double mx22 = m0[2] * x02 + m1[2] * x12 + m2[2] * x22;
1254 
1255       // Xn+1
1256       o0[0] = x00 - 0.5 * (x00 * mx00 + x01 * mx10 + x02 * mx20 - m0[0]);
1257       o0[1] = x01 - 0.5 * (x00 * mx01 + x01 * mx11 + x02 * mx21 - m0[1]);
1258       o0[2] = x02 - 0.5 * (x00 * mx02 + x01 * mx12 + x02 * mx22 - m0[2]);
1259       o1[0] = x10 - 0.5 * (x10 * mx00 + x11 * mx10 + x12 * mx20 - m1[0]);
1260       o1[1] = x11 - 0.5 * (x10 * mx01 + x11 * mx11 + x12 * mx21 - m1[1]);
1261       o1[2] = x12 - 0.5 * (x10 * mx02 + x11 * mx12 + x12 * mx22 - m1[2]);
1262       o2[0] = x20 - 0.5 * (x20 * mx00 + x21 * mx10 + x22 * mx20 - m2[0]);
1263       o2[1] = x21 - 0.5 * (x20 * mx01 + x21 * mx11 + x22 * mx21 - m2[1]);
1264       o2[2] = x22 - 0.5 * (x20 * mx02 + x21 * mx12 + x22 * mx22 - m2[2]);
1265 
1266       // correction on each elements
1267       double corr00 = o0[0] - m0[0];
1268       double corr01 = o0[1] - m0[1];
1269       double corr02 = o0[2] - m0[2];
1270       double corr10 = o1[0] - m1[0];
1271       double corr11 = o1[1] - m1[1];
1272       double corr12 = o1[2] - m1[2];
1273       double corr20 = o2[0] - m2[0];
1274       double corr21 = o2[1] - m2[1];
1275       double corr22 = o2[2] - m2[2];
1276 
1277       // Frobenius norm of the correction
1278       fn1 = corr00 * corr00 + corr01 * corr01 + corr02 * corr02 +
1279             corr10 * corr10 + corr11 * corr11 + corr12 * corr12 +
1280             corr20 * corr20 + corr21 * corr21 + corr22 * corr22;
1281 
1282       // convergence test
1283       if (FastMath.abs(fn1 - fn) <= threshold) {
1284           return o;
1285       }
1286 
1287       // prepare next iteration
1288       x00 = o0[0];
1289       x01 = o0[1];
1290       x02 = o0[2];
1291       x10 = o1[0];
1292       x11 = o1[1];
1293       x12 = o1[2];
1294       x20 = o2[0];
1295       x21 = o2[1];
1296       x22 = o2[2];
1297       fn  = fn1;
1298 
1299     }
1300 
1301     // the algorithm did not converge after 10 iterations
1302     throw new MathIllegalArgumentException(LocalizedGeometryFormats.UNABLE_TO_ORTHOGONOLIZE_MATRIX,
1303                                            i - 1);
1304   }
1305 
1306   /** Compute the <i>distance</i> between two rotations.
1307    * <p>The <i>distance</i> is intended here as a way to check if two
1308    * rotations are almost similar (i.e. they transform vectors the same way)
1309    * or very different. It is mathematically defined as the angle of
1310    * the rotation r that prepended to one of the rotations gives the other
1311    * one: \(r_1(r) = r_2\)
1312    * </p>
1313    * <p>This distance is an angle between 0 and &pi;. Its value is the smallest
1314    * possible upper bound of the angle in radians between r<sub>1</sub>(v)
1315    * and r<sub>2</sub>(v) for all possible vectors v. This upper bound is
1316    * reached for some v. The distance is equal to 0 if and only if the two
1317    * rotations are identical.</p>
1318    * <p>Comparing two rotations should always be done using this value rather
1319    * than for example comparing the components of the quaternions. It is much
1320    * more stable, and has a geometric meaning. Also comparing quaternions
1321    * components is error prone since for example quaternions (0.36, 0.48, -0.48, -0.64)
1322    * and (-0.36, -0.48, 0.48, 0.64) represent exactly the same rotation despite
1323    * their components are different (they are exact opposites).</p>
1324    * @param r1 first rotation
1325    * @param r2 second rotation
1326    * @return <i>distance</i> between r1 and r2
1327    */
1328   public static double distance(Rotation r1, Rotation r2) {
1329       return r1.composeInverseInternal(r2).getAngle();
1330   }
1331 
1332 }