001/*
002 * Licensed to the Apache Software Foundation (ASF) under one or more
003 * contributor license agreements.  See the NOTICE file distributed with
004 * this work for additional information regarding copyright ownership.
005 * The ASF licenses this file to You under the Apache License, Version 2.0
006 * (the "License"); you may not use this file except in compliance with
007 * the License.  You may obtain a copy of the License at
008 *
009 *      http://www.apache.org/licenses/LICENSE-2.0
010 *
011 * Unless required by applicable law or agreed to in writing, software
012 * distributed under the License is distributed on an "AS IS" BASIS,
013 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
014 * See the License for the specific language governing permissions and
015 * limitations under the License.
016 */
017
018package org.apache.commons.math3.geometry.euclidean.threed;
019
020import java.io.Serializable;
021
022import org.apache.commons.math3.exception.MathArithmeticException;
023import org.apache.commons.math3.exception.MathIllegalArgumentException;
024import org.apache.commons.math3.exception.util.LocalizedFormats;
025import org.apache.commons.math3.util.FastMath;
026import org.apache.commons.math3.util.MathArrays;
027
028/**
029 * This class implements rotations in a three-dimensional space.
030 *
031 * <p>Rotations can be represented by several different mathematical
032 * entities (matrices, axe and angle, Cardan or Euler angles,
033 * quaternions). This class presents an higher level abstraction, more
034 * user-oriented and hiding this implementation details. Well, for the
035 * curious, we use quaternions for the internal representation. The
036 * user can build a rotation from any of these representations, and
037 * any of these representations can be retrieved from a
038 * <code>Rotation</code> instance (see the various constructors and
039 * getters). In addition, a rotation can also be built implicitly
040 * from a set of vectors and their image.</p>
041 * <p>This implies that this class can be used to convert from one
042 * representation to another one. For example, converting a rotation
043 * matrix into a set of Cardan angles from can be done using the
044 * following single line of code:</p>
045 * <pre>
046 * double[] angles = new Rotation(matrix, 1.0e-10).getAngles(RotationOrder.XYZ);
047 * </pre>
048 * <p>Focus is oriented on what a rotation <em>do</em> rather than on its
049 * underlying representation. Once it has been built, and regardless of its
050 * internal representation, a rotation is an <em>operator</em> which basically
051 * transforms three dimensional {@link Vector3D vectors} into other three
052 * dimensional {@link Vector3D vectors}. Depending on the application, the
053 * meaning of these vectors may vary and the semantics of the rotation also.</p>
054 * <p>For example in an spacecraft attitude simulation tool, users will often
055 * consider the vectors are fixed (say the Earth direction for example) and the
056 * frames change. The rotation transforms the coordinates of the vector in inertial
057 * frame into the coordinates of the same vector in satellite frame. In this
058 * case, the rotation implicitly defines the relation between the two frames.</p>
059 * <p>Another example could be a telescope control application, where the rotation
060 * would transform the sighting direction at rest into the desired observing
061 * direction when the telescope is pointed towards an object of interest. In this
062 * case the rotation transforms the direction at rest in a topocentric frame
063 * into the sighting direction in the same topocentric frame. This implies in this
064 * case the frame is fixed and the vector moves.</p>
065 * <p>In many case, both approaches will be combined. In our telescope example,
066 * we will probably also need to transform the observing direction in the topocentric
067 * frame into the observing direction in inertial frame taking into account the observatory
068 * location and the Earth rotation, which would essentially be an application of the
069 * first approach.</p>
070 *
071 * <p>These examples show that a rotation is what the user wants it to be. This
072 * class does not push the user towards one specific definition and hence does not
073 * provide methods like <code>projectVectorIntoDestinationFrame</code> or
074 * <code>computeTransformedDirection</code>. It provides simpler and more generic
075 * methods: {@link #applyTo(Vector3D) applyTo(Vector3D)} and {@link
076 * #applyInverseTo(Vector3D) applyInverseTo(Vector3D)}.</p>
077 *
078 * <p>Since a rotation is basically a vectorial operator, several rotations can be
079 * composed together and the composite operation <code>r = r<sub>1</sub> o
080 * r<sub>2</sub></code> (which means that for each vector <code>u</code>,
081 * <code>r(u) = r<sub>1</sub>(r<sub>2</sub>(u))</code>) is also a rotation. Hence
082 * we can consider that in addition to vectors, a rotation can be applied to other
083 * rotations as well (or to itself). With our previous notations, we would say we
084 * can apply <code>r<sub>1</sub></code> to <code>r<sub>2</sub></code> and the result
085 * we get is <code>r = r<sub>1</sub> o r<sub>2</sub></code>. For this purpose, the
086 * class provides the methods: {@link #applyTo(Rotation) applyTo(Rotation)} and
087 * {@link #applyInverseTo(Rotation) applyInverseTo(Rotation)}.</p>
088 *
089 * <p>Rotations are guaranteed to be immutable objects.</p>
090 *
091 * @see Vector3D
092 * @see RotationOrder
093 * @since 1.2
094 */
095
096public class Rotation implements Serializable {
097
098  /** Identity rotation. */
099  public static final Rotation IDENTITY = new Rotation(1.0, 0.0, 0.0, 0.0, false);
100
101  /** Serializable version identifier */
102  private static final long serialVersionUID = -2153622329907944313L;
103
104  /** Scalar coordinate of the quaternion. */
105  private final double q0;
106
107  /** First coordinate of the vectorial part of the quaternion. */
108  private final double q1;
109
110  /** Second coordinate of the vectorial part of the quaternion. */
111  private final double q2;
112
113  /** Third coordinate of the vectorial part of the quaternion. */
114  private final double q3;
115
116  /** Build a rotation from the quaternion coordinates.
117   * <p>A rotation can be built from a <em>normalized</em> quaternion,
118   * i.e. a quaternion for which q<sub>0</sub><sup>2</sup> +
119   * q<sub>1</sub><sup>2</sup> + q<sub>2</sub><sup>2</sup> +
120   * q<sub>3</sub><sup>2</sup> = 1. If the quaternion is not normalized,
121   * the constructor can normalize it in a preprocessing step.</p>
122   * <p>Note that some conventions put the scalar part of the quaternion
123   * as the 4<sup>th</sup> component and the vector part as the first three
124   * components. This is <em>not</em> our convention. We put the scalar part
125   * as the first component.</p>
126   * @param q0 scalar part of the quaternion
127   * @param q1 first coordinate of the vectorial part of the quaternion
128   * @param q2 second coordinate of the vectorial part of the quaternion
129   * @param q3 third coordinate of the vectorial part of the quaternion
130   * @param needsNormalization if true, the coordinates are considered
131   * not to be normalized, a normalization preprocessing step is performed
132   * before using them
133   */
134  public Rotation(double q0, double q1, double q2, double q3,
135                  boolean needsNormalization) {
136
137    if (needsNormalization) {
138      // normalization preprocessing
139      double inv = 1.0 / FastMath.sqrt(q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3);
140      q0 *= inv;
141      q1 *= inv;
142      q2 *= inv;
143      q3 *= inv;
144    }
145
146    this.q0 = q0;
147    this.q1 = q1;
148    this.q2 = q2;
149    this.q3 = q3;
150
151  }
152
153  /** Build a rotation from an axis and an angle.
154   * <p>
155   * Calling this constructor is equivalent to call
156   * {@link #Rotation(Vector3D, double, RotationConvention)
157   * new Rotation(axis, angle, RotationConvention.VECTOR_OPERATOR)}
158   * </p>
159   * @param axis axis around which to rotate
160   * @param angle rotation angle.
161   * @exception MathIllegalArgumentException if the axis norm is zero
162   * @deprecated as of 3.6, replaced with {@link #Rotation(Vector3D, double, RotationConvention)}
163   */
164  @Deprecated
165  public Rotation(Vector3D axis, double angle) throws MathIllegalArgumentException {
166      this(axis, angle, RotationConvention.VECTOR_OPERATOR);
167  }
168
169  /** Build a rotation from an axis and an angle.
170   * @param axis axis around which to rotate
171   * @param angle rotation angle
172   * @param convention convention to use for the semantics of the angle
173   * @exception MathIllegalArgumentException if the axis norm is zero
174   * @since 3.6
175   */
176  public Rotation(final Vector3D axis, final double angle, final RotationConvention convention)
177      throws MathIllegalArgumentException {
178
179    double norm = axis.getNorm();
180    if (norm == 0) {
181      throw new MathIllegalArgumentException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_AXIS);
182    }
183
184    double halfAngle = convention == RotationConvention.VECTOR_OPERATOR ? -0.5 * angle : +0.5 * angle;
185    double coeff = FastMath.sin(halfAngle) / norm;
186
187    q0 = FastMath.cos (halfAngle);
188    q1 = coeff * axis.getX();
189    q2 = coeff * axis.getY();
190    q3 = coeff * axis.getZ();
191
192  }
193
194  /** Build a rotation from a 3X3 matrix.
195
196   * <p>Rotation matrices are orthogonal matrices, i.e. unit matrices
197   * (which are matrices for which m.m<sup>T</sup> = I) with real
198   * coefficients. The module of the determinant of unit matrices is
199   * 1, among the orthogonal 3X3 matrices, only the ones having a
200   * positive determinant (+1) are rotation matrices.</p>
201
202   * <p>When a rotation is defined by a matrix with truncated values
203   * (typically when it is extracted from a technical sheet where only
204   * four to five significant digits are available), the matrix is not
205   * orthogonal anymore. This constructor handles this case
206   * transparently by using a copy of the given matrix and applying a
207   * correction to the copy in order to perfect its orthogonality. If
208   * the Frobenius norm of the correction needed is above the given
209   * threshold, then the matrix is considered to be too far from a
210   * true rotation matrix and an exception is thrown.<p>
211
212   * @param m rotation matrix
213   * @param threshold convergence threshold for the iterative
214   * orthogonality correction (convergence is reached when the
215   * difference between two steps of the Frobenius norm of the
216   * correction is below this threshold)
217
218   * @exception NotARotationMatrixException if the matrix is not a 3X3
219   * matrix, or if it cannot be transformed into an orthogonal matrix
220   * with the given threshold, or if the determinant of the resulting
221   * orthogonal matrix is negative
222
223   */
224  public Rotation(double[][] m, double threshold)
225    throws NotARotationMatrixException {
226
227    // dimension check
228    if ((m.length != 3) || (m[0].length != 3) ||
229        (m[1].length != 3) || (m[2].length != 3)) {
230      throw new NotARotationMatrixException(
231              LocalizedFormats.ROTATION_MATRIX_DIMENSIONS,
232              m.length, m[0].length);
233    }
234
235    // compute a "close" orthogonal matrix
236    double[][] ort = orthogonalizeMatrix(m, threshold);
237
238    // check the sign of the determinant
239    double det = ort[0][0] * (ort[1][1] * ort[2][2] - ort[2][1] * ort[1][2]) -
240                 ort[1][0] * (ort[0][1] * ort[2][2] - ort[2][1] * ort[0][2]) +
241                 ort[2][0] * (ort[0][1] * ort[1][2] - ort[1][1] * ort[0][2]);
242    if (det < 0.0) {
243      throw new NotARotationMatrixException(
244              LocalizedFormats.CLOSEST_ORTHOGONAL_MATRIX_HAS_NEGATIVE_DETERMINANT,
245              det);
246    }
247
248    double[] quat = mat2quat(ort);
249    q0 = quat[0];
250    q1 = quat[1];
251    q2 = quat[2];
252    q3 = quat[3];
253
254  }
255
256  /** Build the rotation that transforms a pair of vectors into another pair.
257
258   * <p>Except for possible scale factors, if the instance were applied to
259   * the pair (u<sub>1</sub>, u<sub>2</sub>) it will produce the pair
260   * (v<sub>1</sub>, v<sub>2</sub>).</p>
261
262   * <p>If the angular separation between u<sub>1</sub> and u<sub>2</sub> is
263   * not the same as the angular separation between v<sub>1</sub> and
264   * v<sub>2</sub>, then a corrected v'<sub>2</sub> will be used rather than
265   * v<sub>2</sub>, the corrected vector will be in the (&pm;v<sub>1</sub>,
266   * +v<sub>2</sub>) half-plane.</p>
267
268   * @param u1 first vector of the origin pair
269   * @param u2 second vector of the origin pair
270   * @param v1 desired image of u1 by the rotation
271   * @param v2 desired image of u2 by the rotation
272   * @exception MathArithmeticException if the norm of one of the vectors is zero,
273   * or if one of the pair is degenerated (i.e. the vectors of the pair are collinear)
274   */
275  public Rotation(Vector3D u1, Vector3D u2, Vector3D v1, Vector3D v2)
276      throws MathArithmeticException {
277
278      // build orthonormalized base from u1, u2
279      // this fails when vectors are null or collinear, which is forbidden to define a rotation
280      final Vector3D u3 = u1.crossProduct(u2).normalize();
281      u2 = u3.crossProduct(u1).normalize();
282      u1 = u1.normalize();
283
284      // build an orthonormalized base from v1, v2
285      // this fails when vectors are null or collinear, which is forbidden to define a rotation
286      final Vector3D v3 = v1.crossProduct(v2).normalize();
287      v2 = v3.crossProduct(v1).normalize();
288      v1 = v1.normalize();
289
290      // buid a matrix transforming the first base into the second one
291      final double[][] m = new double[][] {
292          {
293              MathArrays.linearCombination(u1.getX(), v1.getX(), u2.getX(), v2.getX(), u3.getX(), v3.getX()),
294              MathArrays.linearCombination(u1.getY(), v1.getX(), u2.getY(), v2.getX(), u3.getY(), v3.getX()),
295              MathArrays.linearCombination(u1.getZ(), v1.getX(), u2.getZ(), v2.getX(), u3.getZ(), v3.getX())
296          },
297          {
298              MathArrays.linearCombination(u1.getX(), v1.getY(), u2.getX(), v2.getY(), u3.getX(), v3.getY()),
299              MathArrays.linearCombination(u1.getY(), v1.getY(), u2.getY(), v2.getY(), u3.getY(), v3.getY()),
300              MathArrays.linearCombination(u1.getZ(), v1.getY(), u2.getZ(), v2.getY(), u3.getZ(), v3.getY())
301          },
302          {
303              MathArrays.linearCombination(u1.getX(), v1.getZ(), u2.getX(), v2.getZ(), u3.getX(), v3.getZ()),
304              MathArrays.linearCombination(u1.getY(), v1.getZ(), u2.getY(), v2.getZ(), u3.getY(), v3.getZ()),
305              MathArrays.linearCombination(u1.getZ(), v1.getZ(), u2.getZ(), v2.getZ(), u3.getZ(), v3.getZ())
306          }
307      };
308
309      double[] quat = mat2quat(m);
310      q0 = quat[0];
311      q1 = quat[1];
312      q2 = quat[2];
313      q3 = quat[3];
314
315  }
316
317  /** Build one of the rotations that transform one vector into another one.
318
319   * <p>Except for a possible scale factor, if the instance were
320   * applied to the vector u it will produce the vector v. There is an
321   * infinite number of such rotations, this constructor choose the
322   * one with the smallest associated angle (i.e. the one whose axis
323   * is orthogonal to the (u, v) plane). If u and v are collinear, an
324   * arbitrary rotation axis is chosen.</p>
325
326   * @param u origin vector
327   * @param v desired image of u by the rotation
328   * @exception MathArithmeticException if the norm of one of the vectors is zero
329   */
330  public Rotation(Vector3D u, Vector3D v) throws MathArithmeticException {
331
332    double normProduct = u.getNorm() * v.getNorm();
333    if (normProduct == 0) {
334        throw new MathArithmeticException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_DEFINING_VECTOR);
335    }
336
337    double dot = u.dotProduct(v);
338
339    if (dot < ((2.0e-15 - 1.0) * normProduct)) {
340      // special case u = -v: we select a PI angle rotation around
341      // an arbitrary vector orthogonal to u
342      Vector3D w = u.orthogonal();
343      q0 = 0.0;
344      q1 = -w.getX();
345      q2 = -w.getY();
346      q3 = -w.getZ();
347    } else {
348      // general case: (u, v) defines a plane, we select
349      // the shortest possible rotation: axis orthogonal to this plane
350      q0 = FastMath.sqrt(0.5 * (1.0 + dot / normProduct));
351      double coeff = 1.0 / (2.0 * q0 * normProduct);
352      Vector3D q = v.crossProduct(u);
353      q1 = coeff * q.getX();
354      q2 = coeff * q.getY();
355      q3 = coeff * q.getZ();
356    }
357
358  }
359
360  /** Build a rotation from three Cardan or Euler elementary rotations.
361
362   * <p>
363   * Calling this constructor is equivalent to call
364   * {@link #Rotation(RotationOrder, RotationConvention, double, double, double)
365   * new Rotation(order, RotationConvention.VECTOR_OPERATOR, alpha1, alpha2, alpha3)}
366   * </p>
367
368   * @param order order of rotations to use
369   * @param alpha1 angle of the first elementary rotation
370   * @param alpha2 angle of the second elementary rotation
371   * @param alpha3 angle of the third elementary rotation
372   * @deprecated as of 3.6, replaced with {@link
373   * #Rotation(RotationOrder, RotationConvention, double, double, double)}
374   */
375  @Deprecated
376  public Rotation(RotationOrder order,
377                  double alpha1, double alpha2, double alpha3) {
378      this(order, RotationConvention.VECTOR_OPERATOR, alpha1, alpha2, alpha3);
379  }
380
381  /** Build a rotation from three Cardan or Euler elementary rotations.
382
383   * <p>Cardan rotations are three successive rotations around the
384   * canonical axes X, Y and Z, each axis being used once. There are
385   * 6 such sets of rotations (XYZ, XZY, YXZ, YZX, ZXY and ZYX). Euler
386   * rotations are three successive rotations around the canonical
387   * axes X, Y and Z, the first and last rotations being around the
388   * same axis. There are 6 such sets of rotations (XYX, XZX, YXY,
389   * YZY, ZXZ and ZYZ), the most popular one being ZXZ.</p>
390   * <p>Beware that many people routinely use the term Euler angles even
391   * for what really are Cardan angles (this confusion is especially
392   * widespread in the aerospace business where Roll, Pitch and Yaw angles
393   * are often wrongly tagged as Euler angles).</p>
394
395   * @param order order of rotations to compose, from left to right
396   * (i.e. we will use {@code r1.compose(r2.compose(r3, convention), convention)})
397   * @param convention convention to use for the semantics of the angle
398   * @param alpha1 angle of the first elementary rotation
399   * @param alpha2 angle of the second elementary rotation
400   * @param alpha3 angle of the third elementary rotation
401   * @since 3.6
402   */
403  public Rotation(RotationOrder order, RotationConvention convention,
404                  double alpha1, double alpha2, double alpha3) {
405      Rotation r1 = new Rotation(order.getA1(), alpha1, convention);
406      Rotation r2 = new Rotation(order.getA2(), alpha2, convention);
407      Rotation r3 = new Rotation(order.getA3(), alpha3, convention);
408      Rotation composed = r1.compose(r2.compose(r3, convention), convention);
409      q0 = composed.q0;
410      q1 = composed.q1;
411      q2 = composed.q2;
412      q3 = composed.q3;
413  }
414
415  /** Convert an orthogonal rotation matrix to a quaternion.
416   * @param ort orthogonal rotation matrix
417   * @return quaternion corresponding to the matrix
418   */
419  private static double[] mat2quat(final double[][] ort) {
420
421      final double[] quat = new double[4];
422
423      // There are different ways to compute the quaternions elements
424      // from the matrix. They all involve computing one element from
425      // the diagonal of the matrix, and computing the three other ones
426      // using a formula involving a division by the first element,
427      // which unfortunately can be zero. Since the norm of the
428      // quaternion is 1, we know at least one element has an absolute
429      // value greater or equal to 0.5, so it is always possible to
430      // select the right formula and avoid division by zero and even
431      // numerical inaccuracy. Checking the elements in turn and using
432      // the first one greater than 0.45 is safe (this leads to a simple
433      // test since qi = 0.45 implies 4 qi^2 - 1 = -0.19)
434      double s = ort[0][0] + ort[1][1] + ort[2][2];
435      if (s > -0.19) {
436          // compute q0 and deduce q1, q2 and q3
437          quat[0] = 0.5 * FastMath.sqrt(s + 1.0);
438          double inv = 0.25 / quat[0];
439          quat[1] = inv * (ort[1][2] - ort[2][1]);
440          quat[2] = inv * (ort[2][0] - ort[0][2]);
441          quat[3] = inv * (ort[0][1] - ort[1][0]);
442      } else {
443          s = ort[0][0] - ort[1][1] - ort[2][2];
444          if (s > -0.19) {
445              // compute q1 and deduce q0, q2 and q3
446              quat[1] = 0.5 * FastMath.sqrt(s + 1.0);
447              double inv = 0.25 / quat[1];
448              quat[0] = inv * (ort[1][2] - ort[2][1]);
449              quat[2] = inv * (ort[0][1] + ort[1][0]);
450              quat[3] = inv * (ort[0][2] + ort[2][0]);
451          } else {
452              s = ort[1][1] - ort[0][0] - ort[2][2];
453              if (s > -0.19) {
454                  // compute q2 and deduce q0, q1 and q3
455                  quat[2] = 0.5 * FastMath.sqrt(s + 1.0);
456                  double inv = 0.25 / quat[2];
457                  quat[0] = inv * (ort[2][0] - ort[0][2]);
458                  quat[1] = inv * (ort[0][1] + ort[1][0]);
459                  quat[3] = inv * (ort[2][1] + ort[1][2]);
460              } else {
461                  // compute q3 and deduce q0, q1 and q2
462                  s = ort[2][2] - ort[0][0] - ort[1][1];
463                  quat[3] = 0.5 * FastMath.sqrt(s + 1.0);
464                  double inv = 0.25 / quat[3];
465                  quat[0] = inv * (ort[0][1] - ort[1][0]);
466                  quat[1] = inv * (ort[0][2] + ort[2][0]);
467                  quat[2] = inv * (ort[2][1] + ort[1][2]);
468              }
469          }
470      }
471
472      return quat;
473
474  }
475
476  /** Revert a rotation.
477   * Build a rotation which reverse the effect of another
478   * rotation. This means that if r(u) = v, then r.revert(v) = u. The
479   * instance is not changed.
480   * @return a new rotation whose effect is the reverse of the effect
481   * of the instance
482   */
483  public Rotation revert() {
484    return new Rotation(-q0, q1, q2, q3, false);
485  }
486
487  /** Get the scalar coordinate of the quaternion.
488   * @return scalar coordinate of the quaternion
489   */
490  public double getQ0() {
491    return q0;
492  }
493
494  /** Get the first coordinate of the vectorial part of the quaternion.
495   * @return first coordinate of the vectorial part of the quaternion
496   */
497  public double getQ1() {
498    return q1;
499  }
500
501  /** Get the second coordinate of the vectorial part of the quaternion.
502   * @return second coordinate of the vectorial part of the quaternion
503   */
504  public double getQ2() {
505    return q2;
506  }
507
508  /** Get the third coordinate of the vectorial part of the quaternion.
509   * @return third coordinate of the vectorial part of the quaternion
510   */
511  public double getQ3() {
512    return q3;
513  }
514
515  /** Get the normalized axis of the rotation.
516   * <p>
517   * Calling this method is equivalent to call
518   * {@link #getAxis(RotationConvention) getAxis(RotationConvention.VECTOR_OPERATOR)}
519   * </p>
520   * @return normalized axis of the rotation
521   * @see #Rotation(Vector3D, double, RotationConvention)
522   * @deprecated as of 3.6, replaced with {@link #getAxis(RotationConvention)}
523   */
524  @Deprecated
525  public Vector3D getAxis() {
526    return getAxis(RotationConvention.VECTOR_OPERATOR);
527  }
528
529  /** Get the normalized axis of the rotation.
530   * <p>
531   * Note that as {@link #getAngle()} always returns an angle
532   * between 0 and &pi;, changing the convention changes the
533   * direction of the axis, not the sign of the angle.
534   * </p>
535   * @param convention convention to use for the semantics of the angle
536   * @return normalized axis of the rotation
537   * @see #Rotation(Vector3D, double, RotationConvention)
538   * @since 3.6
539   */
540  public Vector3D getAxis(final RotationConvention convention) {
541    final double squaredSine = q1 * q1 + q2 * q2 + q3 * q3;
542    if (squaredSine == 0) {
543      return convention == RotationConvention.VECTOR_OPERATOR ? Vector3D.PLUS_I : Vector3D.MINUS_I;
544    } else {
545        final double sgn = convention == RotationConvention.VECTOR_OPERATOR ? +1 : -1;
546        if (q0 < 0) {
547            final double inverse = sgn / FastMath.sqrt(squaredSine);
548            return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
549        }
550        final double inverse = -sgn / FastMath.sqrt(squaredSine);
551        return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
552    }
553  }
554
555  /** Get the angle of the rotation.
556   * @return angle of the rotation (between 0 and &pi;)
557   * @see #Rotation(Vector3D, double)
558   */
559  public double getAngle() {
560    if ((q0 < -0.1) || (q0 > 0.1)) {
561      return 2 * FastMath.asin(FastMath.sqrt(q1 * q1 + q2 * q2 + q3 * q3));
562    } else if (q0 < 0) {
563      return 2 * FastMath.acos(-q0);
564    }
565    return 2 * FastMath.acos(q0);
566  }
567
568  /** Get the Cardan or Euler angles corresponding to the instance.
569
570   * <p>
571   * Calling this method is equivalent to call
572   * {@link #getAngles(RotationOrder, RotationConvention)
573   * getAngles(order, RotationConvention.VECTOR_OPERATOR)}
574   * </p>
575
576   * @param order rotation order to use
577   * @return an array of three angles, in the order specified by the set
578   * @exception CardanEulerSingularityException if the rotation is
579   * singular with respect to the angles set specified
580   * @deprecated as of 3.6, replaced with {@link #getAngles(RotationOrder, RotationConvention)}
581   */
582  @Deprecated
583  public double[] getAngles(RotationOrder order)
584      throws CardanEulerSingularityException {
585      return getAngles(order, RotationConvention.VECTOR_OPERATOR);
586  }
587
588  /** Get the Cardan or Euler angles corresponding to the instance.
589
590   * <p>The equations show that each rotation can be defined by two
591   * different values of the Cardan or Euler angles set. For example
592   * if Cardan angles are used, the rotation defined by the angles
593   * a<sub>1</sub>, a<sub>2</sub> and a<sub>3</sub> is the same as
594   * the rotation defined by the angles &pi; + a<sub>1</sub>, &pi;
595   * - a<sub>2</sub> and &pi; + a<sub>3</sub>. This method implements
596   * the following arbitrary choices:</p>
597   * <ul>
598   *   <li>for Cardan angles, the chosen set is the one for which the
599   *   second angle is between -&pi;/2 and &pi;/2 (i.e its cosine is
600   *   positive),</li>
601   *   <li>for Euler angles, the chosen set is the one for which the
602   *   second angle is between 0 and &pi; (i.e its sine is positive).</li>
603   * </ul>
604
605   * <p>Cardan and Euler angle have a very disappointing drawback: all
606   * of them have singularities. This means that if the instance is
607   * too close to the singularities corresponding to the given
608   * rotation order, it will be impossible to retrieve the angles. For
609   * Cardan angles, this is often called gimbal lock. There is
610   * <em>nothing</em> to do to prevent this, it is an intrinsic problem
611   * with Cardan and Euler representation (but not a problem with the
612   * rotation itself, which is perfectly well defined). For Cardan
613   * angles, singularities occur when the second angle is close to
614   * -&pi;/2 or +&pi;/2, for Euler angle singularities occur when the
615   * second angle is close to 0 or &pi;, this implies that the identity
616   * rotation is always singular for Euler angles!</p>
617
618   * @param order rotation order to use
619   * @param convention convention to use for the semantics of the angle
620   * @return an array of three angles, in the order specified by the set
621   * @exception CardanEulerSingularityException if the rotation is
622   * singular with respect to the angles set specified
623   * @since 3.6
624   */
625  public double[] getAngles(RotationOrder order, RotationConvention convention)
626      throws CardanEulerSingularityException {
627
628      if (convention == RotationConvention.VECTOR_OPERATOR) {
629          if (order == RotationOrder.XYZ) {
630
631              // r (Vector3D.plusK) coordinates are :
632              //  sin (theta), -cos (theta) sin (phi), cos (theta) cos (phi)
633              // (-r) (Vector3D.plusI) coordinates are :
634              // cos (psi) cos (theta), -sin (psi) cos (theta), sin (theta)
635              // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
636              Vector3D v1 = applyTo(Vector3D.PLUS_K);
637              Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
638              if  ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
639                  throw new CardanEulerSingularityException(true);
640              }
641              return new double[] {
642                  FastMath.atan2(-(v1.getY()), v1.getZ()),
643                  FastMath.asin(v2.getZ()),
644                  FastMath.atan2(-(v2.getY()), v2.getX())
645              };
646
647          } else if (order == RotationOrder.XZY) {
648
649              // r (Vector3D.plusJ) coordinates are :
650              // -sin (psi), cos (psi) cos (phi), cos (psi) sin (phi)
651              // (-r) (Vector3D.plusI) coordinates are :
652              // cos (theta) cos (psi), -sin (psi), sin (theta) cos (psi)
653              // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
654              Vector3D v1 = applyTo(Vector3D.PLUS_J);
655              Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
656              if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
657                  throw new CardanEulerSingularityException(true);
658              }
659              return new double[] {
660                  FastMath.atan2(v1.getZ(), v1.getY()),
661                 -FastMath.asin(v2.getY()),
662                  FastMath.atan2(v2.getZ(), v2.getX())
663              };
664
665          } else if (order == RotationOrder.YXZ) {
666
667              // r (Vector3D.plusK) coordinates are :
668              //  cos (phi) sin (theta), -sin (phi), cos (phi) cos (theta)
669              // (-r) (Vector3D.plusJ) coordinates are :
670              // sin (psi) cos (phi), cos (psi) cos (phi), -sin (phi)
671              // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
672              Vector3D v1 = applyTo(Vector3D.PLUS_K);
673              Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
674              if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
675                  throw new CardanEulerSingularityException(true);
676              }
677              return new double[] {
678                  FastMath.atan2(v1.getX(), v1.getZ()),
679                 -FastMath.asin(v2.getZ()),
680                  FastMath.atan2(v2.getX(), v2.getY())
681              };
682
683          } else if (order == RotationOrder.YZX) {
684
685              // r (Vector3D.plusI) coordinates are :
686              // cos (psi) cos (theta), sin (psi), -cos (psi) sin (theta)
687              // (-r) (Vector3D.plusJ) coordinates are :
688              // sin (psi), cos (phi) cos (psi), -sin (phi) cos (psi)
689              // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
690              Vector3D v1 = applyTo(Vector3D.PLUS_I);
691              Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
692              if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
693                  throw new CardanEulerSingularityException(true);
694              }
695              return new double[] {
696                  FastMath.atan2(-(v1.getZ()), v1.getX()),
697                  FastMath.asin(v2.getX()),
698                  FastMath.atan2(-(v2.getZ()), v2.getY())
699              };
700
701          } else if (order == RotationOrder.ZXY) {
702
703              // r (Vector3D.plusJ) coordinates are :
704              // -cos (phi) sin (psi), cos (phi) cos (psi), sin (phi)
705              // (-r) (Vector3D.plusK) coordinates are :
706              // -sin (theta) cos (phi), sin (phi), cos (theta) cos (phi)
707              // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
708              Vector3D v1 = applyTo(Vector3D.PLUS_J);
709              Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
710              if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
711                  throw new CardanEulerSingularityException(true);
712              }
713              return new double[] {
714                  FastMath.atan2(-(v1.getX()), v1.getY()),
715                  FastMath.asin(v2.getY()),
716                  FastMath.atan2(-(v2.getX()), v2.getZ())
717              };
718
719          } else if (order == RotationOrder.ZYX) {
720
721              // r (Vector3D.plusI) coordinates are :
722              //  cos (theta) cos (psi), cos (theta) sin (psi), -sin (theta)
723              // (-r) (Vector3D.plusK) coordinates are :
724              // -sin (theta), sin (phi) cos (theta), cos (phi) cos (theta)
725              // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
726              Vector3D v1 = applyTo(Vector3D.PLUS_I);
727              Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
728              if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
729                  throw new CardanEulerSingularityException(true);
730              }
731              return new double[] {
732                  FastMath.atan2(v1.getY(), v1.getX()),
733                 -FastMath.asin(v2.getX()),
734                  FastMath.atan2(v2.getY(), v2.getZ())
735              };
736
737          } else if (order == RotationOrder.XYX) {
738
739              // r (Vector3D.plusI) coordinates are :
740              //  cos (theta), sin (phi1) sin (theta), -cos (phi1) sin (theta)
741              // (-r) (Vector3D.plusI) coordinates are :
742              // cos (theta), sin (theta) sin (phi2), sin (theta) cos (phi2)
743              // and we can choose to have theta in the interval [0 ; PI]
744              Vector3D v1 = applyTo(Vector3D.PLUS_I);
745              Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
746              if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
747                  throw new CardanEulerSingularityException(false);
748              }
749              return new double[] {
750                  FastMath.atan2(v1.getY(), -v1.getZ()),
751                  FastMath.acos(v2.getX()),
752                  FastMath.atan2(v2.getY(), v2.getZ())
753              };
754
755          } else if (order == RotationOrder.XZX) {
756
757              // r (Vector3D.plusI) coordinates are :
758              //  cos (psi), cos (phi1) sin (psi), sin (phi1) sin (psi)
759              // (-r) (Vector3D.plusI) coordinates are :
760              // cos (psi), -sin (psi) cos (phi2), sin (psi) sin (phi2)
761              // and we can choose to have psi in the interval [0 ; PI]
762              Vector3D v1 = applyTo(Vector3D.PLUS_I);
763              Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
764              if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
765                  throw new CardanEulerSingularityException(false);
766              }
767              return new double[] {
768                  FastMath.atan2(v1.getZ(), v1.getY()),
769                  FastMath.acos(v2.getX()),
770                  FastMath.atan2(v2.getZ(), -v2.getY())
771              };
772
773          } else if (order == RotationOrder.YXY) {
774
775              // r (Vector3D.plusJ) coordinates are :
776              //  sin (theta1) sin (phi), cos (phi), cos (theta1) sin (phi)
777              // (-r) (Vector3D.plusJ) coordinates are :
778              // sin (phi) sin (theta2), cos (phi), -sin (phi) cos (theta2)
779              // and we can choose to have phi in the interval [0 ; PI]
780              Vector3D v1 = applyTo(Vector3D.PLUS_J);
781              Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
782              if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
783                  throw new CardanEulerSingularityException(false);
784              }
785              return new double[] {
786                  FastMath.atan2(v1.getX(), v1.getZ()),
787                  FastMath.acos(v2.getY()),
788                  FastMath.atan2(v2.getX(), -v2.getZ())
789              };
790
791          } else if (order == RotationOrder.YZY) {
792
793              // r (Vector3D.plusJ) coordinates are :
794              //  -cos (theta1) sin (psi), cos (psi), sin (theta1) sin (psi)
795              // (-r) (Vector3D.plusJ) coordinates are :
796              // sin (psi) cos (theta2), cos (psi), sin (psi) sin (theta2)
797              // and we can choose to have psi in the interval [0 ; PI]
798              Vector3D v1 = applyTo(Vector3D.PLUS_J);
799              Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
800              if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
801                  throw new CardanEulerSingularityException(false);
802              }
803              return new double[] {
804                  FastMath.atan2(v1.getZ(), -v1.getX()),
805                  FastMath.acos(v2.getY()),
806                  FastMath.atan2(v2.getZ(), v2.getX())
807              };
808
809          } else if (order == RotationOrder.ZXZ) {
810
811              // r (Vector3D.plusK) coordinates are :
812              //  sin (psi1) sin (phi), -cos (psi1) sin (phi), cos (phi)
813              // (-r) (Vector3D.plusK) coordinates are :
814              // sin (phi) sin (psi2), sin (phi) cos (psi2), cos (phi)
815              // and we can choose to have phi in the interval [0 ; PI]
816              Vector3D v1 = applyTo(Vector3D.PLUS_K);
817              Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
818              if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
819                  throw new CardanEulerSingularityException(false);
820              }
821              return new double[] {
822                  FastMath.atan2(v1.getX(), -v1.getY()),
823                  FastMath.acos(v2.getZ()),
824                  FastMath.atan2(v2.getX(), v2.getY())
825              };
826
827          } else { // last possibility is ZYZ
828
829              // r (Vector3D.plusK) coordinates are :
830              //  cos (psi1) sin (theta), sin (psi1) sin (theta), cos (theta)
831              // (-r) (Vector3D.plusK) coordinates are :
832              // -sin (theta) cos (psi2), sin (theta) sin (psi2), cos (theta)
833              // and we can choose to have theta in the interval [0 ; PI]
834              Vector3D v1 = applyTo(Vector3D.PLUS_K);
835              Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
836              if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
837                  throw new CardanEulerSingularityException(false);
838              }
839              return new double[] {
840                  FastMath.atan2(v1.getY(), v1.getX()),
841                  FastMath.acos(v2.getZ()),
842                  FastMath.atan2(v2.getY(), -v2.getX())
843              };
844
845          }
846      } else {
847          if (order == RotationOrder.XYZ) {
848
849              // r (Vector3D.plusI) coordinates are :
850              //  cos (theta) cos (psi), -cos (theta) sin (psi), sin (theta)
851              // (-r) (Vector3D.plusK) coordinates are :
852              // sin (theta), -sin (phi) cos (theta), cos (phi) cos (theta)
853              // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
854              Vector3D v1 = applyTo(Vector3D.PLUS_I);
855              Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
856              if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
857                  throw new CardanEulerSingularityException(true);
858              }
859              return new double[] {
860                  FastMath.atan2(-v2.getY(), v2.getZ()),
861                  FastMath.asin(v2.getX()),
862                  FastMath.atan2(-v1.getY(), v1.getX())
863              };
864
865          } else if (order == RotationOrder.XZY) {
866
867              // r (Vector3D.plusI) coordinates are :
868              // cos (psi) cos (theta), -sin (psi), cos (psi) sin (theta)
869              // (-r) (Vector3D.plusJ) coordinates are :
870              // -sin (psi), cos (phi) cos (psi), sin (phi) cos (psi)
871              // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
872              Vector3D v1 = applyTo(Vector3D.PLUS_I);
873              Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
874              if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
875                  throw new CardanEulerSingularityException(true);
876              }
877              return new double[] {
878                  FastMath.atan2(v2.getZ(), v2.getY()),
879                 -FastMath.asin(v2.getX()),
880                  FastMath.atan2(v1.getZ(), v1.getX())
881              };
882
883          } else if (order == RotationOrder.YXZ) {
884
885              // r (Vector3D.plusJ) coordinates are :
886              // cos (phi) sin (psi), cos (phi) cos (psi), -sin (phi)
887              // (-r) (Vector3D.plusK) coordinates are :
888              // sin (theta) cos (phi), -sin (phi), cos (theta) cos (phi)
889              // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
890              Vector3D v1 = applyTo(Vector3D.PLUS_J);
891              Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
892              if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
893                  throw new CardanEulerSingularityException(true);
894              }
895              return new double[] {
896                  FastMath.atan2(v2.getX(), v2.getZ()),
897                 -FastMath.asin(v2.getY()),
898                  FastMath.atan2(v1.getX(), v1.getY())
899              };
900
901          } else if (order == RotationOrder.YZX) {
902
903              // r (Vector3D.plusJ) coordinates are :
904              // sin (psi), cos (psi) cos (phi), -cos (psi) sin (phi)
905              // (-r) (Vector3D.plusI) coordinates are :
906              // cos (theta) cos (psi), sin (psi), -sin (theta) cos (psi)
907              // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
908              Vector3D v1 = applyTo(Vector3D.PLUS_J);
909              Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
910              if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
911                  throw new CardanEulerSingularityException(true);
912              }
913              return new double[] {
914                  FastMath.atan2(-v2.getZ(), v2.getX()),
915                  FastMath.asin(v2.getY()),
916                  FastMath.atan2(-v1.getZ(), v1.getY())
917              };
918
919          } else if (order == RotationOrder.ZXY) {
920
921              // r (Vector3D.plusK) coordinates are :
922              //  -cos (phi) sin (theta), sin (phi), cos (phi) cos (theta)
923              // (-r) (Vector3D.plusJ) coordinates are :
924              // -sin (psi) cos (phi), cos (psi) cos (phi), sin (phi)
925              // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
926              Vector3D v1 = applyTo(Vector3D.PLUS_K);
927              Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
928              if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
929                  throw new CardanEulerSingularityException(true);
930              }
931              return new double[] {
932                  FastMath.atan2(-v2.getX(), v2.getY()),
933                  FastMath.asin(v2.getZ()),
934                  FastMath.atan2(-v1.getX(), v1.getZ())
935              };
936
937          } else if (order == RotationOrder.ZYX) {
938
939              // r (Vector3D.plusK) coordinates are :
940              //  -sin (theta), cos (theta) sin (phi), cos (theta) cos (phi)
941              // (-r) (Vector3D.plusI) coordinates are :
942              // cos (psi) cos (theta), sin (psi) cos (theta), -sin (theta)
943              // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
944              Vector3D v1 = applyTo(Vector3D.PLUS_K);
945              Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
946              if  ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
947                  throw new CardanEulerSingularityException(true);
948              }
949              return new double[] {
950                  FastMath.atan2(v2.getY(), v2.getX()),
951                 -FastMath.asin(v2.getZ()),
952                  FastMath.atan2(v1.getY(), v1.getZ())
953              };
954
955          } else if (order == RotationOrder.XYX) {
956
957              // r (Vector3D.plusI) coordinates are :
958              //  cos (theta), sin (phi2) sin (theta), cos (phi2) sin (theta)
959              // (-r) (Vector3D.plusI) coordinates are :
960              // cos (theta), sin (theta) sin (phi1), -sin (theta) cos (phi1)
961              // and we can choose to have theta in the interval [0 ; PI]
962              Vector3D v1 = applyTo(Vector3D.PLUS_I);
963              Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
964              if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
965                  throw new CardanEulerSingularityException(false);
966              }
967              return new double[] {
968                  FastMath.atan2(v2.getY(), -v2.getZ()),
969                  FastMath.acos(v2.getX()),
970                  FastMath.atan2(v1.getY(), v1.getZ())
971              };
972
973          } else if (order == RotationOrder.XZX) {
974
975              // r (Vector3D.plusI) coordinates are :
976              //  cos (psi), -cos (phi2) sin (psi), sin (phi2) sin (psi)
977              // (-r) (Vector3D.plusI) coordinates are :
978              // cos (psi), sin (psi) cos (phi1), sin (psi) sin (phi1)
979              // and we can choose to have psi in the interval [0 ; PI]
980              Vector3D v1 = applyTo(Vector3D.PLUS_I);
981              Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
982              if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
983                  throw new CardanEulerSingularityException(false);
984              }
985              return new double[] {
986                  FastMath.atan2(v2.getZ(), v2.getY()),
987                  FastMath.acos(v2.getX()),
988                  FastMath.atan2(v1.getZ(), -v1.getY())
989              };
990
991          } else if (order == RotationOrder.YXY) {
992
993              // r (Vector3D.plusJ) coordinates are :
994              // sin (phi) sin (theta2), cos (phi), -sin (phi) cos (theta2)
995              // (-r) (Vector3D.plusJ) coordinates are :
996              //  sin (theta1) sin (phi), cos (phi), cos (theta1) sin (phi)
997              // and we can choose to have phi in the interval [0 ; PI]
998              Vector3D v1 = applyTo(Vector3D.PLUS_J);
999              Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
1000              if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
1001                  throw new CardanEulerSingularityException(false);
1002              }
1003              return new double[] {
1004                  FastMath.atan2(v2.getX(), v2.getZ()),
1005                  FastMath.acos(v2.getY()),
1006                  FastMath.atan2(v1.getX(), -v1.getZ())
1007              };
1008
1009          } else if (order == RotationOrder.YZY) {
1010
1011              // r (Vector3D.plusJ) coordinates are :
1012              // sin (psi) cos (theta2), cos (psi), sin (psi) sin (theta2)
1013              // (-r) (Vector3D.plusJ) coordinates are :
1014              //  -cos (theta1) sin (psi), cos (psi), sin (theta1) sin (psi)
1015              // and we can choose to have psi in the interval [0 ; PI]
1016              Vector3D v1 = applyTo(Vector3D.PLUS_J);
1017              Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
1018              if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
1019                  throw new CardanEulerSingularityException(false);
1020              }
1021              return new double[] {
1022                  FastMath.atan2(v2.getZ(), -v2.getX()),
1023                  FastMath.acos(v2.getY()),
1024                  FastMath.atan2(v1.getZ(), v1.getX())
1025              };
1026
1027          } else if (order == RotationOrder.ZXZ) {
1028
1029              // r (Vector3D.plusK) coordinates are :
1030              // sin (phi) sin (psi2), sin (phi) cos (psi2), cos (phi)
1031              // (-r) (Vector3D.plusK) coordinates are :
1032              //  sin (psi1) sin (phi), -cos (psi1) sin (phi), cos (phi)
1033              // and we can choose to have phi in the interval [0 ; PI]
1034              Vector3D v1 = applyTo(Vector3D.PLUS_K);
1035              Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
1036              if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
1037                  throw new CardanEulerSingularityException(false);
1038              }
1039              return new double[] {
1040                  FastMath.atan2(v2.getX(), -v2.getY()),
1041                  FastMath.acos(v2.getZ()),
1042                  FastMath.atan2(v1.getX(), v1.getY())
1043              };
1044
1045          } else { // last possibility is ZYZ
1046
1047              // r (Vector3D.plusK) coordinates are :
1048              // -sin (theta) cos (psi2), sin (theta) sin (psi2), cos (theta)
1049              // (-r) (Vector3D.plusK) coordinates are :
1050              //  cos (psi1) sin (theta), sin (psi1) sin (theta), cos (theta)
1051              // and we can choose to have theta in the interval [0 ; PI]
1052              Vector3D v1 = applyTo(Vector3D.PLUS_K);
1053              Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
1054              if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
1055                  throw new CardanEulerSingularityException(false);
1056              }
1057              return new double[] {
1058                  FastMath.atan2(v2.getY(), v2.getX()),
1059                  FastMath.acos(v2.getZ()),
1060                  FastMath.atan2(v1.getY(), -v1.getX())
1061              };
1062
1063          }
1064      }
1065
1066  }
1067
1068  /** Get the 3X3 matrix corresponding to the instance
1069   * @return the matrix corresponding to the instance
1070   */
1071  public double[][] getMatrix() {
1072
1073    // products
1074    double q0q0  = q0 * q0;
1075    double q0q1  = q0 * q1;
1076    double q0q2  = q0 * q2;
1077    double q0q3  = q0 * q3;
1078    double q1q1  = q1 * q1;
1079    double q1q2  = q1 * q2;
1080    double q1q3  = q1 * q3;
1081    double q2q2  = q2 * q2;
1082    double q2q3  = q2 * q3;
1083    double q3q3  = q3 * q3;
1084
1085    // create the matrix
1086    double[][] m = new double[3][];
1087    m[0] = new double[3];
1088    m[1] = new double[3];
1089    m[2] = new double[3];
1090
1091    m [0][0] = 2.0 * (q0q0 + q1q1) - 1.0;
1092    m [1][0] = 2.0 * (q1q2 - q0q3);
1093    m [2][0] = 2.0 * (q1q3 + q0q2);
1094
1095    m [0][1] = 2.0 * (q1q2 + q0q3);
1096    m [1][1] = 2.0 * (q0q0 + q2q2) - 1.0;
1097    m [2][1] = 2.0 * (q2q3 - q0q1);
1098
1099    m [0][2] = 2.0 * (q1q3 - q0q2);
1100    m [1][2] = 2.0 * (q2q3 + q0q1);
1101    m [2][2] = 2.0 * (q0q0 + q3q3) - 1.0;
1102
1103    return m;
1104
1105  }
1106
1107  /** Apply the rotation to a vector.
1108   * @param u vector to apply the rotation to
1109   * @return a new vector which is the image of u by the rotation
1110   */
1111  public Vector3D applyTo(Vector3D u) {
1112
1113    double x = u.getX();
1114    double y = u.getY();
1115    double z = u.getZ();
1116
1117    double s = q1 * x + q2 * y + q3 * z;
1118
1119    return new Vector3D(2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x,
1120                        2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y,
1121                        2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - z);
1122
1123  }
1124
1125  /** Apply the rotation to a vector stored in an array.
1126   * @param in an array with three items which stores vector to rotate
1127   * @param out an array with three items to put result to (it can be the same
1128   * array as in)
1129   */
1130  public void applyTo(final double[] in, final double[] out) {
1131
1132      final double x = in[0];
1133      final double y = in[1];
1134      final double z = in[2];
1135
1136      final double s = q1 * x + q2 * y + q3 * z;
1137
1138      out[0] = 2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x;
1139      out[1] = 2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y;
1140      out[2] = 2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - z;
1141
1142  }
1143
1144  /** Apply the inverse of the rotation to a vector.
1145   * @param u vector to apply the inverse of the rotation to
1146   * @return a new vector which such that u is its image by the rotation
1147   */
1148  public Vector3D applyInverseTo(Vector3D u) {
1149
1150    double x = u.getX();
1151    double y = u.getY();
1152    double z = u.getZ();
1153
1154    double s = q1 * x + q2 * y + q3 * z;
1155    double m0 = -q0;
1156
1157    return new Vector3D(2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x,
1158                        2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y,
1159                        2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - z);
1160
1161  }
1162
1163  /** Apply the inverse of the rotation to a vector stored in an array.
1164   * @param in an array with three items which stores vector to rotate
1165   * @param out an array with three items to put result to (it can be the same
1166   * array as in)
1167   */
1168  public void applyInverseTo(final double[] in, final double[] out) {
1169
1170      final double x = in[0];
1171      final double y = in[1];
1172      final double z = in[2];
1173
1174      final double s = q1 * x + q2 * y + q3 * z;
1175      final double m0 = -q0;
1176
1177      out[0] = 2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x;
1178      out[1] = 2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y;
1179      out[2] = 2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - z;
1180
1181  }
1182
1183  /** Apply the instance to another rotation.
1184   * <p>
1185   * Calling this method is equivalent to call
1186   * {@link #compose(Rotation, RotationConvention)
1187   * compose(r, RotationConvention.VECTOR_OPERATOR)}.
1188   * </p>
1189   * @param r rotation to apply the rotation to
1190   * @return a new rotation which is the composition of r by the instance
1191   */
1192  public Rotation applyTo(Rotation r) {
1193    return compose(r, RotationConvention.VECTOR_OPERATOR);
1194  }
1195
1196  /** Compose the instance with another rotation.
1197   * <p>
1198   * If the semantics of the rotations composition corresponds to a
1199   * {@link RotationConvention#VECTOR_OPERATOR vector operator} convention,
1200   * applying the instance to a rotation is computing the composition
1201   * in an order compliant with the following rule : let {@code u} be any
1202   * vector and {@code v} its image by {@code r1} (i.e.
1203   * {@code r1.applyTo(u) = v}). Let {@code w} be the image of {@code v} by
1204   * rotation {@code r2} (i.e. {@code r2.applyTo(v) = w}). Then
1205   * {@code w = comp.applyTo(u)}, where
1206   * {@code comp = r2.compose(r1, RotationConvention.VECTOR_OPERATOR)}.
1207   * </p>
1208   * <p>
1209   * If the semantics of the rotations composition corresponds to a
1210   * {@link RotationConvention#FRAME_TRANSFORM frame transform} convention,
1211   * the application order will be reversed. So keeping the exact same
1212   * meaning of all {@code r1}, {@code r2}, {@code u}, {@code v}, {@code w}
1213   * and  {@code comp} as above, {@code comp} could also be computed as
1214   * {@code comp = r1.compose(r2, RotationConvention.FRAME_TRANSFORM)}.
1215   * </p>
1216   * @param r rotation to apply the rotation to
1217   * @param convention convention to use for the semantics of the angle
1218   * @return a new rotation which is the composition of r by the instance
1219   */
1220  public Rotation compose(final Rotation r, final RotationConvention convention) {
1221    return convention == RotationConvention.VECTOR_OPERATOR ?
1222           composeInternal(r) : r.composeInternal(this);
1223  }
1224
1225  /** Compose the instance with another rotation using vector operator convention.
1226   * @param r rotation to apply the rotation to
1227   * @return a new rotation which is the composition of r by the instance
1228   * using vector operator convention
1229   */
1230  private Rotation composeInternal(final Rotation r) {
1231    return new Rotation(r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),
1232                        r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),
1233                        r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),
1234                        r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),
1235                        false);
1236  }
1237
1238  /** Apply the inverse of the instance to another rotation.
1239   * <p>
1240   * Calling this method is equivalent to call
1241   * {@link #composeInverse(Rotation, RotationConvention)
1242   * composeInverse(r, RotationConvention.VECTOR_OPERATOR)}.
1243   * </p>
1244   * @param r rotation to apply the rotation to
1245   * @return a new rotation which is the composition of r by the inverse
1246   * of the instance
1247   */
1248  public Rotation applyInverseTo(Rotation r) {
1249    return composeInverse(r, RotationConvention.VECTOR_OPERATOR);
1250  }
1251
1252  /** Compose the inverse of the instance with another rotation.
1253   * <p>
1254   * If the semantics of the rotations composition corresponds to a
1255   * {@link RotationConvention#VECTOR_OPERATOR vector operator} convention,
1256   * applying the inverse of the instance to a rotation is computing
1257   * the composition in an order compliant with the following rule :
1258   * let {@code u} be any vector and {@code v} its image by {@code r1}
1259   * (i.e. {@code r1.applyTo(u) = v}). Let {@code w} be the inverse image
1260   * of {@code v} by {@code r2} (i.e. {@code r2.applyInverseTo(v) = w}).
1261   * Then {@code w = comp.applyTo(u)}, where
1262   * {@code comp = r2.composeInverse(r1)}.
1263   * </p>
1264   * <p>
1265   * If the semantics of the rotations composition corresponds to a
1266   * {@link RotationConvention#FRAME_TRANSFORM frame transform} convention,
1267   * the application order will be reversed, which means it is the
1268   * <em>innermost</em> rotation that will be reversed. So keeping the exact same
1269   * meaning of all {@code r1}, {@code r2}, {@code u}, {@code v}, {@code w}
1270   * and  {@code comp} as above, {@code comp} could also be computed as
1271   * {@code comp = r1.revert().composeInverse(r2.revert(), RotationConvention.FRAME_TRANSFORM)}.
1272   * </p>
1273   * @param r rotation to apply the rotation to
1274   * @param convention convention to use for the semantics of the angle
1275   * @return a new rotation which is the composition of r by the inverse
1276   * of the instance
1277   */
1278  public Rotation composeInverse(final Rotation r, final RotationConvention convention) {
1279    return convention == RotationConvention.VECTOR_OPERATOR ?
1280           composeInverseInternal(r) : r.composeInternal(revert());
1281  }
1282
1283  /** Compose the inverse of the instance with another rotation
1284   * using vector operator convention.
1285   * @param r rotation to apply the rotation to
1286   * @return a new rotation which is the composition of r by the inverse
1287   * of the instance using vector operator convention
1288   */
1289  private Rotation composeInverseInternal(Rotation r) {
1290    return new Rotation(-r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),
1291                        -r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),
1292                        -r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),
1293                        -r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),
1294                        false);
1295  }
1296
1297  /** Perfect orthogonality on a 3X3 matrix.
1298   * @param m initial matrix (not exactly orthogonal)
1299   * @param threshold convergence threshold for the iterative
1300   * orthogonality correction (convergence is reached when the
1301   * difference between two steps of the Frobenius norm of the
1302   * correction is below this threshold)
1303   * @return an orthogonal matrix close to m
1304   * @exception NotARotationMatrixException if the matrix cannot be
1305   * orthogonalized with the given threshold after 10 iterations
1306   */
1307  private double[][] orthogonalizeMatrix(double[][] m, double threshold)
1308    throws NotARotationMatrixException {
1309    double[] m0 = m[0];
1310    double[] m1 = m[1];
1311    double[] m2 = m[2];
1312    double x00 = m0[0];
1313    double x01 = m0[1];
1314    double x02 = m0[2];
1315    double x10 = m1[0];
1316    double x11 = m1[1];
1317    double x12 = m1[2];
1318    double x20 = m2[0];
1319    double x21 = m2[1];
1320    double x22 = m2[2];
1321    double fn = 0;
1322    double fn1;
1323
1324    double[][] o = new double[3][3];
1325    double[] o0 = o[0];
1326    double[] o1 = o[1];
1327    double[] o2 = o[2];
1328
1329    // iterative correction: Xn+1 = Xn - 0.5 * (Xn.Mt.Xn - M)
1330    int i = 0;
1331    while (++i < 11) {
1332
1333      // Mt.Xn
1334      double mx00 = m0[0] * x00 + m1[0] * x10 + m2[0] * x20;
1335      double mx10 = m0[1] * x00 + m1[1] * x10 + m2[1] * x20;
1336      double mx20 = m0[2] * x00 + m1[2] * x10 + m2[2] * x20;
1337      double mx01 = m0[0] * x01 + m1[0] * x11 + m2[0] * x21;
1338      double mx11 = m0[1] * x01 + m1[1] * x11 + m2[1] * x21;
1339      double mx21 = m0[2] * x01 + m1[2] * x11 + m2[2] * x21;
1340      double mx02 = m0[0] * x02 + m1[0] * x12 + m2[0] * x22;
1341      double mx12 = m0[1] * x02 + m1[1] * x12 + m2[1] * x22;
1342      double mx22 = m0[2] * x02 + m1[2] * x12 + m2[2] * x22;
1343
1344      // Xn+1
1345      o0[0] = x00 - 0.5 * (x00 * mx00 + x01 * mx10 + x02 * mx20 - m0[0]);
1346      o0[1] = x01 - 0.5 * (x00 * mx01 + x01 * mx11 + x02 * mx21 - m0[1]);
1347      o0[2] = x02 - 0.5 * (x00 * mx02 + x01 * mx12 + x02 * mx22 - m0[2]);
1348      o1[0] = x10 - 0.5 * (x10 * mx00 + x11 * mx10 + x12 * mx20 - m1[0]);
1349      o1[1] = x11 - 0.5 * (x10 * mx01 + x11 * mx11 + x12 * mx21 - m1[1]);
1350      o1[2] = x12 - 0.5 * (x10 * mx02 + x11 * mx12 + x12 * mx22 - m1[2]);
1351      o2[0] = x20 - 0.5 * (x20 * mx00 + x21 * mx10 + x22 * mx20 - m2[0]);
1352      o2[1] = x21 - 0.5 * (x20 * mx01 + x21 * mx11 + x22 * mx21 - m2[1]);
1353      o2[2] = x22 - 0.5 * (x20 * mx02 + x21 * mx12 + x22 * mx22 - m2[2]);
1354
1355      // correction on each elements
1356      double corr00 = o0[0] - m0[0];
1357      double corr01 = o0[1] - m0[1];
1358      double corr02 = o0[2] - m0[2];
1359      double corr10 = o1[0] - m1[0];
1360      double corr11 = o1[1] - m1[1];
1361      double corr12 = o1[2] - m1[2];
1362      double corr20 = o2[0] - m2[0];
1363      double corr21 = o2[1] - m2[1];
1364      double corr22 = o2[2] - m2[2];
1365
1366      // Frobenius norm of the correction
1367      fn1 = corr00 * corr00 + corr01 * corr01 + corr02 * corr02 +
1368            corr10 * corr10 + corr11 * corr11 + corr12 * corr12 +
1369            corr20 * corr20 + corr21 * corr21 + corr22 * corr22;
1370
1371      // convergence test
1372      if (FastMath.abs(fn1 - fn) <= threshold) {
1373          return o;
1374      }
1375
1376      // prepare next iteration
1377      x00 = o0[0];
1378      x01 = o0[1];
1379      x02 = o0[2];
1380      x10 = o1[0];
1381      x11 = o1[1];
1382      x12 = o1[2];
1383      x20 = o2[0];
1384      x21 = o2[1];
1385      x22 = o2[2];
1386      fn  = fn1;
1387
1388    }
1389
1390    // the algorithm did not converge after 10 iterations
1391    throw new NotARotationMatrixException(
1392            LocalizedFormats.UNABLE_TO_ORTHOGONOLIZE_MATRIX,
1393            i - 1);
1394  }
1395
1396  /** Compute the <i>distance</i> between two rotations.
1397   * <p>The <i>distance</i> is intended here as a way to check if two
1398   * rotations are almost similar (i.e. they transform vectors the same way)
1399   * or very different. It is mathematically defined as the angle of
1400   * the rotation r that prepended to one of the rotations gives the other
1401   * one:</p>
1402   * <pre>
1403   *        r<sub>1</sub>(r) = r<sub>2</sub>
1404   * </pre>
1405   * <p>This distance is an angle between 0 and &pi;. Its value is the smallest
1406   * possible upper bound of the angle in radians between r<sub>1</sub>(v)
1407   * and r<sub>2</sub>(v) for all possible vectors v. This upper bound is
1408   * reached for some v. The distance is equal to 0 if and only if the two
1409   * rotations are identical.</p>
1410   * <p>Comparing two rotations should always be done using this value rather
1411   * than for example comparing the components of the quaternions. It is much
1412   * more stable, and has a geometric meaning. Also comparing quaternions
1413   * components is error prone since for example quaternions (0.36, 0.48, -0.48, -0.64)
1414   * and (-0.36, -0.48, 0.48, 0.64) represent exactly the same rotation despite
1415   * their components are different (they are exact opposites).</p>
1416   * @param r1 first rotation
1417   * @param r2 second rotation
1418   * @return <i>distance</i> between r1 and r2
1419   */
1420  public static double distance(Rotation r1, Rotation r2) {
1421      return r1.composeInverseInternal(r2).getAngle();
1422  }
1423
1424}