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.math.geometry;
019    
020    import java.io.Serializable;
021    
022    import org.apache.commons.math.MathRuntimeException;
023    import org.apache.commons.math.exception.util.LocalizedFormats;
024    import org.apache.commons.math.util.FastMath;
025    
026    /**
027     * This class implements rotations in a three-dimensional space.
028     *
029     * <p>Rotations can be represented by several different mathematical
030     * entities (matrices, axe and angle, Cardan or Euler angles,
031     * quaternions). This class presents an higher level abstraction, more
032     * user-oriented and hiding this implementation details. Well, for the
033     * curious, we use quaternions for the internal representation. The
034     * user can build a rotation from any of these representations, and
035     * any of these representations can be retrieved from a
036     * <code>Rotation</code> instance (see the various constructors and
037     * getters). In addition, a rotation can also be built implicitly
038     * from a set of vectors and their image.</p>
039     * <p>This implies that this class can be used to convert from one
040     * representation to another one. For example, converting a rotation
041     * matrix into a set of Cardan angles from can be done using the
042     * following single line of code:</p>
043     * <pre>
044     * double[] angles = new Rotation(matrix, 1.0e-10).getAngles(RotationOrder.XYZ);
045     * </pre>
046     * <p>Focus is oriented on what a rotation <em>do</em> rather than on its
047     * underlying representation. Once it has been built, and regardless of its
048     * internal representation, a rotation is an <em>operator</em> which basically
049     * transforms three dimensional {@link Vector3D vectors} into other three
050     * dimensional {@link Vector3D vectors}. Depending on the application, the
051     * meaning of these vectors may vary and the semantics of the rotation also.</p>
052     * <p>For example in an spacecraft attitude simulation tool, users will often
053     * consider the vectors are fixed (say the Earth direction for example) and the
054     * frames change. The rotation transforms the coordinates of the vector in inertial
055     * frame into the coordinates of the same vector in satellite frame. In this
056     * case, the rotation implicitly defines the relation between the two frames.</p>
057     * <p>Another example could be a telescope control application, where the rotation
058     * would transform the sighting direction at rest into the desired observing
059     * direction when the telescope is pointed towards an object of interest. In this
060     * case the rotation transforms the direction at rest in a topocentric frame
061     * into the sighting direction in the same topocentric frame. This implies in this
062     * case the frame is fixed and the vector moves.</p>
063     * <p>In many case, both approaches will be combined. In our telescope example,
064     * we will probably also need to transform the observing direction in the topocentric
065     * frame into the observing direction in inertial frame taking into account the observatory
066     * location and the Earth rotation, which would essentially be an application of the
067     * first approach.</p>
068     *
069     * <p>These examples show that a rotation is what the user wants it to be. This
070     * class does not push the user towards one specific definition and hence does not
071     * provide methods like <code>projectVectorIntoDestinationFrame</code> or
072     * <code>computeTransformedDirection</code>. It provides simpler and more generic
073     * methods: {@link #applyTo(Vector3D) applyTo(Vector3D)} and {@link
074     * #applyInverseTo(Vector3D) applyInverseTo(Vector3D)}.</p>
075     *
076     * <p>Since a rotation is basically a vectorial operator, several rotations can be
077     * composed together and the composite operation <code>r = r<sub>1</sub> o
078     * r<sub>2</sub></code> (which means that for each vector <code>u</code>,
079     * <code>r(u) = r<sub>1</sub>(r<sub>2</sub>(u))</code>) is also a rotation. Hence
080     * we can consider that in addition to vectors, a rotation can be applied to other
081     * rotations as well (or to itself). With our previous notations, we would say we
082     * can apply <code>r<sub>1</sub></code> to <code>r<sub>2</sub></code> and the result
083     * we get is <code>r = r<sub>1</sub> o r<sub>2</sub></code>. For this purpose, the
084     * class provides the methods: {@link #applyTo(Rotation) applyTo(Rotation)} and
085     * {@link #applyInverseTo(Rotation) applyInverseTo(Rotation)}.</p>
086     *
087     * <p>Rotations are guaranteed to be immutable objects.</p>
088     *
089     * @version $Revision: 1067500 $ $Date: 2011-02-05 21:11:30 +0100 (sam. 05 f??vr. 2011) $
090     * @see Vector3D
091     * @see RotationOrder
092     * @since 1.2
093     */
094    
095    public class Rotation implements Serializable {
096    
097      /** Identity rotation. */
098      public static final Rotation IDENTITY = new Rotation(1.0, 0.0, 0.0, 0.0, false);
099    
100      /** Serializable version identifier */
101      private static final long serialVersionUID = -2153622329907944313L;
102    
103      /** Scalar coordinate of the quaternion. */
104      private final double q0;
105    
106      /** First coordinate of the vectorial part of the quaternion. */
107      private final double q1;
108    
109      /** Second coordinate of the vectorial part of the quaternion. */
110      private final double q2;
111    
112      /** Third coordinate of the vectorial part of the quaternion. */
113      private final double q3;
114    
115      /** Build a rotation from the quaternion coordinates.
116       * <p>A rotation can be built from a <em>normalized</em> quaternion,
117       * i.e. a quaternion for which q<sub>0</sub><sup>2</sup> +
118       * q<sub>1</sub><sup>2</sup> + q<sub>2</sub><sup>2</sup> +
119       * q<sub>3</sub><sup>2</sup> = 1. If the quaternion is not normalized,
120       * the constructor can normalize it in a preprocessing step.</p>
121       * <p>Note that some conventions put the scalar part of the quaternion
122       * as the 4<sup>th</sup> component and the vector part as the first three
123       * components. This is <em>not</em> our convention. We put the scalar part
124       * as the first component.</p>
125       * @param q0 scalar part of the quaternion
126       * @param q1 first coordinate of the vectorial part of the quaternion
127       * @param q2 second coordinate of the vectorial part of the quaternion
128       * @param q3 third coordinate of the vectorial part of the quaternion
129       * @param needsNormalization if true, the coordinates are considered
130       * not to be normalized, a normalization preprocessing step is performed
131       * before using them
132       */
133      public Rotation(double q0, double q1, double q2, double q3,
134                      boolean needsNormalization) {
135    
136        if (needsNormalization) {
137          // normalization preprocessing
138          double inv = 1.0 / FastMath.sqrt(q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3);
139          q0 *= inv;
140          q1 *= inv;
141          q2 *= inv;
142          q3 *= inv;
143        }
144    
145        this.q0 = q0;
146        this.q1 = q1;
147        this.q2 = q2;
148        this.q3 = q3;
149    
150      }
151    
152      /** Build a rotation from an axis and an angle.
153       * <p>We use the convention that angles are oriented according to
154       * the effect of the rotation on vectors around the axis. That means
155       * that if (i, j, k) is a direct frame and if we first provide +k as
156       * the axis and &pi;/2 as the angle to this constructor, and then
157       * {@link #applyTo(Vector3D) apply} the instance to +i, we will get
158       * +j.</p>
159       * <p>Another way to represent our convention is to say that a rotation
160       * of angle &theta; about the unit vector (x, y, z) is the same as the
161       * rotation build from quaternion components { cos(-&theta;/2),
162       * x * sin(-&theta;/2), y * sin(-&theta;/2), z * sin(-&theta;/2) }.
163       * Note the minus sign on the angle!</p>
164       * <p>On the one hand this convention is consistent with a vectorial
165       * perspective (moving vectors in fixed frames), on the other hand it
166       * is different from conventions with a frame perspective (fixed vectors
167       * viewed from different frames) like the ones used for example in spacecraft
168       * attitude community or in the graphics community.</p>
169       * @param axis axis around which to rotate
170       * @param angle rotation angle.
171       * @exception ArithmeticException if the axis norm is zero
172       */
173      public Rotation(Vector3D axis, double angle) {
174    
175        double norm = axis.getNorm();
176        if (norm == 0) {
177          throw MathRuntimeException.createArithmeticException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_AXIS);
178        }
179    
180        double halfAngle = -0.5 * angle;
181        double coeff = FastMath.sin(halfAngle) / norm;
182    
183        q0 = FastMath.cos (halfAngle);
184        q1 = coeff * axis.getX();
185        q2 = coeff * axis.getY();
186        q3 = coeff * axis.getZ();
187    
188      }
189    
190      /** Build a rotation from a 3X3 matrix.
191    
192       * <p>Rotation matrices are orthogonal matrices, i.e. unit matrices
193       * (which are matrices for which m.m<sup>T</sup> = I) with real
194       * coefficients. The module of the determinant of unit matrices is
195       * 1, among the orthogonal 3X3 matrices, only the ones having a
196       * positive determinant (+1) are rotation matrices.</p>
197       *
198       * <p>When a rotation is defined by a matrix with truncated values
199       * (typically when it is extracted from a technical sheet where only
200       * four to five significant digits are available), the matrix is not
201       * orthogonal anymore. This constructor handles this case
202       * transparently by using a copy of the given matrix and applying a
203       * correction to the copy in order to perfect its orthogonality. If
204       * the Frobenius norm of the correction needed is above the given
205       * threshold, then the matrix is considered to be too far from a
206       * true rotation matrix and an exception is thrown.<p>
207       *
208       * @param m rotation matrix
209       * @param threshold convergence threshold for the iterative
210       * orthogonality correction (convergence is reached when the
211       * difference between two steps of the Frobenius norm of the
212       * correction is below this threshold)
213       *
214       * @exception NotARotationMatrixException if the matrix is not a 3X3
215       * matrix, or if it cannot be transformed into an orthogonal matrix
216       * with the given threshold, or if the determinant of the resulting
217       * orthogonal matrix is negative
218       *
219       */
220      public Rotation(double[][] m, double threshold)
221        throws NotARotationMatrixException {
222    
223        // dimension check
224        if ((m.length != 3) || (m[0].length != 3) ||
225            (m[1].length != 3) || (m[2].length != 3)) {
226          throw new NotARotationMatrixException(
227                  LocalizedFormats.ROTATION_MATRIX_DIMENSIONS,
228                  m.length, m[0].length);
229        }
230    
231        // compute a "close" orthogonal matrix
232        double[][] ort = orthogonalizeMatrix(m, threshold);
233    
234        // check the sign of the determinant
235        double det = ort[0][0] * (ort[1][1] * ort[2][2] - ort[2][1] * ort[1][2]) -
236                     ort[1][0] * (ort[0][1] * ort[2][2] - ort[2][1] * ort[0][2]) +
237                     ort[2][0] * (ort[0][1] * ort[1][2] - ort[1][1] * ort[0][2]);
238        if (det < 0.0) {
239          throw new NotARotationMatrixException(
240                  LocalizedFormats.CLOSEST_ORTHOGONAL_MATRIX_HAS_NEGATIVE_DETERMINANT,
241                  det);
242        }
243    
244        // There are different ways to compute the quaternions elements
245        // from the matrix. They all involve computing one element from
246        // the diagonal of the matrix, and computing the three other ones
247        // using a formula involving a division by the first element,
248        // which unfortunately can be zero. Since the norm of the
249        // quaternion is 1, we know at least one element has an absolute
250        // value greater or equal to 0.5, so it is always possible to
251        // select the right formula and avoid division by zero and even
252        // numerical inaccuracy. Checking the elements in turn and using
253        // the first one greater than 0.45 is safe (this leads to a simple
254        // test since qi = 0.45 implies 4 qi^2 - 1 = -0.19)
255        double s = ort[0][0] + ort[1][1] + ort[2][2];
256        if (s > -0.19) {
257          // compute q0 and deduce q1, q2 and q3
258          q0 = 0.5 * FastMath.sqrt(s + 1.0);
259          double inv = 0.25 / q0;
260          q1 = inv * (ort[1][2] - ort[2][1]);
261          q2 = inv * (ort[2][0] - ort[0][2]);
262          q3 = inv * (ort[0][1] - ort[1][0]);
263        } else {
264          s = ort[0][0] - ort[1][1] - ort[2][2];
265          if (s > -0.19) {
266            // compute q1 and deduce q0, q2 and q3
267            q1 = 0.5 * FastMath.sqrt(s + 1.0);
268            double inv = 0.25 / q1;
269            q0 = inv * (ort[1][2] - ort[2][1]);
270            q2 = inv * (ort[0][1] + ort[1][0]);
271            q3 = inv * (ort[0][2] + ort[2][0]);
272          } else {
273            s = ort[1][1] - ort[0][0] - ort[2][2];
274            if (s > -0.19) {
275              // compute q2 and deduce q0, q1 and q3
276              q2 = 0.5 * FastMath.sqrt(s + 1.0);
277              double inv = 0.25 / q2;
278              q0 = inv * (ort[2][0] - ort[0][2]);
279              q1 = inv * (ort[0][1] + ort[1][0]);
280              q3 = inv * (ort[2][1] + ort[1][2]);
281            } else {
282              // compute q3 and deduce q0, q1 and q2
283              s = ort[2][2] - ort[0][0] - ort[1][1];
284              q3 = 0.5 * FastMath.sqrt(s + 1.0);
285              double inv = 0.25 / q3;
286              q0 = inv * (ort[0][1] - ort[1][0]);
287              q1 = inv * (ort[0][2] + ort[2][0]);
288              q2 = inv * (ort[2][1] + ort[1][2]);
289            }
290          }
291        }
292    
293      }
294    
295      /** Build the rotation that transforms a pair of vector into another pair.
296    
297       * <p>Except for possible scale factors, if the instance were applied to
298       * the pair (u<sub>1</sub>, u<sub>2</sub>) it will produce the pair
299       * (v<sub>1</sub>, v<sub>2</sub>).</p>
300       *
301       * <p>If the angular separation between u<sub>1</sub> and u<sub>2</sub> is
302       * not the same as the angular separation between v<sub>1</sub> and
303       * v<sub>2</sub>, then a corrected v'<sub>2</sub> will be used rather than
304       * v<sub>2</sub>, the corrected vector will be in the (v<sub>1</sub>,
305       * v<sub>2</sub>) plane.</p>
306       *
307       * @param u1 first vector of the origin pair
308       * @param u2 second vector of the origin pair
309       * @param v1 desired image of u1 by the rotation
310       * @param v2 desired image of u2 by the rotation
311       * @exception IllegalArgumentException if the norm of one of the vectors is zero
312       */
313      public Rotation(Vector3D u1, Vector3D u2, Vector3D v1, Vector3D v2) {
314    
315      // norms computation
316      double u1u1 = Vector3D.dotProduct(u1, u1);
317      double u2u2 = Vector3D.dotProduct(u2, u2);
318      double v1v1 = Vector3D.dotProduct(v1, v1);
319      double v2v2 = Vector3D.dotProduct(v2, v2);
320      if ((u1u1 == 0) || (u2u2 == 0) || (v1v1 == 0) || (v2v2 == 0)) {
321        throw MathRuntimeException.createIllegalArgumentException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_DEFINING_VECTOR);
322      }
323    
324      double u1x = u1.getX();
325      double u1y = u1.getY();
326      double u1z = u1.getZ();
327    
328      double u2x = u2.getX();
329      double u2y = u2.getY();
330      double u2z = u2.getZ();
331    
332      // normalize v1 in order to have (v1'|v1') = (u1|u1)
333      double coeff = FastMath.sqrt (u1u1 / v1v1);
334      double v1x   = coeff * v1.getX();
335      double v1y   = coeff * v1.getY();
336      double v1z   = coeff * v1.getZ();
337      v1 = new Vector3D(v1x, v1y, v1z);
338    
339      // adjust v2 in order to have (u1|u2) = (v1|v2) and (v2'|v2') = (u2|u2)
340      double u1u2   = Vector3D.dotProduct(u1, u2);
341      double v1v2   = Vector3D.dotProduct(v1, v2);
342      double coeffU = u1u2 / u1u1;
343      double coeffV = v1v2 / u1u1;
344      double beta   = FastMath.sqrt((u2u2 - u1u2 * coeffU) / (v2v2 - v1v2 * coeffV));
345      double alpha  = coeffU - beta * coeffV;
346      double v2x    = alpha * v1x + beta * v2.getX();
347      double v2y    = alpha * v1y + beta * v2.getY();
348      double v2z    = alpha * v1z + beta * v2.getZ();
349      v2 = new Vector3D(v2x, v2y, v2z);
350    
351      // preliminary computation (we use explicit formulation instead
352      // of relying on the Vector3D class in order to avoid building lots
353      // of temporary objects)
354      Vector3D uRef = u1;
355      Vector3D vRef = v1;
356      double dx1 = v1x - u1.getX();
357      double dy1 = v1y - u1.getY();
358      double dz1 = v1z - u1.getZ();
359      double dx2 = v2x - u2.getX();
360      double dy2 = v2y - u2.getY();
361      double dz2 = v2z - u2.getZ();
362      Vector3D k = new Vector3D(dy1 * dz2 - dz1 * dy2,
363                                dz1 * dx2 - dx1 * dz2,
364                                dx1 * dy2 - dy1 * dx2);
365      double c = k.getX() * (u1y * u2z - u1z * u2y) +
366                 k.getY() * (u1z * u2x - u1x * u2z) +
367                 k.getZ() * (u1x * u2y - u1y * u2x);
368    
369      if (c == 0) {
370        // the (q1, q2, q3) vector is in the (u1, u2) plane
371        // we try other vectors
372        Vector3D u3 = Vector3D.crossProduct(u1, u2);
373        Vector3D v3 = Vector3D.crossProduct(v1, v2);
374        double u3x  = u3.getX();
375        double u3y  = u3.getY();
376        double u3z  = u3.getZ();
377        double v3x  = v3.getX();
378        double v3y  = v3.getY();
379        double v3z  = v3.getZ();
380    
381        double dx3 = v3x - u3x;
382        double dy3 = v3y - u3y;
383        double dz3 = v3z - u3z;
384        k = new Vector3D(dy1 * dz3 - dz1 * dy3,
385                         dz1 * dx3 - dx1 * dz3,
386                         dx1 * dy3 - dy1 * dx3);
387        c = k.getX() * (u1y * u3z - u1z * u3y) +
388            k.getY() * (u1z * u3x - u1x * u3z) +
389            k.getZ() * (u1x * u3y - u1y * u3x);
390    
391        if (c == 0) {
392          // the (q1, q2, q3) vector is aligned with u1:
393          // we try (u2, u3) and (v2, v3)
394          k = new Vector3D(dy2 * dz3 - dz2 * dy3,
395                           dz2 * dx3 - dx2 * dz3,
396                           dx2 * dy3 - dy2 * dx3);
397          c = k.getX() * (u2y * u3z - u2z * u3y) +
398              k.getY() * (u2z * u3x - u2x * u3z) +
399              k.getZ() * (u2x * u3y - u2y * u3x);
400    
401          if (c == 0) {
402            // the (q1, q2, q3) vector is aligned with everything
403            // this is really the identity rotation
404            q0 = 1.0;
405            q1 = 0.0;
406            q2 = 0.0;
407            q3 = 0.0;
408            return;
409          }
410    
411          // we will have to use u2 and v2 to compute the scalar part
412          uRef = u2;
413          vRef = v2;
414    
415        }
416    
417      }
418    
419      // compute the vectorial part
420      c = FastMath.sqrt(c);
421      double inv = 1.0 / (c + c);
422      q1 = inv * k.getX();
423      q2 = inv * k.getY();
424      q3 = inv * k.getZ();
425    
426      // compute the scalar part
427       k = new Vector3D(uRef.getY() * q3 - uRef.getZ() * q2,
428                        uRef.getZ() * q1 - uRef.getX() * q3,
429                        uRef.getX() * q2 - uRef.getY() * q1);
430       c = Vector3D.dotProduct(k, k);
431      q0 = Vector3D.dotProduct(vRef, k) / (c + c);
432    
433      }
434    
435      /** Build one of the rotations that transform one vector into another one.
436    
437       * <p>Except for a possible scale factor, if the instance were
438       * applied to the vector u it will produce the vector v. There is an
439       * infinite number of such rotations, this constructor choose the
440       * one with the smallest associated angle (i.e. the one whose axis
441       * is orthogonal to the (u, v) plane). If u and v are colinear, an
442       * arbitrary rotation axis is chosen.</p>
443       *
444       * @param u origin vector
445       * @param v desired image of u by the rotation
446       * @exception IllegalArgumentException if the norm of one of the vectors is zero
447       */
448      public Rotation(Vector3D u, Vector3D v) {
449    
450        double normProduct = u.getNorm() * v.getNorm();
451        if (normProduct == 0) {
452            throw MathRuntimeException.createIllegalArgumentException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_DEFINING_VECTOR);
453        }
454    
455        double dot = Vector3D.dotProduct(u, v);
456    
457        if (dot < ((2.0e-15 - 1.0) * normProduct)) {
458          // special case u = -v: we select a PI angle rotation around
459          // an arbitrary vector orthogonal to u
460          Vector3D w = u.orthogonal();
461          q0 = 0.0;
462          q1 = -w.getX();
463          q2 = -w.getY();
464          q3 = -w.getZ();
465        } else {
466          // general case: (u, v) defines a plane, we select
467          // the shortest possible rotation: axis orthogonal to this plane
468          q0 = FastMath.sqrt(0.5 * (1.0 + dot / normProduct));
469          double coeff = 1.0 / (2.0 * q0 * normProduct);
470          q1 = coeff * (v.getY() * u.getZ() - v.getZ() * u.getY());
471          q2 = coeff * (v.getZ() * u.getX() - v.getX() * u.getZ());
472          q3 = coeff * (v.getX() * u.getY() - v.getY() * u.getX());
473        }
474    
475      }
476    
477      /** Build a rotation from three Cardan or Euler elementary rotations.
478    
479       * <p>Cardan rotations are three successive rotations around the
480       * canonical axes X, Y and Z, each axis being used once. There are
481       * 6 such sets of rotations (XYZ, XZY, YXZ, YZX, ZXY and ZYX). Euler
482       * rotations are three successive rotations around the canonical
483       * axes X, Y and Z, the first and last rotations being around the
484       * same axis. There are 6 such sets of rotations (XYX, XZX, YXY,
485       * YZY, ZXZ and ZYZ), the most popular one being ZXZ.</p>
486       * <p>Beware that many people routinely use the term Euler angles even
487       * for what really are Cardan angles (this confusion is especially
488       * widespread in the aerospace business where Roll, Pitch and Yaw angles
489       * are often wrongly tagged as Euler angles).</p>
490       *
491       * @param order order of rotations to use
492       * @param alpha1 angle of the first elementary rotation
493       * @param alpha2 angle of the second elementary rotation
494       * @param alpha3 angle of the third elementary rotation
495       */
496      public Rotation(RotationOrder order,
497                      double alpha1, double alpha2, double alpha3) {
498        Rotation r1 = new Rotation(order.getA1(), alpha1);
499        Rotation r2 = new Rotation(order.getA2(), alpha2);
500        Rotation r3 = new Rotation(order.getA3(), alpha3);
501        Rotation composed = r1.applyTo(r2.applyTo(r3));
502        q0 = composed.q0;
503        q1 = composed.q1;
504        q2 = composed.q2;
505        q3 = composed.q3;
506      }
507    
508      /** Revert a rotation.
509       * Build a rotation which reverse the effect of another
510       * rotation. This means that if r(u) = v, then r.revert(v) = u. The
511       * instance is not changed.
512       * @return a new rotation whose effect is the reverse of the effect
513       * of the instance
514       */
515      public Rotation revert() {
516        return new Rotation(-q0, q1, q2, q3, false);
517      }
518    
519      /** Get the scalar coordinate of the quaternion.
520       * @return scalar coordinate of the quaternion
521       */
522      public double getQ0() {
523        return q0;
524      }
525    
526      /** Get the first coordinate of the vectorial part of the quaternion.
527       * @return first coordinate of the vectorial part of the quaternion
528       */
529      public double getQ1() {
530        return q1;
531      }
532    
533      /** Get the second coordinate of the vectorial part of the quaternion.
534       * @return second coordinate of the vectorial part of the quaternion
535       */
536      public double getQ2() {
537        return q2;
538      }
539    
540      /** Get the third coordinate of the vectorial part of the quaternion.
541       * @return third coordinate of the vectorial part of the quaternion
542       */
543      public double getQ3() {
544        return q3;
545      }
546    
547      /** Get the normalized axis of the rotation.
548       * @return normalized axis of the rotation
549       * @see #Rotation(Vector3D, double)
550       */
551      public Vector3D getAxis() {
552        double squaredSine = q1 * q1 + q2 * q2 + q3 * q3;
553        if (squaredSine == 0) {
554          return new Vector3D(1, 0, 0);
555        } else if (q0 < 0) {
556          double inverse = 1 / FastMath.sqrt(squaredSine);
557          return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
558        }
559        double inverse = -1 / FastMath.sqrt(squaredSine);
560        return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
561      }
562    
563      /** Get the angle of the rotation.
564       * @return angle of the rotation (between 0 and &pi;)
565       * @see #Rotation(Vector3D, double)
566       */
567      public double getAngle() {
568        if ((q0 < -0.1) || (q0 > 0.1)) {
569          return 2 * FastMath.asin(FastMath.sqrt(q1 * q1 + q2 * q2 + q3 * q3));
570        } else if (q0 < 0) {
571          return 2 * FastMath.acos(-q0);
572        }
573        return 2 * FastMath.acos(q0);
574      }
575    
576      /** Get the Cardan or Euler angles corresponding to the instance.
577    
578       * <p>The equations show that each rotation can be defined by two
579       * different values of the Cardan or Euler angles set. For example
580       * if Cardan angles are used, the rotation defined by the angles
581       * a<sub>1</sub>, a<sub>2</sub> and a<sub>3</sub> is the same as
582       * the rotation defined by the angles &pi; + a<sub>1</sub>, &pi;
583       * - a<sub>2</sub> and &pi; + a<sub>3</sub>. This method implements
584       * the following arbitrary choices:</p>
585       * <ul>
586       *   <li>for Cardan angles, the chosen set is the one for which the
587       *   second angle is between -&pi;/2 and &pi;/2 (i.e its cosine is
588       *   positive),</li>
589       *   <li>for Euler angles, the chosen set is the one for which the
590       *   second angle is between 0 and &pi; (i.e its sine is positive).</li>
591       * </ul>
592       *
593       * <p>Cardan and Euler angle have a very disappointing drawback: all
594       * of them have singularities. This means that if the instance is
595       * too close to the singularities corresponding to the given
596       * rotation order, it will be impossible to retrieve the angles. For
597       * Cardan angles, this is often called gimbal lock. There is
598       * <em>nothing</em> to do to prevent this, it is an intrinsic problem
599       * with Cardan and Euler representation (but not a problem with the
600       * rotation itself, which is perfectly well defined). For Cardan
601       * angles, singularities occur when the second angle is close to
602       * -&pi;/2 or +&pi;/2, for Euler angle singularities occur when the
603       * second angle is close to 0 or &pi;, this implies that the identity
604       * rotation is always singular for Euler angles!</p>
605       *
606       * @param order rotation order to use
607       * @return an array of three angles, in the order specified by the set
608       * @exception CardanEulerSingularityException if the rotation is
609       * singular with respect to the angles set specified
610       */
611      public double[] getAngles(RotationOrder order)
612        throws CardanEulerSingularityException {
613    
614        if (order == RotationOrder.XYZ) {
615    
616          // r (Vector3D.plusK) coordinates are :
617          //  sin (theta), -cos (theta) sin (phi), cos (theta) cos (phi)
618          // (-r) (Vector3D.plusI) coordinates are :
619          // cos (psi) cos (theta), -sin (psi) cos (theta), sin (theta)
620          // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
621          Vector3D v1 = applyTo(Vector3D.PLUS_K);
622          Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
623          if  ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
624            throw new CardanEulerSingularityException(true);
625          }
626          return new double[] {
627            FastMath.atan2(-(v1.getY()), v1.getZ()),
628            FastMath.asin(v2.getZ()),
629            FastMath.atan2(-(v2.getY()), v2.getX())
630          };
631    
632        } else if (order == RotationOrder.XZY) {
633    
634          // r (Vector3D.plusJ) coordinates are :
635          // -sin (psi), cos (psi) cos (phi), cos (psi) sin (phi)
636          // (-r) (Vector3D.plusI) coordinates are :
637          // cos (theta) cos (psi), -sin (psi), sin (theta) cos (psi)
638          // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
639          Vector3D v1 = applyTo(Vector3D.PLUS_J);
640          Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
641          if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
642            throw new CardanEulerSingularityException(true);
643          }
644          return new double[] {
645            FastMath.atan2(v1.getZ(), v1.getY()),
646           -FastMath.asin(v2.getY()),
647            FastMath.atan2(v2.getZ(), v2.getX())
648          };
649    
650        } else if (order == RotationOrder.YXZ) {
651    
652          // r (Vector3D.plusK) coordinates are :
653          //  cos (phi) sin (theta), -sin (phi), cos (phi) cos (theta)
654          // (-r) (Vector3D.plusJ) coordinates are :
655          // sin (psi) cos (phi), cos (psi) cos (phi), -sin (phi)
656          // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
657          Vector3D v1 = applyTo(Vector3D.PLUS_K);
658          Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
659          if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
660            throw new CardanEulerSingularityException(true);
661          }
662          return new double[] {
663            FastMath.atan2(v1.getX(), v1.getZ()),
664           -FastMath.asin(v2.getZ()),
665            FastMath.atan2(v2.getX(), v2.getY())
666          };
667    
668        } else if (order == RotationOrder.YZX) {
669    
670          // r (Vector3D.plusI) coordinates are :
671          // cos (psi) cos (theta), sin (psi), -cos (psi) sin (theta)
672          // (-r) (Vector3D.plusJ) coordinates are :
673          // sin (psi), cos (phi) cos (psi), -sin (phi) cos (psi)
674          // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
675          Vector3D v1 = applyTo(Vector3D.PLUS_I);
676          Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
677          if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
678            throw new CardanEulerSingularityException(true);
679          }
680          return new double[] {
681            FastMath.atan2(-(v1.getZ()), v1.getX()),
682            FastMath.asin(v2.getX()),
683            FastMath.atan2(-(v2.getZ()), v2.getY())
684          };
685    
686        } else if (order == RotationOrder.ZXY) {
687    
688          // r (Vector3D.plusJ) coordinates are :
689          // -cos (phi) sin (psi), cos (phi) cos (psi), sin (phi)
690          // (-r) (Vector3D.plusK) coordinates are :
691          // -sin (theta) cos (phi), sin (phi), cos (theta) cos (phi)
692          // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
693          Vector3D v1 = applyTo(Vector3D.PLUS_J);
694          Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
695          if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
696            throw new CardanEulerSingularityException(true);
697          }
698          return new double[] {
699            FastMath.atan2(-(v1.getX()), v1.getY()),
700            FastMath.asin(v2.getY()),
701            FastMath.atan2(-(v2.getX()), v2.getZ())
702          };
703    
704        } else if (order == RotationOrder.ZYX) {
705    
706          // r (Vector3D.plusI) coordinates are :
707          //  cos (theta) cos (psi), cos (theta) sin (psi), -sin (theta)
708          // (-r) (Vector3D.plusK) coordinates are :
709          // -sin (theta), sin (phi) cos (theta), cos (phi) cos (theta)
710          // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
711          Vector3D v1 = applyTo(Vector3D.PLUS_I);
712          Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
713          if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
714            throw new CardanEulerSingularityException(true);
715          }
716          return new double[] {
717            FastMath.atan2(v1.getY(), v1.getX()),
718           -FastMath.asin(v2.getX()),
719            FastMath.atan2(v2.getY(), v2.getZ())
720          };
721    
722        } else if (order == RotationOrder.XYX) {
723    
724          // r (Vector3D.plusI) coordinates are :
725          //  cos (theta), sin (phi1) sin (theta), -cos (phi1) sin (theta)
726          // (-r) (Vector3D.plusI) coordinates are :
727          // cos (theta), sin (theta) sin (phi2), sin (theta) cos (phi2)
728          // and we can choose to have theta in the interval [0 ; PI]
729          Vector3D v1 = applyTo(Vector3D.PLUS_I);
730          Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
731          if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
732            throw new CardanEulerSingularityException(false);
733          }
734          return new double[] {
735            FastMath.atan2(v1.getY(), -v1.getZ()),
736            FastMath.acos(v2.getX()),
737            FastMath.atan2(v2.getY(), v2.getZ())
738          };
739    
740        } else if (order == RotationOrder.XZX) {
741    
742          // r (Vector3D.plusI) coordinates are :
743          //  cos (psi), cos (phi1) sin (psi), sin (phi1) sin (psi)
744          // (-r) (Vector3D.plusI) coordinates are :
745          // cos (psi), -sin (psi) cos (phi2), sin (psi) sin (phi2)
746          // and we can choose to have psi in the interval [0 ; PI]
747          Vector3D v1 = applyTo(Vector3D.PLUS_I);
748          Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
749          if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
750            throw new CardanEulerSingularityException(false);
751          }
752          return new double[] {
753            FastMath.atan2(v1.getZ(), v1.getY()),
754            FastMath.acos(v2.getX()),
755            FastMath.atan2(v2.getZ(), -v2.getY())
756          };
757    
758        } else if (order == RotationOrder.YXY) {
759    
760          // r (Vector3D.plusJ) coordinates are :
761          //  sin (theta1) sin (phi), cos (phi), cos (theta1) sin (phi)
762          // (-r) (Vector3D.plusJ) coordinates are :
763          // sin (phi) sin (theta2), cos (phi), -sin (phi) cos (theta2)
764          // and we can choose to have phi in the interval [0 ; PI]
765          Vector3D v1 = applyTo(Vector3D.PLUS_J);
766          Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
767          if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
768            throw new CardanEulerSingularityException(false);
769          }
770          return new double[] {
771            FastMath.atan2(v1.getX(), v1.getZ()),
772            FastMath.acos(v2.getY()),
773            FastMath.atan2(v2.getX(), -v2.getZ())
774          };
775    
776        } else if (order == RotationOrder.YZY) {
777    
778          // r (Vector3D.plusJ) coordinates are :
779          //  -cos (theta1) sin (psi), cos (psi), sin (theta1) sin (psi)
780          // (-r) (Vector3D.plusJ) coordinates are :
781          // sin (psi) cos (theta2), cos (psi), sin (psi) sin (theta2)
782          // and we can choose to have psi in the interval [0 ; PI]
783          Vector3D v1 = applyTo(Vector3D.PLUS_J);
784          Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
785          if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
786            throw new CardanEulerSingularityException(false);
787          }
788          return new double[] {
789            FastMath.atan2(v1.getZ(), -v1.getX()),
790            FastMath.acos(v2.getY()),
791            FastMath.atan2(v2.getZ(), v2.getX())
792          };
793    
794        } else if (order == RotationOrder.ZXZ) {
795    
796          // r (Vector3D.plusK) coordinates are :
797          //  sin (psi1) sin (phi), -cos (psi1) sin (phi), cos (phi)
798          // (-r) (Vector3D.plusK) coordinates are :
799          // sin (phi) sin (psi2), sin (phi) cos (psi2), cos (phi)
800          // and we can choose to have phi in the interval [0 ; PI]
801          Vector3D v1 = applyTo(Vector3D.PLUS_K);
802          Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
803          if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
804            throw new CardanEulerSingularityException(false);
805          }
806          return new double[] {
807            FastMath.atan2(v1.getX(), -v1.getY()),
808            FastMath.acos(v2.getZ()),
809            FastMath.atan2(v2.getX(), v2.getY())
810          };
811    
812        } else { // last possibility is ZYZ
813    
814          // r (Vector3D.plusK) coordinates are :
815          //  cos (psi1) sin (theta), sin (psi1) sin (theta), cos (theta)
816          // (-r) (Vector3D.plusK) coordinates are :
817          // -sin (theta) cos (psi2), sin (theta) sin (psi2), cos (theta)
818          // and we can choose to have theta in the interval [0 ; PI]
819          Vector3D v1 = applyTo(Vector3D.PLUS_K);
820          Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
821          if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
822            throw new CardanEulerSingularityException(false);
823          }
824          return new double[] {
825            FastMath.atan2(v1.getY(), v1.getX()),
826            FastMath.acos(v2.getZ()),
827            FastMath.atan2(v2.getY(), -v2.getX())
828          };
829    
830        }
831    
832      }
833    
834      /** Get the 3X3 matrix corresponding to the instance
835       * @return the matrix corresponding to the instance
836       */
837      public double[][] getMatrix() {
838    
839        // products
840        double q0q0  = q0 * q0;
841        double q0q1  = q0 * q1;
842        double q0q2  = q0 * q2;
843        double q0q3  = q0 * q3;
844        double q1q1  = q1 * q1;
845        double q1q2  = q1 * q2;
846        double q1q3  = q1 * q3;
847        double q2q2  = q2 * q2;
848        double q2q3  = q2 * q3;
849        double q3q3  = q3 * q3;
850    
851        // create the matrix
852        double[][] m = new double[3][];
853        m[0] = new double[3];
854        m[1] = new double[3];
855        m[2] = new double[3];
856    
857        m [0][0] = 2.0 * (q0q0 + q1q1) - 1.0;
858        m [1][0] = 2.0 * (q1q2 - q0q3);
859        m [2][0] = 2.0 * (q1q3 + q0q2);
860    
861        m [0][1] = 2.0 * (q1q2 + q0q3);
862        m [1][1] = 2.0 * (q0q0 + q2q2) - 1.0;
863        m [2][1] = 2.0 * (q2q3 - q0q1);
864    
865        m [0][2] = 2.0 * (q1q3 - q0q2);
866        m [1][2] = 2.0 * (q2q3 + q0q1);
867        m [2][2] = 2.0 * (q0q0 + q3q3) - 1.0;
868    
869        return m;
870    
871      }
872    
873      /** Apply the rotation to a vector.
874       * @param u vector to apply the rotation to
875       * @return a new vector which is the image of u by the rotation
876       */
877      public Vector3D applyTo(Vector3D u) {
878    
879        double x = u.getX();
880        double y = u.getY();
881        double z = u.getZ();
882    
883        double s = q1 * x + q2 * y + q3 * z;
884    
885        return new Vector3D(2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x,
886                            2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y,
887                            2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - z);
888    
889      }
890    
891      /** Apply the inverse of the rotation to a vector.
892       * @param u vector to apply the inverse of the rotation to
893       * @return a new vector which such that u is its image by the rotation
894       */
895      public Vector3D applyInverseTo(Vector3D u) {
896    
897        double x = u.getX();
898        double y = u.getY();
899        double z = u.getZ();
900    
901        double s = q1 * x + q2 * y + q3 * z;
902        double m0 = -q0;
903    
904        return new Vector3D(2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x,
905                            2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y,
906                            2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - z);
907    
908      }
909    
910      /** Apply the instance to another rotation.
911       * Applying the instance to a rotation is computing the composition
912       * in an order compliant with the following rule : let u be any
913       * vector and v its image by r (i.e. r.applyTo(u) = v), let w be the image
914       * of v by the instance (i.e. applyTo(v) = w), then w = comp.applyTo(u),
915       * where comp = applyTo(r).
916       * @param r rotation to apply the rotation to
917       * @return a new rotation which is the composition of r by the instance
918       */
919      public Rotation applyTo(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      /** Apply the inverse of the instance to another rotation.
928       * Applying the inverse of the instance to a rotation is computing
929       * the composition in an order compliant with the following rule :
930       * let u be any vector and v its image by r (i.e. r.applyTo(u) = v),
931       * let w be the inverse image of v by the instance
932       * (i.e. applyInverseTo(v) = w), then w = comp.applyTo(u), where
933       * comp = applyInverseTo(r).
934       * @param r rotation to apply the rotation to
935       * @return a new rotation which is the composition of r by the inverse
936       * of the instance
937       */
938      public Rotation applyInverseTo(Rotation r) {
939        return new Rotation(-r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),
940                            -r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),
941                            -r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),
942                            -r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),
943                            false);
944      }
945    
946      /** Perfect orthogonality on a 3X3 matrix.
947       * @param m initial matrix (not exactly orthogonal)
948       * @param threshold convergence threshold for the iterative
949       * orthogonality correction (convergence is reached when the
950       * difference between two steps of the Frobenius norm of the
951       * correction is below this threshold)
952       * @return an orthogonal matrix close to m
953       * @exception NotARotationMatrixException if the matrix cannot be
954       * orthogonalized with the given threshold after 10 iterations
955       */
956      private double[][] orthogonalizeMatrix(double[][] m, double threshold)
957        throws NotARotationMatrixException {
958        double[] m0 = m[0];
959        double[] m1 = m[1];
960        double[] m2 = m[2];
961        double x00 = m0[0];
962        double x01 = m0[1];
963        double x02 = m0[2];
964        double x10 = m1[0];
965        double x11 = m1[1];
966        double x12 = m1[2];
967        double x20 = m2[0];
968        double x21 = m2[1];
969        double x22 = m2[2];
970        double fn = 0;
971        double fn1;
972    
973        double[][] o = new double[3][3];
974        double[] o0 = o[0];
975        double[] o1 = o[1];
976        double[] o2 = o[2];
977    
978        // iterative correction: Xn+1 = Xn - 0.5 * (Xn.Mt.Xn - M)
979        int i = 0;
980        while (++i < 11) {
981    
982          // Mt.Xn
983          double mx00 = m0[0] * x00 + m1[0] * x10 + m2[0] * x20;
984          double mx10 = m0[1] * x00 + m1[1] * x10 + m2[1] * x20;
985          double mx20 = m0[2] * x00 + m1[2] * x10 + m2[2] * x20;
986          double mx01 = m0[0] * x01 + m1[0] * x11 + m2[0] * x21;
987          double mx11 = m0[1] * x01 + m1[1] * x11 + m2[1] * x21;
988          double mx21 = m0[2] * x01 + m1[2] * x11 + m2[2] * x21;
989          double mx02 = m0[0] * x02 + m1[0] * x12 + m2[0] * x22;
990          double mx12 = m0[1] * x02 + m1[1] * x12 + m2[1] * x22;
991          double mx22 = m0[2] * x02 + m1[2] * x12 + m2[2] * x22;
992    
993          // Xn+1
994          o0[0] = x00 - 0.5 * (x00 * mx00 + x01 * mx10 + x02 * mx20 - m0[0]);
995          o0[1] = x01 - 0.5 * (x00 * mx01 + x01 * mx11 + x02 * mx21 - m0[1]);
996          o0[2] = x02 - 0.5 * (x00 * mx02 + x01 * mx12 + x02 * mx22 - m0[2]);
997          o1[0] = x10 - 0.5 * (x10 * mx00 + x11 * mx10 + x12 * mx20 - m1[0]);
998          o1[1] = x11 - 0.5 * (x10 * mx01 + x11 * mx11 + x12 * mx21 - m1[1]);
999          o1[2] = x12 - 0.5 * (x10 * mx02 + x11 * mx12 + x12 * mx22 - m1[2]);
1000          o2[0] = x20 - 0.5 * (x20 * mx00 + x21 * mx10 + x22 * mx20 - m2[0]);
1001          o2[1] = x21 - 0.5 * (x20 * mx01 + x21 * mx11 + x22 * mx21 - m2[1]);
1002          o2[2] = x22 - 0.5 * (x20 * mx02 + x21 * mx12 + x22 * mx22 - m2[2]);
1003    
1004          // correction on each elements
1005          double corr00 = o0[0] - m0[0];
1006          double corr01 = o0[1] - m0[1];
1007          double corr02 = o0[2] - m0[2];
1008          double corr10 = o1[0] - m1[0];
1009          double corr11 = o1[1] - m1[1];
1010          double corr12 = o1[2] - m1[2];
1011          double corr20 = o2[0] - m2[0];
1012          double corr21 = o2[1] - m2[1];
1013          double corr22 = o2[2] - m2[2];
1014    
1015          // Frobenius norm of the correction
1016          fn1 = corr00 * corr00 + corr01 * corr01 + corr02 * corr02 +
1017                corr10 * corr10 + corr11 * corr11 + corr12 * corr12 +
1018                corr20 * corr20 + corr21 * corr21 + corr22 * corr22;
1019    
1020          // convergence test
1021          if (FastMath.abs(fn1 - fn) <= threshold)
1022            return o;
1023    
1024          // prepare next iteration
1025          x00 = o0[0];
1026          x01 = o0[1];
1027          x02 = o0[2];
1028          x10 = o1[0];
1029          x11 = o1[1];
1030          x12 = o1[2];
1031          x20 = o2[0];
1032          x21 = o2[1];
1033          x22 = o2[2];
1034          fn  = fn1;
1035    
1036        }
1037    
1038        // the algorithm did not converge after 10 iterations
1039        throw new NotARotationMatrixException(
1040                LocalizedFormats.UNABLE_TO_ORTHOGONOLIZE_MATRIX,
1041                i - 1);
1042      }
1043    
1044      /** Compute the <i>distance</i> between two rotations.
1045       * <p>The <i>distance</i> is intended here as a way to check if two
1046       * rotations are almost similar (i.e. they transform vectors the same way)
1047       * or very different. It is mathematically defined as the angle of
1048       * the rotation r that prepended to one of the rotations gives the other
1049       * one:</p>
1050       * <pre>
1051       *        r<sub>1</sub>(r) = r<sub>2</sub>
1052       * </pre>
1053       * <p>This distance is an angle between 0 and &pi;. Its value is the smallest
1054       * possible upper bound of the angle in radians between r<sub>1</sub>(v)
1055       * and r<sub>2</sub>(v) for all possible vectors v. This upper bound is
1056       * reached for some v. The distance is equal to 0 if and only if the two
1057       * rotations are identical.</p>
1058       * <p>Comparing two rotations should always be done using this value rather
1059       * than for example comparing the components of the quaternions. It is much
1060       * more stable, and has a geometric meaning. Also comparing quaternions
1061       * components is error prone since for example quaternions (0.36, 0.48, -0.48, -0.64)
1062       * and (-0.36, -0.48, 0.48, 0.64) represent exactly the same rotation despite
1063       * their components are different (they are exact opposites).</p>
1064       * @param r1 first rotation
1065       * @param r2 second rotation
1066       * @return <i>distance</i> between r1 and r2
1067       */
1068      public static double distance(Rotation r1, Rotation r2) {
1069          return r1.applyInverseTo(r2).getAngle();
1070      }
1071    
1072    }