001    /*
002     * Licensed to the Apache Software Foundation (ASF) under one or more
003     * contributor license agreements.  See the NOTICE file distributed with
004     * this work for additional information regarding copyright ownership.
005     * The ASF licenses this file to You under the Apache License, Version 2.0
006     * (the "License"); you may not use this file except in compliance with
007     * the License.  You may obtain a copy of the License at
008     *
009     *      http://www.apache.org/licenses/LICENSE-2.0
010     *
011     * Unless required by applicable law or agreed to in writing, software
012     * distributed under the License is distributed on an "AS IS" BASIS,
013     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
014     * See the License for the specific language governing permissions and
015     * limitations under the License.
016     */
017    
018    package org.apache.commons.math3.geometry.euclidean.threed;
019    
020    import java.io.Serializable;
021    
022    import org.apache.commons.math3.exception.MathArithmeticException;
023    import org.apache.commons.math3.exception.MathIllegalArgumentException;
024    import org.apache.commons.math3.exception.util.LocalizedFormats;
025    import org.apache.commons.math3.util.FastMath;
026    import org.apache.commons.math3.util.MathArrays;
027    
028    /**
029     * This class implements rotations in a three-dimensional space.
030     *
031     * <p>Rotations can be represented by several different mathematical
032     * entities (matrices, axe and angle, Cardan or Euler angles,
033     * quaternions). This class presents an higher level abstraction, more
034     * user-oriented and hiding this implementation details. Well, for the
035     * curious, we use quaternions for the internal representation. The
036     * user can build a rotation from any of these representations, and
037     * any of these representations can be retrieved from a
038     * <code>Rotation</code> instance (see the various constructors and
039     * getters). In addition, a rotation can also be built implicitly
040     * from a set of vectors and their image.</p>
041     * <p>This implies that this class can be used to convert from one
042     * representation to another one. For example, converting a rotation
043     * matrix into a set of Cardan angles from can be done using the
044     * following single line of code:</p>
045     * <pre>
046     * double[] angles = new Rotation(matrix, 1.0e-10).getAngles(RotationOrder.XYZ);
047     * </pre>
048     * <p>Focus is oriented on what a rotation <em>do</em> rather than on its
049     * underlying representation. Once it has been built, and regardless of its
050     * internal representation, a rotation is an <em>operator</em> which basically
051     * transforms three dimensional {@link Vector3D vectors} into other three
052     * dimensional {@link Vector3D vectors}. Depending on the application, the
053     * meaning of these vectors may vary and the semantics of the rotation also.</p>
054     * <p>For example in an spacecraft attitude simulation tool, users will often
055     * consider the vectors are fixed (say the Earth direction for example) and the
056     * frames change. The rotation transforms the coordinates of the vector in inertial
057     * frame into the coordinates of the same vector in satellite frame. In this
058     * case, the rotation implicitly defines the relation between the two frames.</p>
059     * <p>Another example could be a telescope control application, where the rotation
060     * would transform the sighting direction at rest into the desired observing
061     * direction when the telescope is pointed towards an object of interest. In this
062     * case the rotation transforms the direction at rest in a topocentric frame
063     * into the sighting direction in the same topocentric frame. This implies in this
064     * case the frame is fixed and the vector moves.</p>
065     * <p>In many case, both approaches will be combined. In our telescope example,
066     * we will probably also need to transform the observing direction in the topocentric
067     * frame into the observing direction in inertial frame taking into account the observatory
068     * location and the Earth rotation, which would essentially be an application of the
069     * first approach.</p>
070     *
071     * <p>These examples show that a rotation is what the user wants it to be. This
072     * class does not push the user towards one specific definition and hence does not
073     * provide methods like <code>projectVectorIntoDestinationFrame</code> or
074     * <code>computeTransformedDirection</code>. It provides simpler and more generic
075     * methods: {@link #applyTo(Vector3D) applyTo(Vector3D)} and {@link
076     * #applyInverseTo(Vector3D) applyInverseTo(Vector3D)}.</p>
077     *
078     * <p>Since a rotation is basically a vectorial operator, several rotations can be
079     * composed together and the composite operation <code>r = r<sub>1</sub> o
080     * r<sub>2</sub></code> (which means that for each vector <code>u</code>,
081     * <code>r(u) = r<sub>1</sub>(r<sub>2</sub>(u))</code>) is also a rotation. Hence
082     * we can consider that in addition to vectors, a rotation can be applied to other
083     * rotations as well (or to itself). With our previous notations, we would say we
084     * can apply <code>r<sub>1</sub></code> to <code>r<sub>2</sub></code> and the result
085     * we get is <code>r = r<sub>1</sub> o r<sub>2</sub></code>. For this purpose, the
086     * class provides the methods: {@link #applyTo(Rotation) applyTo(Rotation)} and
087     * {@link #applyInverseTo(Rotation) applyInverseTo(Rotation)}.</p>
088     *
089     * <p>Rotations are guaranteed to be immutable objects.</p>
090     *
091     * @version $Id: Rotation.java 1416643 2012-12-03 19:37:14Z tn $
092     * @see Vector3D
093     * @see RotationOrder
094     * @since 1.2
095     */
096    
097    public class Rotation implements Serializable {
098    
099      /** Identity rotation. */
100      public static final Rotation IDENTITY = new Rotation(1.0, 0.0, 0.0, 0.0, false);
101    
102      /** Serializable version identifier */
103      private static final long serialVersionUID = -2153622329907944313L;
104    
105      /** Scalar coordinate of the quaternion. */
106      private final double q0;
107    
108      /** First coordinate of the vectorial part of the quaternion. */
109      private final double q1;
110    
111      /** Second coordinate of the vectorial part of the quaternion. */
112      private final double q2;
113    
114      /** Third coordinate of the vectorial part of the quaternion. */
115      private final double q3;
116    
117      /** Build a rotation from the quaternion coordinates.
118       * <p>A rotation can be built from a <em>normalized</em> quaternion,
119       * i.e. a quaternion for which q<sub>0</sub><sup>2</sup> +
120       * q<sub>1</sub><sup>2</sup> + q<sub>2</sub><sup>2</sup> +
121       * q<sub>3</sub><sup>2</sup> = 1. If the quaternion is not normalized,
122       * the constructor can normalize it in a preprocessing step.</p>
123       * <p>Note that some conventions put the scalar part of the quaternion
124       * as the 4<sup>th</sup> component and the vector part as the first three
125       * components. This is <em>not</em> our convention. We put the scalar part
126       * as the first component.</p>
127       * @param q0 scalar part of the quaternion
128       * @param q1 first coordinate of the vectorial part of the quaternion
129       * @param q2 second coordinate of the vectorial part of the quaternion
130       * @param q3 third coordinate of the vectorial part of the quaternion
131       * @param needsNormalization if true, the coordinates are considered
132       * not to be normalized, a normalization preprocessing step is performed
133       * before using them
134       */
135      public Rotation(double q0, double q1, double q2, double q3,
136                      boolean needsNormalization) {
137    
138        if (needsNormalization) {
139          // normalization preprocessing
140          double inv = 1.0 / FastMath.sqrt(q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3);
141          q0 *= inv;
142          q1 *= inv;
143          q2 *= inv;
144          q3 *= inv;
145        }
146    
147        this.q0 = q0;
148        this.q1 = q1;
149        this.q2 = q2;
150        this.q3 = q3;
151    
152      }
153    
154      /** Build a rotation from an axis and an angle.
155       * <p>We use the convention that angles are oriented according to
156       * the effect of the rotation on vectors around the axis. That means
157       * that if (i, j, k) is a direct frame and if we first provide +k as
158       * the axis and &pi;/2 as the angle to this constructor, and then
159       * {@link #applyTo(Vector3D) apply} the instance to +i, we will get
160       * +j.</p>
161       * <p>Another way to represent our convention is to say that a rotation
162       * of angle &theta; about the unit vector (x, y, z) is the same as the
163       * rotation build from quaternion components { cos(-&theta;/2),
164       * x * sin(-&theta;/2), y * sin(-&theta;/2), z * sin(-&theta;/2) }.
165       * Note the minus sign on the angle!</p>
166       * <p>On the one hand this convention is consistent with a vectorial
167       * perspective (moving vectors in fixed frames), on the other hand it
168       * is different from conventions with a frame perspective (fixed vectors
169       * viewed from different frames) like the ones used for example in spacecraft
170       * attitude community or in the graphics community.</p>
171       * @param axis axis around which to rotate
172       * @param angle rotation angle.
173       * @exception MathIllegalArgumentException if the axis norm is zero
174       */
175      public Rotation(Vector3D axis, double angle) throws MathIllegalArgumentException {
176    
177        double norm = axis.getNorm();
178        if (norm == 0) {
179          throw new MathIllegalArgumentException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_AXIS);
180        }
181    
182        double halfAngle = -0.5 * angle;
183        double coeff = FastMath.sin(halfAngle) / norm;
184    
185        q0 = FastMath.cos (halfAngle);
186        q1 = coeff * axis.getX();
187        q2 = coeff * axis.getY();
188        q3 = coeff * axis.getZ();
189    
190      }
191    
192      /** Build a rotation from a 3X3 matrix.
193    
194       * <p>Rotation matrices are orthogonal matrices, i.e. unit matrices
195       * (which are matrices for which m.m<sup>T</sup> = I) with real
196       * coefficients. The module of the determinant of unit matrices is
197       * 1, among the orthogonal 3X3 matrices, only the ones having a
198       * positive determinant (+1) are rotation matrices.</p>
199    
200       * <p>When a rotation is defined by a matrix with truncated values
201       * (typically when it is extracted from a technical sheet where only
202       * four to five significant digits are available), the matrix is not
203       * orthogonal anymore. This constructor handles this case
204       * transparently by using a copy of the given matrix and applying a
205       * correction to the copy in order to perfect its orthogonality. If
206       * the Frobenius norm of the correction needed is above the given
207       * threshold, then the matrix is considered to be too far from a
208       * true rotation matrix and an exception is thrown.<p>
209    
210       * @param m rotation matrix
211       * @param threshold convergence threshold for the iterative
212       * orthogonality correction (convergence is reached when the
213       * difference between two steps of the Frobenius norm of the
214       * correction is below this threshold)
215    
216       * @exception NotARotationMatrixException if the matrix is not a 3X3
217       * matrix, or if it cannot be transformed into an orthogonal matrix
218       * with the given threshold, or if the determinant of the resulting
219       * orthogonal matrix is negative
220    
221       */
222      public Rotation(double[][] m, double threshold)
223        throws NotARotationMatrixException {
224    
225        // dimension check
226        if ((m.length != 3) || (m[0].length != 3) ||
227            (m[1].length != 3) || (m[2].length != 3)) {
228          throw new NotARotationMatrixException(
229                  LocalizedFormats.ROTATION_MATRIX_DIMENSIONS,
230                  m.length, m[0].length);
231        }
232    
233        // compute a "close" orthogonal matrix
234        double[][] ort = orthogonalizeMatrix(m, threshold);
235    
236        // check the sign of the determinant
237        double det = ort[0][0] * (ort[1][1] * ort[2][2] - ort[2][1] * ort[1][2]) -
238                     ort[1][0] * (ort[0][1] * ort[2][2] - ort[2][1] * ort[0][2]) +
239                     ort[2][0] * (ort[0][1] * ort[1][2] - ort[1][1] * ort[0][2]);
240        if (det < 0.0) {
241          throw new NotARotationMatrixException(
242                  LocalizedFormats.CLOSEST_ORTHOGONAL_MATRIX_HAS_NEGATIVE_DETERMINANT,
243                  det);
244        }
245    
246        double[] quat = mat2quat(ort);
247        q0 = quat[0];
248        q1 = quat[1];
249        q2 = quat[2];
250        q3 = quat[3];
251    
252      }
253    
254      /** Build the rotation that transforms a pair of vector into another pair.
255    
256       * <p>Except for possible scale factors, if the instance were applied to
257       * the pair (u<sub>1</sub>, u<sub>2</sub>) it will produce the pair
258       * (v<sub>1</sub>, v<sub>2</sub>).</p>
259    
260       * <p>If the angular separation between u<sub>1</sub> and u<sub>2</sub> is
261       * not the same as the angular separation between v<sub>1</sub> and
262       * v<sub>2</sub>, then a corrected v'<sub>2</sub> will be used rather than
263       * v<sub>2</sub>, the corrected vector will be in the (v<sub>1</sub>,
264       * v<sub>2</sub>) plane.</p>
265    
266       * @param u1 first vector of the origin pair
267       * @param u2 second vector of the origin pair
268       * @param v1 desired image of u1 by the rotation
269       * @param v2 desired image of u2 by the rotation
270       * @exception MathArithmeticException if the norm of one of the vectors is zero,
271       * or if one of the pair is degenerated (i.e. the vectors of the pair are colinear)
272       */
273      public Rotation(Vector3D u1, Vector3D u2, Vector3D v1, Vector3D v2)
274          throws MathArithmeticException {
275    
276          // build orthonormalized base from u1, u2
277          // this fails when vectors are null or colinear, which is forbidden to define a rotation
278          final Vector3D u3 = u1.crossProduct(u2).normalize();
279          u2 = u3.crossProduct(u1).normalize();
280          u1 = u1.normalize();
281    
282          // build an orthonormalized base from v1, v2
283          // this fails when vectors are null or colinear, which is forbidden to define a rotation
284          final Vector3D v3 = v1.crossProduct(v2).normalize();
285          v2 = v3.crossProduct(v1).normalize();
286          v1 = v1.normalize();
287    
288          // buid a matrix transforming the first base into the second one
289          final double[][] m = new double[][] {
290              {
291                  MathArrays.linearCombination(u1.getX(), v1.getX(), u2.getX(), v2.getX(), u3.getX(), v3.getX()),
292                  MathArrays.linearCombination(u1.getY(), v1.getX(), u2.getY(), v2.getX(), u3.getY(), v3.getX()),
293                  MathArrays.linearCombination(u1.getZ(), v1.getX(), u2.getZ(), v2.getX(), u3.getZ(), v3.getX())
294              },
295              {
296                  MathArrays.linearCombination(u1.getX(), v1.getY(), u2.getX(), v2.getY(), u3.getX(), v3.getY()),
297                  MathArrays.linearCombination(u1.getY(), v1.getY(), u2.getY(), v2.getY(), u3.getY(), v3.getY()),
298                  MathArrays.linearCombination(u1.getZ(), v1.getY(), u2.getZ(), v2.getY(), u3.getZ(), v3.getY())
299              },
300              {
301                  MathArrays.linearCombination(u1.getX(), v1.getZ(), u2.getX(), v2.getZ(), u3.getX(), v3.getZ()),
302                  MathArrays.linearCombination(u1.getY(), v1.getZ(), u2.getY(), v2.getZ(), u3.getY(), v3.getZ()),
303                  MathArrays.linearCombination(u1.getZ(), v1.getZ(), u2.getZ(), v2.getZ(), u3.getZ(), v3.getZ())
304              }
305          };
306    
307          double[] quat = mat2quat(m);
308          q0 = quat[0];
309          q1 = quat[1];
310          q2 = quat[2];
311          q3 = quat[3];
312    
313      }
314    
315      /** Build one of the rotations that transform one vector into another one.
316    
317       * <p>Except for a possible scale factor, if the instance were
318       * applied to the vector u it will produce the vector v. There is an
319       * infinite number of such rotations, this constructor choose the
320       * one with the smallest associated angle (i.e. the one whose axis
321       * is orthogonal to the (u, v) plane). If u and v are colinear, an
322       * arbitrary rotation axis is chosen.</p>
323    
324       * @param u origin vector
325       * @param v desired image of u by the rotation
326       * @exception MathArithmeticException if the norm of one of the vectors is zero
327       */
328      public Rotation(Vector3D u, Vector3D v) throws MathArithmeticException {
329    
330        double normProduct = u.getNorm() * v.getNorm();
331        if (normProduct == 0) {
332            throw new MathArithmeticException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_DEFINING_VECTOR);
333        }
334    
335        double dot = u.dotProduct(v);
336    
337        if (dot < ((2.0e-15 - 1.0) * normProduct)) {
338          // special case u = -v: we select a PI angle rotation around
339          // an arbitrary vector orthogonal to u
340          Vector3D w = u.orthogonal();
341          q0 = 0.0;
342          q1 = -w.getX();
343          q2 = -w.getY();
344          q3 = -w.getZ();
345        } else {
346          // general case: (u, v) defines a plane, we select
347          // the shortest possible rotation: axis orthogonal to this plane
348          q0 = FastMath.sqrt(0.5 * (1.0 + dot / normProduct));
349          double coeff = 1.0 / (2.0 * q0 * normProduct);
350          Vector3D q = v.crossProduct(u);
351          q1 = coeff * q.getX();
352          q2 = coeff * q.getY();
353          q3 = coeff * q.getZ();
354        }
355    
356      }
357    
358      /** Build a rotation from three Cardan or Euler elementary rotations.
359    
360       * <p>Cardan rotations are three successive rotations around the
361       * canonical axes X, Y and Z, each axis being used once. There are
362       * 6 such sets of rotations (XYZ, XZY, YXZ, YZX, ZXY and ZYX). Euler
363       * rotations are three successive rotations around the canonical
364       * axes X, Y and Z, the first and last rotations being around the
365       * same axis. There are 6 such sets of rotations (XYX, XZX, YXY,
366       * YZY, ZXZ and ZYZ), the most popular one being ZXZ.</p>
367       * <p>Beware that many people routinely use the term Euler angles even
368       * for what really are Cardan angles (this confusion is especially
369       * widespread in the aerospace business where Roll, Pitch and Yaw angles
370       * are often wrongly tagged as Euler angles).</p>
371    
372       * @param order order of rotations to use
373       * @param alpha1 angle of the first elementary rotation
374       * @param alpha2 angle of the second elementary rotation
375       * @param alpha3 angle of the third elementary rotation
376       */
377      public Rotation(RotationOrder order,
378                      double alpha1, double alpha2, double alpha3) {
379          Rotation r1 = new Rotation(order.getA1(), alpha1);
380          Rotation r2 = new Rotation(order.getA2(), alpha2);
381          Rotation r3 = new Rotation(order.getA3(), alpha3);
382          Rotation composed = r1.applyTo(r2.applyTo(r3));
383          q0 = composed.q0;
384          q1 = composed.q1;
385          q2 = composed.q2;
386          q3 = composed.q3;
387      }
388    
389      /** Convert an orthogonal rotation matrix to a quaternion.
390       * @param ort orthogonal rotation matrix
391       * @return quaternion corresponding to the matrix
392       */
393      private static double[] mat2quat(final double[][] ort) {
394    
395          final double[] quat = new double[4];
396    
397          // There are different ways to compute the quaternions elements
398          // from the matrix. They all involve computing one element from
399          // the diagonal of the matrix, and computing the three other ones
400          // using a formula involving a division by the first element,
401          // which unfortunately can be zero. Since the norm of the
402          // quaternion is 1, we know at least one element has an absolute
403          // value greater or equal to 0.5, so it is always possible to
404          // select the right formula and avoid division by zero and even
405          // numerical inaccuracy. Checking the elements in turn and using
406          // the first one greater than 0.45 is safe (this leads to a simple
407          // test since qi = 0.45 implies 4 qi^2 - 1 = -0.19)
408          double s = ort[0][0] + ort[1][1] + ort[2][2];
409          if (s > -0.19) {
410              // compute q0 and deduce q1, q2 and q3
411              quat[0] = 0.5 * FastMath.sqrt(s + 1.0);
412              double inv = 0.25 / quat[0];
413              quat[1] = inv * (ort[1][2] - ort[2][1]);
414              quat[2] = inv * (ort[2][0] - ort[0][2]);
415              quat[3] = inv * (ort[0][1] - ort[1][0]);
416          } else {
417              s = ort[0][0] - ort[1][1] - ort[2][2];
418              if (s > -0.19) {
419                  // compute q1 and deduce q0, q2 and q3
420                  quat[1] = 0.5 * FastMath.sqrt(s + 1.0);
421                  double inv = 0.25 / quat[1];
422                  quat[0] = inv * (ort[1][2] - ort[2][1]);
423                  quat[2] = inv * (ort[0][1] + ort[1][0]);
424                  quat[3] = inv * (ort[0][2] + ort[2][0]);
425              } else {
426                  s = ort[1][1] - ort[0][0] - ort[2][2];
427                  if (s > -0.19) {
428                      // compute q2 and deduce q0, q1 and q3
429                      quat[2] = 0.5 * FastMath.sqrt(s + 1.0);
430                      double inv = 0.25 / quat[2];
431                      quat[0] = inv * (ort[2][0] - ort[0][2]);
432                      quat[1] = inv * (ort[0][1] + ort[1][0]);
433                      quat[3] = inv * (ort[2][1] + ort[1][2]);
434                  } else {
435                      // compute q3 and deduce q0, q1 and q2
436                      s = ort[2][2] - ort[0][0] - ort[1][1];
437                      quat[3] = 0.5 * FastMath.sqrt(s + 1.0);
438                      double inv = 0.25 / quat[3];
439                      quat[0] = inv * (ort[0][1] - ort[1][0]);
440                      quat[1] = inv * (ort[0][2] + ort[2][0]);
441                      quat[2] = inv * (ort[2][1] + ort[1][2]);
442                  }
443              }
444          }
445    
446          return quat;
447    
448      }
449    
450      /** Revert a rotation.
451       * Build a rotation which reverse the effect of another
452       * rotation. This means that if r(u) = v, then r.revert(v) = u. The
453       * instance is not changed.
454       * @return a new rotation whose effect is the reverse of the effect
455       * of the instance
456       */
457      public Rotation revert() {
458        return new Rotation(-q0, q1, q2, q3, false);
459      }
460    
461      /** Get the scalar coordinate of the quaternion.
462       * @return scalar coordinate of the quaternion
463       */
464      public double getQ0() {
465        return q0;
466      }
467    
468      /** Get the first coordinate of the vectorial part of the quaternion.
469       * @return first coordinate of the vectorial part of the quaternion
470       */
471      public double getQ1() {
472        return q1;
473      }
474    
475      /** Get the second coordinate of the vectorial part of the quaternion.
476       * @return second coordinate of the vectorial part of the quaternion
477       */
478      public double getQ2() {
479        return q2;
480      }
481    
482      /** Get the third coordinate of the vectorial part of the quaternion.
483       * @return third coordinate of the vectorial part of the quaternion
484       */
485      public double getQ3() {
486        return q3;
487      }
488    
489      /** Get the normalized axis of the rotation.
490       * @return normalized axis of the rotation
491       * @see #Rotation(Vector3D, double)
492       */
493      public Vector3D getAxis() {
494        double squaredSine = q1 * q1 + q2 * q2 + q3 * q3;
495        if (squaredSine == 0) {
496          return new Vector3D(1, 0, 0);
497        } else if (q0 < 0) {
498          double inverse = 1 / FastMath.sqrt(squaredSine);
499          return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
500        }
501        double inverse = -1 / FastMath.sqrt(squaredSine);
502        return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
503      }
504    
505      /** Get the angle of the rotation.
506       * @return angle of the rotation (between 0 and &pi;)
507       * @see #Rotation(Vector3D, double)
508       */
509      public double getAngle() {
510        if ((q0 < -0.1) || (q0 > 0.1)) {
511          return 2 * FastMath.asin(FastMath.sqrt(q1 * q1 + q2 * q2 + q3 * q3));
512        } else if (q0 < 0) {
513          return 2 * FastMath.acos(-q0);
514        }
515        return 2 * FastMath.acos(q0);
516      }
517    
518      /** Get the Cardan or Euler angles corresponding to the instance.
519    
520       * <p>The equations show that each rotation can be defined by two
521       * different values of the Cardan or Euler angles set. For example
522       * if Cardan angles are used, the rotation defined by the angles
523       * a<sub>1</sub>, a<sub>2</sub> and a<sub>3</sub> is the same as
524       * the rotation defined by the angles &pi; + a<sub>1</sub>, &pi;
525       * - a<sub>2</sub> and &pi; + a<sub>3</sub>. This method implements
526       * the following arbitrary choices:</p>
527       * <ul>
528       *   <li>for Cardan angles, the chosen set is the one for which the
529       *   second angle is between -&pi;/2 and &pi;/2 (i.e its cosine is
530       *   positive),</li>
531       *   <li>for Euler angles, the chosen set is the one for which the
532       *   second angle is between 0 and &pi; (i.e its sine is positive).</li>
533       * </ul>
534    
535       * <p>Cardan and Euler angle have a very disappointing drawback: all
536       * of them have singularities. This means that if the instance is
537       * too close to the singularities corresponding to the given
538       * rotation order, it will be impossible to retrieve the angles. For
539       * Cardan angles, this is often called gimbal lock. There is
540       * <em>nothing</em> to do to prevent this, it is an intrinsic problem
541       * with Cardan and Euler representation (but not a problem with the
542       * rotation itself, which is perfectly well defined). For Cardan
543       * angles, singularities occur when the second angle is close to
544       * -&pi;/2 or +&pi;/2, for Euler angle singularities occur when the
545       * second angle is close to 0 or &pi;, this implies that the identity
546       * rotation is always singular for Euler angles!</p>
547    
548       * @param order rotation order to use
549       * @return an array of three angles, in the order specified by the set
550       * @exception CardanEulerSingularityException if the rotation is
551       * singular with respect to the angles set specified
552       */
553      public double[] getAngles(RotationOrder order)
554        throws CardanEulerSingularityException {
555    
556        if (order == RotationOrder.XYZ) {
557    
558          // r (Vector3D.plusK) coordinates are :
559          //  sin (theta), -cos (theta) sin (phi), cos (theta) cos (phi)
560          // (-r) (Vector3D.plusI) coordinates are :
561          // cos (psi) cos (theta), -sin (psi) cos (theta), sin (theta)
562          // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
563          Vector3D v1 = applyTo(Vector3D.PLUS_K);
564          Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
565          if  ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
566            throw new CardanEulerSingularityException(true);
567          }
568          return new double[] {
569            FastMath.atan2(-(v1.getY()), v1.getZ()),
570            FastMath.asin(v2.getZ()),
571            FastMath.atan2(-(v2.getY()), v2.getX())
572          };
573    
574        } else if (order == RotationOrder.XZY) {
575    
576          // r (Vector3D.plusJ) coordinates are :
577          // -sin (psi), cos (psi) cos (phi), cos (psi) sin (phi)
578          // (-r) (Vector3D.plusI) coordinates are :
579          // cos (theta) cos (psi), -sin (psi), sin (theta) cos (psi)
580          // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
581          Vector3D v1 = applyTo(Vector3D.PLUS_J);
582          Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
583          if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
584            throw new CardanEulerSingularityException(true);
585          }
586          return new double[] {
587            FastMath.atan2(v1.getZ(), v1.getY()),
588           -FastMath.asin(v2.getY()),
589            FastMath.atan2(v2.getZ(), v2.getX())
590          };
591    
592        } else if (order == RotationOrder.YXZ) {
593    
594          // r (Vector3D.plusK) coordinates are :
595          //  cos (phi) sin (theta), -sin (phi), cos (phi) cos (theta)
596          // (-r) (Vector3D.plusJ) coordinates are :
597          // sin (psi) cos (phi), cos (psi) cos (phi), -sin (phi)
598          // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
599          Vector3D v1 = applyTo(Vector3D.PLUS_K);
600          Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
601          if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
602            throw new CardanEulerSingularityException(true);
603          }
604          return new double[] {
605            FastMath.atan2(v1.getX(), v1.getZ()),
606           -FastMath.asin(v2.getZ()),
607            FastMath.atan2(v2.getX(), v2.getY())
608          };
609    
610        } else if (order == RotationOrder.YZX) {
611    
612          // r (Vector3D.plusI) coordinates are :
613          // cos (psi) cos (theta), sin (psi), -cos (psi) sin (theta)
614          // (-r) (Vector3D.plusJ) coordinates are :
615          // sin (psi), cos (phi) cos (psi), -sin (phi) cos (psi)
616          // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
617          Vector3D v1 = applyTo(Vector3D.PLUS_I);
618          Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
619          if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
620            throw new CardanEulerSingularityException(true);
621          }
622          return new double[] {
623            FastMath.atan2(-(v1.getZ()), v1.getX()),
624            FastMath.asin(v2.getX()),
625            FastMath.atan2(-(v2.getZ()), v2.getY())
626          };
627    
628        } else if (order == RotationOrder.ZXY) {
629    
630          // r (Vector3D.plusJ) coordinates are :
631          // -cos (phi) sin (psi), cos (phi) cos (psi), sin (phi)
632          // (-r) (Vector3D.plusK) coordinates are :
633          // -sin (theta) cos (phi), sin (phi), cos (theta) cos (phi)
634          // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
635          Vector3D v1 = applyTo(Vector3D.PLUS_J);
636          Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
637          if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
638            throw new CardanEulerSingularityException(true);
639          }
640          return new double[] {
641            FastMath.atan2(-(v1.getX()), v1.getY()),
642            FastMath.asin(v2.getY()),
643            FastMath.atan2(-(v2.getX()), v2.getZ())
644          };
645    
646        } else if (order == RotationOrder.ZYX) {
647    
648          // r (Vector3D.plusI) coordinates are :
649          //  cos (theta) cos (psi), cos (theta) sin (psi), -sin (theta)
650          // (-r) (Vector3D.plusK) coordinates are :
651          // -sin (theta), sin (phi) cos (theta), cos (phi) cos (theta)
652          // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
653          Vector3D v1 = applyTo(Vector3D.PLUS_I);
654          Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
655          if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
656            throw new CardanEulerSingularityException(true);
657          }
658          return new double[] {
659            FastMath.atan2(v1.getY(), v1.getX()),
660           -FastMath.asin(v2.getX()),
661            FastMath.atan2(v2.getY(), v2.getZ())
662          };
663    
664        } else if (order == RotationOrder.XYX) {
665    
666          // r (Vector3D.plusI) coordinates are :
667          //  cos (theta), sin (phi1) sin (theta), -cos (phi1) sin (theta)
668          // (-r) (Vector3D.plusI) coordinates are :
669          // cos (theta), sin (theta) sin (phi2), sin (theta) cos (phi2)
670          // and we can choose to have theta in the interval [0 ; PI]
671          Vector3D v1 = applyTo(Vector3D.PLUS_I);
672          Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
673          if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
674            throw new CardanEulerSingularityException(false);
675          }
676          return new double[] {
677            FastMath.atan2(v1.getY(), -v1.getZ()),
678            FastMath.acos(v2.getX()),
679            FastMath.atan2(v2.getY(), v2.getZ())
680          };
681    
682        } else if (order == RotationOrder.XZX) {
683    
684          // r (Vector3D.plusI) coordinates are :
685          //  cos (psi), cos (phi1) sin (psi), sin (phi1) sin (psi)
686          // (-r) (Vector3D.plusI) coordinates are :
687          // cos (psi), -sin (psi) cos (phi2), sin (psi) sin (phi2)
688          // and we can choose to have psi in the interval [0 ; PI]
689          Vector3D v1 = applyTo(Vector3D.PLUS_I);
690          Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
691          if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
692            throw new CardanEulerSingularityException(false);
693          }
694          return new double[] {
695            FastMath.atan2(v1.getZ(), v1.getY()),
696            FastMath.acos(v2.getX()),
697            FastMath.atan2(v2.getZ(), -v2.getY())
698          };
699    
700        } else if (order == RotationOrder.YXY) {
701    
702          // r (Vector3D.plusJ) coordinates are :
703          //  sin (theta1) sin (phi), cos (phi), cos (theta1) sin (phi)
704          // (-r) (Vector3D.plusJ) coordinates are :
705          // sin (phi) sin (theta2), cos (phi), -sin (phi) cos (theta2)
706          // and we can choose to have phi in the interval [0 ; PI]
707          Vector3D v1 = applyTo(Vector3D.PLUS_J);
708          Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
709          if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
710            throw new CardanEulerSingularityException(false);
711          }
712          return new double[] {
713            FastMath.atan2(v1.getX(), v1.getZ()),
714            FastMath.acos(v2.getY()),
715            FastMath.atan2(v2.getX(), -v2.getZ())
716          };
717    
718        } else if (order == RotationOrder.YZY) {
719    
720          // r (Vector3D.plusJ) coordinates are :
721          //  -cos (theta1) sin (psi), cos (psi), sin (theta1) sin (psi)
722          // (-r) (Vector3D.plusJ) coordinates are :
723          // sin (psi) cos (theta2), cos (psi), sin (psi) sin (theta2)
724          // and we can choose to have psi in the interval [0 ; PI]
725          Vector3D v1 = applyTo(Vector3D.PLUS_J);
726          Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
727          if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
728            throw new CardanEulerSingularityException(false);
729          }
730          return new double[] {
731            FastMath.atan2(v1.getZ(), -v1.getX()),
732            FastMath.acos(v2.getY()),
733            FastMath.atan2(v2.getZ(), v2.getX())
734          };
735    
736        } else if (order == RotationOrder.ZXZ) {
737    
738          // r (Vector3D.plusK) coordinates are :
739          //  sin (psi1) sin (phi), -cos (psi1) sin (phi), cos (phi)
740          // (-r) (Vector3D.plusK) coordinates are :
741          // sin (phi) sin (psi2), sin (phi) cos (psi2), cos (phi)
742          // and we can choose to have phi in the interval [0 ; PI]
743          Vector3D v1 = applyTo(Vector3D.PLUS_K);
744          Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
745          if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
746            throw new CardanEulerSingularityException(false);
747          }
748          return new double[] {
749            FastMath.atan2(v1.getX(), -v1.getY()),
750            FastMath.acos(v2.getZ()),
751            FastMath.atan2(v2.getX(), v2.getY())
752          };
753    
754        } else { // last possibility is ZYZ
755    
756          // r (Vector3D.plusK) coordinates are :
757          //  cos (psi1) sin (theta), sin (psi1) sin (theta), cos (theta)
758          // (-r) (Vector3D.plusK) coordinates are :
759          // -sin (theta) cos (psi2), sin (theta) sin (psi2), cos (theta)
760          // and we can choose to have theta in the interval [0 ; PI]
761          Vector3D v1 = applyTo(Vector3D.PLUS_K);
762          Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
763          if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
764            throw new CardanEulerSingularityException(false);
765          }
766          return new double[] {
767            FastMath.atan2(v1.getY(), v1.getX()),
768            FastMath.acos(v2.getZ()),
769            FastMath.atan2(v2.getY(), -v2.getX())
770          };
771    
772        }
773    
774      }
775    
776      /** Get the 3X3 matrix corresponding to the instance
777       * @return the matrix corresponding to the instance
778       */
779      public double[][] getMatrix() {
780    
781        // products
782        double q0q0  = q0 * q0;
783        double q0q1  = q0 * q1;
784        double q0q2  = q0 * q2;
785        double q0q3  = q0 * q3;
786        double q1q1  = q1 * q1;
787        double q1q2  = q1 * q2;
788        double q1q3  = q1 * q3;
789        double q2q2  = q2 * q2;
790        double q2q3  = q2 * q3;
791        double q3q3  = q3 * q3;
792    
793        // create the matrix
794        double[][] m = new double[3][];
795        m[0] = new double[3];
796        m[1] = new double[3];
797        m[2] = new double[3];
798    
799        m [0][0] = 2.0 * (q0q0 + q1q1) - 1.0;
800        m [1][0] = 2.0 * (q1q2 - q0q3);
801        m [2][0] = 2.0 * (q1q3 + q0q2);
802    
803        m [0][1] = 2.0 * (q1q2 + q0q3);
804        m [1][1] = 2.0 * (q0q0 + q2q2) - 1.0;
805        m [2][1] = 2.0 * (q2q3 - q0q1);
806    
807        m [0][2] = 2.0 * (q1q3 - q0q2);
808        m [1][2] = 2.0 * (q2q3 + q0q1);
809        m [2][2] = 2.0 * (q0q0 + q3q3) - 1.0;
810    
811        return m;
812    
813      }
814    
815      /** Apply the rotation to a vector.
816       * @param u vector to apply the rotation to
817       * @return a new vector which is the image of u by the rotation
818       */
819      public Vector3D applyTo(Vector3D u) {
820    
821        double x = u.getX();
822        double y = u.getY();
823        double z = u.getZ();
824    
825        double s = q1 * x + q2 * y + q3 * z;
826    
827        return new Vector3D(2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x,
828                            2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y,
829                            2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - z);
830    
831      }
832    
833      /** Apply the rotation to a vector stored in an array.
834       * @param in an array with three items which stores vector to rotate
835       * @param out an array with three items to put result to (it can be the same
836       * array as in)
837       */
838      public void applyTo(final double[] in, final double[] out) {
839    
840          final double x = in[0];
841          final double y = in[1];
842          final double z = in[2];
843    
844          final double s = q1 * x + q2 * y + q3 * z;
845    
846          out[0] = 2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x;
847          out[1] = 2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y;
848          out[2] = 2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - z;
849    
850      }
851    
852      /** Apply the inverse of the rotation to a vector.
853       * @param u vector to apply the inverse of the rotation to
854       * @return a new vector which such that u is its image by the rotation
855       */
856      public Vector3D applyInverseTo(Vector3D u) {
857    
858        double x = u.getX();
859        double y = u.getY();
860        double z = u.getZ();
861    
862        double s = q1 * x + q2 * y + q3 * z;
863        double m0 = -q0;
864    
865        return new Vector3D(2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x,
866                            2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y,
867                            2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - z);
868    
869      }
870    
871      /** Apply the inverse of the rotation to a vector stored in an array.
872       * @param in an array with three items which stores vector to rotate
873       * @param out an array with three items to put result to (it can be the same
874       * array as in)
875       */
876      public void applyInverseTo(final double[] in, final double[] out) {
877    
878          final double x = in[0];
879          final double y = in[1];
880          final double z = in[2];
881    
882          final double s = q1 * x + q2 * y + q3 * z;
883          final double m0 = -q0;
884    
885          out[0] = 2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x;
886          out[1] = 2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y;
887          out[2] = 2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - z;
888    
889      }
890    
891      /** Apply the instance to another rotation.
892       * Applying the instance to a rotation is computing the composition
893       * in an order compliant with the following rule : let u be any
894       * vector and v its image by r (i.e. r.applyTo(u) = v), let w be the image
895       * of v by the instance (i.e. applyTo(v) = w), then w = comp.applyTo(u),
896       * where comp = applyTo(r).
897       * @param r rotation to apply the rotation to
898       * @return a new rotation which is the composition of r by the instance
899       */
900      public Rotation applyTo(Rotation r) {
901        return new Rotation(r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),
902                            r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),
903                            r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),
904                            r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),
905                            false);
906      }
907    
908      /** Apply the inverse of the instance to another rotation.
909       * Applying the inverse of the instance to a rotation is computing
910       * the composition in an order compliant with the following rule :
911       * let u be any vector and v its image by r (i.e. r.applyTo(u) = v),
912       * let w be the inverse image of v by the instance
913       * (i.e. applyInverseTo(v) = w), then w = comp.applyTo(u), where
914       * comp = applyInverseTo(r).
915       * @param r rotation to apply the rotation to
916       * @return a new rotation which is the composition of r by the inverse
917       * of the instance
918       */
919      public Rotation applyInverseTo(Rotation r) {
920        return new Rotation(-r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),
921                            -r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),
922                            -r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),
923                            -r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),
924                            false);
925      }
926    
927      /** Perfect orthogonality on a 3X3 matrix.
928       * @param m initial matrix (not exactly orthogonal)
929       * @param threshold convergence threshold for the iterative
930       * orthogonality correction (convergence is reached when the
931       * difference between two steps of the Frobenius norm of the
932       * correction is below this threshold)
933       * @return an orthogonal matrix close to m
934       * @exception NotARotationMatrixException if the matrix cannot be
935       * orthogonalized with the given threshold after 10 iterations
936       */
937      private double[][] orthogonalizeMatrix(double[][] m, double threshold)
938        throws NotARotationMatrixException {
939        double[] m0 = m[0];
940        double[] m1 = m[1];
941        double[] m2 = m[2];
942        double x00 = m0[0];
943        double x01 = m0[1];
944        double x02 = m0[2];
945        double x10 = m1[0];
946        double x11 = m1[1];
947        double x12 = m1[2];
948        double x20 = m2[0];
949        double x21 = m2[1];
950        double x22 = m2[2];
951        double fn = 0;
952        double fn1;
953    
954        double[][] o = new double[3][3];
955        double[] o0 = o[0];
956        double[] o1 = o[1];
957        double[] o2 = o[2];
958    
959        // iterative correction: Xn+1 = Xn - 0.5 * (Xn.Mt.Xn - M)
960        int i = 0;
961        while (++i < 11) {
962    
963          // Mt.Xn
964          double mx00 = m0[0] * x00 + m1[0] * x10 + m2[0] * x20;
965          double mx10 = m0[1] * x00 + m1[1] * x10 + m2[1] * x20;
966          double mx20 = m0[2] * x00 + m1[2] * x10 + m2[2] * x20;
967          double mx01 = m0[0] * x01 + m1[0] * x11 + m2[0] * x21;
968          double mx11 = m0[1] * x01 + m1[1] * x11 + m2[1] * x21;
969          double mx21 = m0[2] * x01 + m1[2] * x11 + m2[2] * x21;
970          double mx02 = m0[0] * x02 + m1[0] * x12 + m2[0] * x22;
971          double mx12 = m0[1] * x02 + m1[1] * x12 + m2[1] * x22;
972          double mx22 = m0[2] * x02 + m1[2] * x12 + m2[2] * x22;
973    
974          // Xn+1
975          o0[0] = x00 - 0.5 * (x00 * mx00 + x01 * mx10 + x02 * mx20 - m0[0]);
976          o0[1] = x01 - 0.5 * (x00 * mx01 + x01 * mx11 + x02 * mx21 - m0[1]);
977          o0[2] = x02 - 0.5 * (x00 * mx02 + x01 * mx12 + x02 * mx22 - m0[2]);
978          o1[0] = x10 - 0.5 * (x10 * mx00 + x11 * mx10 + x12 * mx20 - m1[0]);
979          o1[1] = x11 - 0.5 * (x10 * mx01 + x11 * mx11 + x12 * mx21 - m1[1]);
980          o1[2] = x12 - 0.5 * (x10 * mx02 + x11 * mx12 + x12 * mx22 - m1[2]);
981          o2[0] = x20 - 0.5 * (x20 * mx00 + x21 * mx10 + x22 * mx20 - m2[0]);
982          o2[1] = x21 - 0.5 * (x20 * mx01 + x21 * mx11 + x22 * mx21 - m2[1]);
983          o2[2] = x22 - 0.5 * (x20 * mx02 + x21 * mx12 + x22 * mx22 - m2[2]);
984    
985          // correction on each elements
986          double corr00 = o0[0] - m0[0];
987          double corr01 = o0[1] - m0[1];
988          double corr02 = o0[2] - m0[2];
989          double corr10 = o1[0] - m1[0];
990          double corr11 = o1[1] - m1[1];
991          double corr12 = o1[2] - m1[2];
992          double corr20 = o2[0] - m2[0];
993          double corr21 = o2[1] - m2[1];
994          double corr22 = o2[2] - m2[2];
995    
996          // Frobenius norm of the correction
997          fn1 = corr00 * corr00 + corr01 * corr01 + corr02 * corr02 +
998                corr10 * corr10 + corr11 * corr11 + corr12 * corr12 +
999                corr20 * corr20 + corr21 * corr21 + corr22 * corr22;
1000    
1001          // convergence test
1002          if (FastMath.abs(fn1 - fn) <= threshold) {
1003              return o;
1004          }
1005    
1006          // prepare next iteration
1007          x00 = o0[0];
1008          x01 = o0[1];
1009          x02 = o0[2];
1010          x10 = o1[0];
1011          x11 = o1[1];
1012          x12 = o1[2];
1013          x20 = o2[0];
1014          x21 = o2[1];
1015          x22 = o2[2];
1016          fn  = fn1;
1017    
1018        }
1019    
1020        // the algorithm did not converge after 10 iterations
1021        throw new NotARotationMatrixException(
1022                LocalizedFormats.UNABLE_TO_ORTHOGONOLIZE_MATRIX,
1023                i - 1);
1024      }
1025    
1026      /** Compute the <i>distance</i> between two rotations.
1027       * <p>The <i>distance</i> is intended here as a way to check if two
1028       * rotations are almost similar (i.e. they transform vectors the same way)
1029       * or very different. It is mathematically defined as the angle of
1030       * the rotation r that prepended to one of the rotations gives the other
1031       * one:</p>
1032       * <pre>
1033       *        r<sub>1</sub>(r) = r<sub>2</sub>
1034       * </pre>
1035       * <p>This distance is an angle between 0 and &pi;. Its value is the smallest
1036       * possible upper bound of the angle in radians between r<sub>1</sub>(v)
1037       * and r<sub>2</sub>(v) for all possible vectors v. This upper bound is
1038       * reached for some v. The distance is equal to 0 if and only if the two
1039       * rotations are identical.</p>
1040       * <p>Comparing two rotations should always be done using this value rather
1041       * than for example comparing the components of the quaternions. It is much
1042       * more stable, and has a geometric meaning. Also comparing quaternions
1043       * components is error prone since for example quaternions (0.36, 0.48, -0.48, -0.64)
1044       * and (-0.36, -0.48, 0.48, 0.64) represent exactly the same rotation despite
1045       * their components are different (they are exact opposites).</p>
1046       * @param r1 first rotation
1047       * @param r2 second rotation
1048       * @return <i>distance</i> between r1 and r2
1049       */
1050      public static double distance(Rotation r1, Rotation r2) {
1051          return r1.applyInverseTo(r2).getAngle();
1052      }
1053    
1054    }