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    
024    /**
025     * This class implements rotations in a three-dimensional space.
026     *
027     * <p>Rotations can be represented by several different mathematical
028     * entities (matrices, axe and angle, Cardan or Euler angles,
029     * quaternions). This class presents an higher level abstraction, more
030     * user-oriented and hiding this implementation details. Well, for the
031     * curious, we use quaternions for the internal representation. The
032     * user can build a rotation from any of these representations, and
033     * any of these representations can be retrieved from a
034     * <code>Rotation</code> instance (see the various constructors and
035     * getters). In addition, a rotation can also be built implicitely
036     * from a set of vectors and their image.</p>
037     * <p>This implies that this class can be used to convert from one
038     * representation to another one. For example, converting a rotation
039     * matrix into a set of Cardan angles from can be done using the
040     * followong single line of code:</p>
041     * <pre>
042     * double[] angles = new Rotation(matrix, 1.0e-10).getAngles(RotationOrder.XYZ);
043     * </pre>
044     * <p>Focus is oriented on what a rotation <em>do</em> rather than on its
045     * underlying representation. Once it has been built, and regardless of its
046     * internal representation, a rotation is an <em>operator</em> which basically
047     * transforms three dimensional {@link Vector3D vectors} into other three
048     * dimensional {@link Vector3D vectors}. Depending on the application, the
049     * meaning of these vectors may vary and the semantics of the rotation also.</p>
050     * <p>For example in an spacecraft attitude simulation tool, users will often
051     * consider the vectors are fixed (say the Earth direction for example) and the
052     * rotation transforms the coordinates coordinates of this vector in inertial
053     * frame into the coordinates of the same vector in satellite frame. In this
054     * case, the rotation implicitely defines the relation between the two frames.
055     * Another example could be a telescope control application, where the rotation
056     * would transform the sighting direction at rest into the desired observing
057     * direction when the telescope is pointed towards an object of interest. In this
058     * case the rotation transforms the directionf at rest in a topocentric frame
059     * into the sighting direction in the same topocentric frame. In many case, both
060     * approaches will be combined, in our telescope example, we will probably also
061     * need to transform the observing direction in the topocentric frame into the
062     * observing direction in inertial frame taking into account the observatory
063     * location and the Earth rotation.</p>
064     *
065     * <p>These examples show that a rotation is what the user wants it to be, so this
066     * class does not push the user towards one specific definition and hence does not
067     * provide methods like <code>projectVectorIntoDestinationFrame</code> or
068     * <code>computeTransformedDirection</code>. It provides simpler and more generic
069     * methods: {@link #applyTo(Vector3D) applyTo(Vector3D)} and {@link
070     * #applyInverseTo(Vector3D) applyInverseTo(Vector3D)}.</p>
071     *
072     * <p>Since a rotation is basically a vectorial operator, several rotations can be
073     * composed together and the composite operation <code>r = r<sub>1</sub> o
074     * r<sub>2</sub></code> (which means that for each vector <code>u</code>,
075     * <code>r(u) = r<sub>1</sub>(r<sub>2</sub>(u))</code>) is also a rotation. Hence
076     * we can consider that in addition to vectors, a rotation can be applied to other
077     * rotations as well (or to itself). With our previous notations, we would say we
078     * can apply <code>r<sub>1</sub></code> to <code>r<sub>2</sub></code> and the result
079     * we get is <code>r = r<sub>1</sub> o r<sub>2</sub></code>. For this purpose, the
080     * class provides the methods: {@link #applyTo(Rotation) applyTo(Rotation)} and
081     * {@link #applyInverseTo(Rotation) applyInverseTo(Rotation)}.</p>
082     *
083     * <p>Rotations are guaranteed to be immutable objects.</p>
084     *
085     * @version $Revision: 772119 $ $Date: 2009-05-06 05:43:28 -0400 (Wed, 06 May 2009) $
086     * @see Vector3D
087     * @see RotationOrder
088     * @since 1.2
089     */
090    
091    public class Rotation implements Serializable {
092    
093      /** Identity rotation. */
094      public static final Rotation IDENTITY = new Rotation(1.0, 0.0, 0.0, 0.0, false);
095    
096      /** Serializable version identifier */
097      private static final long serialVersionUID = -2153622329907944313L;
098    
099      /** Scalar coordinate of the quaternion. */
100      private final double q0;
101    
102      /** First coordinate of the vectorial part of the quaternion. */
103      private final double q1;
104    
105      /** Second coordinate of the vectorial part of the quaternion. */
106      private final double q2;
107    
108      /** Third coordinate of the vectorial part of the quaternion. */
109      private final double q3;
110    
111      /** Build a rotation from the quaternion coordinates.
112       * <p>A rotation can be built from a <em>normalized</em> quaternion,
113       * i.e. a quaternion for which q<sub>0</sub><sup>2</sup> +
114       * q<sub>1</sub><sup>2</sup> + q<sub>2</sub><sup>2</sup> +
115       * q<sub>3</sub><sup>2</sup> = 1. If the quaternion is not normalized,
116       * the constructor can normalize it in a preprocessing step.</p>
117       * @param q0 scalar part of the quaternion
118       * @param q1 first coordinate of the vectorial part of the quaternion
119       * @param q2 second coordinate of the vectorial part of the quaternion
120       * @param q3 third coordinate of the vectorial part of the quaternion
121       * @param needsNormalization if true, the coordinates are considered
122       * not to be normalized, a normalization preprocessing step is performed
123       * before using them
124       */
125      public Rotation(double q0, double q1, double q2, double q3,
126                      boolean needsNormalization) {
127    
128        if (needsNormalization) {
129          // normalization preprocessing
130          double inv = 1.0 / Math.sqrt(q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3);
131          q0 *= inv;
132          q1 *= inv;
133          q2 *= inv;
134          q3 *= inv;
135        }
136    
137        this.q0 = q0;
138        this.q1 = q1;
139        this.q2 = q2;
140        this.q3 = q3;
141    
142      }
143    
144      /** Build a rotation from an axis and an angle.
145       * <p>We use the convention that angles are oriented according to
146       * the effect of the rotation on vectors around the axis. That means
147       * that if (i, j, k) is a direct frame and if we first provide +k as
148       * the axis and PI/2 as the angle to this constructor, and then
149       * {@link #applyTo(Vector3D) apply} the instance to +i, we will get
150       * +j.</p>
151       * @param axis axis around which to rotate
152       * @param angle rotation angle.
153       * @exception ArithmeticException if the axis norm is zero
154       */
155      public Rotation(Vector3D axis, double angle) {
156    
157        double norm = axis.getNorm();
158        if (norm == 0) {
159          throw MathRuntimeException.createArithmeticException("zero norm for rotation axis");
160        }
161    
162        double halfAngle = -0.5 * angle;
163        double coeff = Math.sin(halfAngle) / norm;
164    
165        q0 = Math.cos (halfAngle);
166        q1 = coeff * axis.getX();
167        q2 = coeff * axis.getY();
168        q3 = coeff * axis.getZ();
169    
170      }
171    
172      /** Build a rotation from a 3X3 matrix.
173    
174       * <p>Rotation matrices are orthogonal matrices, i.e. unit matrices
175       * (which are matrices for which m.m<sup>T</sup> = I) with real
176       * coefficients. The module of the determinant of unit matrices is
177       * 1, among the orthogonal 3X3 matrices, only the ones having a
178       * positive determinant (+1) are rotation matrices.</p>
179    
180       * <p>When a rotation is defined by a matrix with truncated values
181       * (typically when it is extracted from a technical sheet where only
182       * four to five significant digits are available), the matrix is not
183       * orthogonal anymore. This constructor handles this case
184       * transparently by using a copy of the given matrix and applying a
185       * correction to the copy in order to perfect its orthogonality. If
186       * the Frobenius norm of the correction needed is above the given
187       * threshold, then the matrix is considered to be too far from a
188       * true rotation matrix and an exception is thrown.<p>
189    
190       * @param m rotation matrix
191       * @param threshold convergence threshold for the iterative
192       * orthogonality correction (convergence is reached when the
193       * difference between two steps of the Frobenius norm of the
194       * correction is below this threshold)
195    
196       * @exception NotARotationMatrixException if the matrix is not a 3X3
197       * matrix, or if it cannot be transformed into an orthogonal matrix
198       * with the given threshold, or if the determinant of the resulting
199       * orthogonal matrix is negative
200    
201       */
202      public Rotation(double[][] m, double threshold)
203        throws NotARotationMatrixException {
204    
205        // dimension check
206        if ((m.length != 3) || (m[0].length != 3) ||
207            (m[1].length != 3) || (m[2].length != 3)) {
208          throw new NotARotationMatrixException(
209                  "a {0}x{1} matrix cannot be a rotation matrix",
210                  m.length, m[0].length);
211        }
212    
213        // compute a "close" orthogonal matrix
214        double[][] ort = orthogonalizeMatrix(m, threshold);
215    
216        // check the sign of the determinant
217        double det = ort[0][0] * (ort[1][1] * ort[2][2] - ort[2][1] * ort[1][2]) -
218                     ort[1][0] * (ort[0][1] * ort[2][2] - ort[2][1] * ort[0][2]) +
219                     ort[2][0] * (ort[0][1] * ort[1][2] - ort[1][1] * ort[0][2]);
220        if (det < 0.0) {
221          throw new NotARotationMatrixException(
222                  "the closest orthogonal matrix has a negative determinant {0}",
223                  det);
224        }
225    
226        // There are different ways to compute the quaternions elements
227        // from the matrix. They all involve computing one element from
228        // the diagonal of the matrix, and computing the three other ones
229        // using a formula involving a division by the first element,
230        // which unfortunately can be zero. Since the norm of the
231        // quaternion is 1, we know at least one element has an absolute
232        // value greater or equal to 0.5, so it is always possible to
233        // select the right formula and avoid division by zero and even
234        // numerical inaccuracy. Checking the elements in turn and using
235        // the first one greater than 0.45 is safe (this leads to a simple
236        // test since qi = 0.45 implies 4 qi^2 - 1 = -0.19)
237        double s = ort[0][0] + ort[1][1] + ort[2][2];
238        if (s > -0.19) {
239          // compute q0 and deduce q1, q2 and q3
240          q0 = 0.5 * Math.sqrt(s + 1.0);
241          double inv = 0.25 / q0;
242          q1 = inv * (ort[1][2] - ort[2][1]);
243          q2 = inv * (ort[2][0] - ort[0][2]);
244          q3 = inv * (ort[0][1] - ort[1][0]);
245        } else {
246          s = ort[0][0] - ort[1][1] - ort[2][2];
247          if (s > -0.19) {
248            // compute q1 and deduce q0, q2 and q3
249            q1 = 0.5 * Math.sqrt(s + 1.0);
250            double inv = 0.25 / q1;
251            q0 = inv * (ort[1][2] - ort[2][1]);
252            q2 = inv * (ort[0][1] + ort[1][0]);
253            q3 = inv * (ort[0][2] + ort[2][0]);
254          } else {
255            s = ort[1][1] - ort[0][0] - ort[2][2];
256            if (s > -0.19) {
257              // compute q2 and deduce q0, q1 and q3
258              q2 = 0.5 * Math.sqrt(s + 1.0);
259              double inv = 0.25 / q2;
260              q0 = inv * (ort[2][0] - ort[0][2]);
261              q1 = inv * (ort[0][1] + ort[1][0]);
262              q3 = inv * (ort[2][1] + ort[1][2]);
263            } else {
264              // compute q3 and deduce q0, q1 and q2
265              s = ort[2][2] - ort[0][0] - ort[1][1];
266              q3 = 0.5 * Math.sqrt(s + 1.0);
267              double inv = 0.25 / q3;
268              q0 = inv * (ort[0][1] - ort[1][0]);
269              q1 = inv * (ort[0][2] + ort[2][0]);
270              q2 = inv * (ort[2][1] + ort[1][2]);
271            }
272          }
273        }
274    
275      }
276    
277      /** Build the rotation that transforms a pair of vector into another pair.
278    
279       * <p>Except for possible scale factors, if the instance were applied to
280       * the pair (u<sub>1</sub>, u<sub>2</sub>) it will produce the pair
281       * (v<sub>1</sub>, v<sub>2</sub>).</p>
282    
283       * <p>If the angular separation between u<sub>1</sub> and u<sub>2</sub> is
284       * not the same as the angular separation between v<sub>1</sub> and
285       * v<sub>2</sub>, then a corrected v'<sub>2</sub> will be used rather than
286       * v<sub>2</sub>, the corrected vector will be in the (v<sub>1</sub>,
287       * v<sub>2</sub>) plane.</p>
288    
289       * @param u1 first vector of the origin pair
290       * @param u2 second vector of the origin pair
291       * @param v1 desired image of u1 by the rotation
292       * @param v2 desired image of u2 by the rotation
293       * @exception IllegalArgumentException if the norm of one of the vectors is zero
294       */
295      public Rotation(Vector3D u1, Vector3D u2, Vector3D v1, Vector3D v2) {
296    
297      // norms computation
298      double u1u1 = Vector3D.dotProduct(u1, u1);
299      double u2u2 = Vector3D.dotProduct(u2, u2);
300      double v1v1 = Vector3D.dotProduct(v1, v1);
301      double v2v2 = Vector3D.dotProduct(v2, v2);
302      if ((u1u1 == 0) || (u2u2 == 0) || (v1v1 == 0) || (v2v2 == 0)) {
303        throw MathRuntimeException.createIllegalArgumentException("zero norm for rotation defining vector");
304      }
305    
306      double u1x = u1.getX();
307      double u1y = u1.getY();
308      double u1z = u1.getZ();
309    
310      double u2x = u2.getX();
311      double u2y = u2.getY();
312      double u2z = u2.getZ();
313    
314      // normalize v1 in order to have (v1'|v1') = (u1|u1)
315      double coeff = Math.sqrt (u1u1 / v1v1);
316      double v1x   = coeff * v1.getX();
317      double v1y   = coeff * v1.getY();
318      double v1z   = coeff * v1.getZ();
319      v1 = new Vector3D(v1x, v1y, v1z);
320    
321      // adjust v2 in order to have (u1|u2) = (v1|v2) and (v2'|v2') = (u2|u2)
322      double u1u2   = Vector3D.dotProduct(u1, u2);
323      double v1v2   = Vector3D.dotProduct(v1, v2);
324      double coeffU = u1u2 / u1u1;
325      double coeffV = v1v2 / u1u1;
326      double beta   = Math.sqrt((u2u2 - u1u2 * coeffU) / (v2v2 - v1v2 * coeffV));
327      double alpha  = coeffU - beta * coeffV;
328      double v2x    = alpha * v1x + beta * v2.getX();
329      double v2y    = alpha * v1y + beta * v2.getY();
330      double v2z    = alpha * v1z + beta * v2.getZ();
331      v2 = new Vector3D(v2x, v2y, v2z);
332    
333      // preliminary computation (we use explicit formulation instead
334      // of relying on the Vector3D class in order to avoid building lots
335      // of temporary objects)
336      Vector3D uRef = u1;
337      Vector3D vRef = v1;
338      double dx1 = v1x - u1.getX();
339      double dy1 = v1y - u1.getY();
340      double dz1 = v1z - u1.getZ();
341      double dx2 = v2x - u2.getX();
342      double dy2 = v2y - u2.getY();
343      double dz2 = v2z - u2.getZ();
344      Vector3D k = new Vector3D(dy1 * dz2 - dz1 * dy2,
345                                dz1 * dx2 - dx1 * dz2,
346                                dx1 * dy2 - dy1 * dx2);
347      double c = k.getX() * (u1y * u2z - u1z * u2y) +
348                 k.getY() * (u1z * u2x - u1x * u2z) +
349                 k.getZ() * (u1x * u2y - u1y * u2x);
350    
351      if (c == 0) {
352        // the (q1, q2, q3) vector is in the (u1, u2) plane
353        // we try other vectors
354        Vector3D u3 = Vector3D.crossProduct(u1, u2);
355        Vector3D v3 = Vector3D.crossProduct(v1, v2);
356        double u3x  = u3.getX();
357        double u3y  = u3.getY();
358        double u3z  = u3.getZ();
359        double v3x  = v3.getX();
360        double v3y  = v3.getY();
361        double v3z  = v3.getZ();
362    
363        double dx3 = v3x - u3x;
364        double dy3 = v3y - u3y;
365        double dz3 = v3z - u3z;
366        k = new Vector3D(dy1 * dz3 - dz1 * dy3,
367                         dz1 * dx3 - dx1 * dz3,
368                         dx1 * dy3 - dy1 * dx3);
369        c = k.getX() * (u1y * u3z - u1z * u3y) +
370            k.getY() * (u1z * u3x - u1x * u3z) +
371            k.getZ() * (u1x * u3y - u1y * u3x);
372    
373        if (c == 0) {
374          // the (q1, q2, q3) vector is aligned with u1:
375          // we try (u2, u3) and (v2, v3)
376          k = new Vector3D(dy2 * dz3 - dz2 * dy3,
377                           dz2 * dx3 - dx2 * dz3,
378                           dx2 * dy3 - dy2 * dx3);
379          c = k.getX() * (u2y * u3z - u2z * u3y) +
380              k.getY() * (u2z * u3x - u2x * u3z) +
381              k.getZ() * (u2x * u3y - u2y * u3x);
382    
383          if (c == 0) {
384            // the (q1, q2, q3) vector is aligned with everything
385            // this is really the identity rotation
386            q0 = 1.0;
387            q1 = 0.0;
388            q2 = 0.0;
389            q3 = 0.0;
390            return;
391          }
392    
393          // we will have to use u2 and v2 to compute the scalar part
394          uRef = u2;
395          vRef = v2;
396    
397        }
398    
399      }
400    
401      // compute the vectorial part
402      c = Math.sqrt(c);
403      double inv = 1.0 / (c + c);
404      q1 = inv * k.getX();
405      q2 = inv * k.getY();
406      q3 = inv * k.getZ();
407    
408      // compute the scalar part
409       k = new Vector3D(uRef.getY() * q3 - uRef.getZ() * q2,
410                        uRef.getZ() * q1 - uRef.getX() * q3,
411                        uRef.getX() * q2 - uRef.getY() * q1);
412       c = Vector3D.dotProduct(k, k);
413      q0 = Vector3D.dotProduct(vRef, k) / (c + c);
414    
415      }
416    
417      /** Build one of the rotations that transform one vector into another one.
418    
419       * <p>Except for a possible scale factor, if the instance were
420       * applied to the vector u it will produce the vector v. There is an
421       * infinite number of such rotations, this constructor choose the
422       * one with the smallest associated angle (i.e. the one whose axis
423       * is orthogonal to the (u, v) plane). If u and v are colinear, an
424       * arbitrary rotation axis is chosen.</p>
425    
426       * @param u origin vector
427       * @param v desired image of u by the rotation
428       * @exception IllegalArgumentException if the norm of one of the vectors is zero
429       */
430      public Rotation(Vector3D u, Vector3D v) {
431    
432        double normProduct = u.getNorm() * v.getNorm();
433        if (normProduct == 0) {
434            throw MathRuntimeException.createIllegalArgumentException("zero norm for rotation defining vector");
435        }
436    
437        double dot = Vector3D.dotProduct(u, v);
438    
439        if (dot < ((2.0e-15 - 1.0) * normProduct)) {
440          // special case u = -v: we select a PI angle rotation around
441          // an arbitrary vector orthogonal to u
442          Vector3D w = u.orthogonal();
443          q0 = 0.0;
444          q1 = -w.getX();
445          q2 = -w.getY();
446          q3 = -w.getZ();
447        } else {
448          // general case: (u, v) defines a plane, we select
449          // the shortest possible rotation: axis orthogonal to this plane
450          q0 = Math.sqrt(0.5 * (1.0 + dot / normProduct));
451          double coeff = 1.0 / (2.0 * q0 * normProduct);
452          q1 = coeff * (v.getY() * u.getZ() - v.getZ() * u.getY());
453          q2 = coeff * (v.getZ() * u.getX() - v.getX() * u.getZ());
454          q3 = coeff * (v.getX() * u.getY() - v.getY() * u.getX());
455        }
456    
457      }
458    
459      /** Build a rotation from three Cardan or Euler elementary rotations.
460    
461       * <p>Cardan rotations are three successive rotations around the
462       * canonical axes X, Y and Z, each axis being used once. There are
463       * 6 such sets of rotations (XYZ, XZY, YXZ, YZX, ZXY and ZYX). Euler
464       * rotations are three successive rotations around the canonical
465       * axes X, Y and Z, the first and last rotations being around the
466       * same axis. There are 6 such sets of rotations (XYX, XZX, YXY,
467       * YZY, ZXZ and ZYZ), the most popular one being ZXZ.</p>
468       * <p>Beware that many people routinely use the term Euler angles even
469       * for what really are Cardan angles (this confusion is especially
470       * widespread in the aerospace business where Roll, Pitch and Yaw angles
471       * are often wrongly tagged as Euler angles).</p>
472    
473       * @param order order of rotations to use
474       * @param alpha1 angle of the first elementary rotation
475       * @param alpha2 angle of the second elementary rotation
476       * @param alpha3 angle of the third elementary rotation
477       */
478      public Rotation(RotationOrder order,
479                      double alpha1, double alpha2, double alpha3) {
480        Rotation r1 = new Rotation(order.getA1(), alpha1);
481        Rotation r2 = new Rotation(order.getA2(), alpha2);
482        Rotation r3 = new Rotation(order.getA3(), alpha3);
483        Rotation composed = r1.applyTo(r2.applyTo(r3));
484        q0 = composed.q0;
485        q1 = composed.q1;
486        q2 = composed.q2;
487        q3 = composed.q3;
488      }
489    
490      /** Revert a rotation.
491       * Build a rotation which reverse the effect of another
492       * rotation. This means that if r(u) = v, then r.revert(v) = u. The
493       * instance is not changed.
494       * @return a new rotation whose effect is the reverse of the effect
495       * of the instance
496       */
497      public Rotation revert() {
498        return new Rotation(-q0, q1, q2, q3, false);
499      }
500    
501      /** Get the scalar coordinate of the quaternion.
502       * @return scalar coordinate of the quaternion
503       */
504      public double getQ0() {
505        return q0;
506      }
507    
508      /** Get the first coordinate of the vectorial part of the quaternion.
509       * @return first coordinate of the vectorial part of the quaternion
510       */
511      public double getQ1() {
512        return q1;
513      }
514    
515      /** Get the second coordinate of the vectorial part of the quaternion.
516       * @return second coordinate of the vectorial part of the quaternion
517       */
518      public double getQ2() {
519        return q2;
520      }
521    
522      /** Get the third coordinate of the vectorial part of the quaternion.
523       * @return third coordinate of the vectorial part of the quaternion
524       */
525      public double getQ3() {
526        return q3;
527      }
528    
529      /** Get the normalized axis of the rotation.
530       * @return normalized axis of the rotation
531       */
532      public Vector3D getAxis() {
533        double squaredSine = q1 * q1 + q2 * q2 + q3 * q3;
534        if (squaredSine == 0) {
535          return new Vector3D(1, 0, 0);
536        } else if (q0 < 0) {
537          double inverse = 1 / Math.sqrt(squaredSine);
538          return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
539        }
540        double inverse = -1 / Math.sqrt(squaredSine);
541        return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
542      }
543    
544      /** Get the angle of the rotation.
545       * @return angle of the rotation (between 0 and &pi;)
546       */
547      public double getAngle() {
548        if ((q0 < -0.1) || (q0 > 0.1)) {
549          return 2 * Math.asin(Math.sqrt(q1 * q1 + q2 * q2 + q3 * q3));
550        } else if (q0 < 0) {
551          return 2 * Math.acos(-q0);
552        }
553        return 2 * Math.acos(q0);
554      }
555    
556      /** Get the Cardan or Euler angles corresponding to the instance.
557    
558       * <p>The equations show that each rotation can be defined by two
559       * different values of the Cardan or Euler angles set. For example
560       * if Cardan angles are used, the rotation defined by the angles
561       * a<sub>1</sub>, a<sub>2</sub> and a<sub>3</sub> is the same as
562       * the rotation defined by the angles &pi; + a<sub>1</sub>, &pi;
563       * - a<sub>2</sub> and &pi; + a<sub>3</sub>. This method implements
564       * the following arbitrary choices:</p>
565       * <ul>
566       *   <li>for Cardan angles, the chosen set is the one for which the
567       *   second angle is between -&pi;/2 and &pi;/2 (i.e its cosine is
568       *   positive),</li>
569       *   <li>for Euler angles, the chosen set is the one for which the
570       *   second angle is between 0 and &pi; (i.e its sine is positive).</li>
571       * </ul>
572    
573       * <p>Cardan and Euler angle have a very disappointing drawback: all
574       * of them have singularities. This means that if the instance is
575       * too close to the singularities corresponding to the given
576       * rotation order, it will be impossible to retrieve the angles. For
577       * Cardan angles, this is often called gimbal lock. There is
578       * <em>nothing</em> to do to prevent this, it is an intrinsic problem
579       * with Cardan and Euler representation (but not a problem with the
580       * rotation itself, which is perfectly well defined). For Cardan
581       * angles, singularities occur when the second angle is close to
582       * -&pi;/2 or +&pi;/2, for Euler angle singularities occur when the
583       * second angle is close to 0 or &pi;, this implies that the identity
584       * rotation is always singular for Euler angles!</p>
585    
586       * @param order rotation order to use
587       * @return an array of three angles, in the order specified by the set
588       * @exception CardanEulerSingularityException if the rotation is
589       * singular with respect to the angles set specified
590       */
591      public double[] getAngles(RotationOrder order)
592        throws CardanEulerSingularityException {
593    
594        if (order == RotationOrder.XYZ) {
595    
596          // r (Vector3D.plusK) coordinates are :
597          //  sin (theta), -cos (theta) sin (phi), cos (theta) cos (phi)
598          // (-r) (Vector3D.plusI) coordinates are :
599          // cos (psi) cos (theta), -sin (psi) cos (theta), sin (theta)
600          // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
601          Vector3D v1 = applyTo(Vector3D.PLUS_K);
602          Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
603          if  ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
604            throw new CardanEulerSingularityException(true);
605          }
606          return new double[] {
607            Math.atan2(-(v1.getY()), v1.getZ()),
608            Math.asin(v2.getZ()),
609            Math.atan2(-(v2.getY()), v2.getX())
610          };
611    
612        } else if (order == RotationOrder.XZY) {
613    
614          // r (Vector3D.plusJ) coordinates are :
615          // -sin (psi), cos (psi) cos (phi), cos (psi) sin (phi)
616          // (-r) (Vector3D.plusI) coordinates are :
617          // cos (theta) cos (psi), -sin (psi), sin (theta) cos (psi)
618          // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
619          Vector3D v1 = applyTo(Vector3D.PLUS_J);
620          Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
621          if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
622            throw new CardanEulerSingularityException(true);
623          }
624          return new double[] {
625            Math.atan2(v1.getZ(), v1.getY()),
626           -Math.asin(v2.getY()),
627            Math.atan2(v2.getZ(), v2.getX())
628          };
629    
630        } else if (order == RotationOrder.YXZ) {
631    
632          // r (Vector3D.plusK) coordinates are :
633          //  cos (phi) sin (theta), -sin (phi), cos (phi) cos (theta)
634          // (-r) (Vector3D.plusJ) coordinates are :
635          // sin (psi) cos (phi), cos (psi) cos (phi), -sin (phi)
636          // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
637          Vector3D v1 = applyTo(Vector3D.PLUS_K);
638          Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
639          if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
640            throw new CardanEulerSingularityException(true);
641          }
642          return new double[] {
643            Math.atan2(v1.getX(), v1.getZ()),
644           -Math.asin(v2.getZ()),
645            Math.atan2(v2.getX(), v2.getY())
646          };
647    
648        } else if (order == RotationOrder.YZX) {
649    
650          // r (Vector3D.plusI) coordinates are :
651          // cos (psi) cos (theta), sin (psi), -cos (psi) sin (theta)
652          // (-r) (Vector3D.plusJ) coordinates are :
653          // sin (psi), cos (phi) cos (psi), -sin (phi) cos (psi)
654          // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
655          Vector3D v1 = applyTo(Vector3D.PLUS_I);
656          Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
657          if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
658            throw new CardanEulerSingularityException(true);
659          }
660          return new double[] {
661            Math.atan2(-(v1.getZ()), v1.getX()),
662            Math.asin(v2.getX()),
663            Math.atan2(-(v2.getZ()), v2.getY())
664          };
665    
666        } else if (order == RotationOrder.ZXY) {
667    
668          // r (Vector3D.plusJ) coordinates are :
669          // -cos (phi) sin (psi), cos (phi) cos (psi), sin (phi)
670          // (-r) (Vector3D.plusK) coordinates are :
671          // -sin (theta) cos (phi), sin (phi), cos (theta) cos (phi)
672          // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
673          Vector3D v1 = applyTo(Vector3D.PLUS_J);
674          Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
675          if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
676            throw new CardanEulerSingularityException(true);
677          }
678          return new double[] {
679            Math.atan2(-(v1.getX()), v1.getY()),
680            Math.asin(v2.getY()),
681            Math.atan2(-(v2.getX()), v2.getZ())
682          };
683    
684        } else if (order == RotationOrder.ZYX) {
685    
686          // r (Vector3D.plusI) coordinates are :
687          //  cos (theta) cos (psi), cos (theta) sin (psi), -sin (theta)
688          // (-r) (Vector3D.plusK) coordinates are :
689          // -sin (theta), sin (phi) cos (theta), cos (phi) cos (theta)
690          // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
691          Vector3D v1 = applyTo(Vector3D.PLUS_I);
692          Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
693          if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
694            throw new CardanEulerSingularityException(true);
695          }
696          return new double[] {
697            Math.atan2(v1.getY(), v1.getX()),
698           -Math.asin(v2.getX()),
699            Math.atan2(v2.getY(), v2.getZ())
700          };
701    
702        } else if (order == RotationOrder.XYX) {
703    
704          // r (Vector3D.plusI) coordinates are :
705          //  cos (theta), sin (phi1) sin (theta), -cos (phi1) sin (theta)
706          // (-r) (Vector3D.plusI) coordinates are :
707          // cos (theta), sin (theta) sin (phi2), sin (theta) cos (phi2)
708          // and we can choose to have theta in the interval [0 ; PI]
709          Vector3D v1 = applyTo(Vector3D.PLUS_I);
710          Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
711          if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
712            throw new CardanEulerSingularityException(false);
713          }
714          return new double[] {
715            Math.atan2(v1.getY(), -v1.getZ()),
716            Math.acos(v2.getX()),
717            Math.atan2(v2.getY(), v2.getZ())
718          };
719    
720        } else if (order == RotationOrder.XZX) {
721    
722          // r (Vector3D.plusI) coordinates are :
723          //  cos (psi), cos (phi1) sin (psi), sin (phi1) sin (psi)
724          // (-r) (Vector3D.plusI) coordinates are :
725          // cos (psi), -sin (psi) cos (phi2), sin (psi) sin (phi2)
726          // and we can choose to have psi in the interval [0 ; PI]
727          Vector3D v1 = applyTo(Vector3D.PLUS_I);
728          Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
729          if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
730            throw new CardanEulerSingularityException(false);
731          }
732          return new double[] {
733            Math.atan2(v1.getZ(), v1.getY()),
734            Math.acos(v2.getX()),
735            Math.atan2(v2.getZ(), -v2.getY())
736          };
737    
738        } else if (order == RotationOrder.YXY) {
739    
740          // r (Vector3D.plusJ) coordinates are :
741          //  sin (theta1) sin (phi), cos (phi), cos (theta1) sin (phi)
742          // (-r) (Vector3D.plusJ) coordinates are :
743          // sin (phi) sin (theta2), cos (phi), -sin (phi) cos (theta2)
744          // and we can choose to have phi in the interval [0 ; PI]
745          Vector3D v1 = applyTo(Vector3D.PLUS_J);
746          Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
747          if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
748            throw new CardanEulerSingularityException(false);
749          }
750          return new double[] {
751            Math.atan2(v1.getX(), v1.getZ()),
752            Math.acos(v2.getY()),
753            Math.atan2(v2.getX(), -v2.getZ())
754          };
755    
756        } else if (order == RotationOrder.YZY) {
757    
758          // r (Vector3D.plusJ) coordinates are :
759          //  -cos (theta1) sin (psi), cos (psi), sin (theta1) sin (psi)
760          // (-r) (Vector3D.plusJ) coordinates are :
761          // sin (psi) cos (theta2), cos (psi), sin (psi) sin (theta2)
762          // and we can choose to have psi in the interval [0 ; PI]
763          Vector3D v1 = applyTo(Vector3D.PLUS_J);
764          Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
765          if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
766            throw new CardanEulerSingularityException(false);
767          }
768          return new double[] {
769            Math.atan2(v1.getZ(), -v1.getX()),
770            Math.acos(v2.getY()),
771            Math.atan2(v2.getZ(), v2.getX())
772          };
773    
774        } else if (order == RotationOrder.ZXZ) {
775    
776          // r (Vector3D.plusK) coordinates are :
777          //  sin (psi1) sin (phi), -cos (psi1) sin (phi), cos (phi)
778          // (-r) (Vector3D.plusK) coordinates are :
779          // sin (phi) sin (psi2), sin (phi) cos (psi2), cos (phi)
780          // and we can choose to have phi in the interval [0 ; PI]
781          Vector3D v1 = applyTo(Vector3D.PLUS_K);
782          Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
783          if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
784            throw new CardanEulerSingularityException(false);
785          }
786          return new double[] {
787            Math.atan2(v1.getX(), -v1.getY()),
788            Math.acos(v2.getZ()),
789            Math.atan2(v2.getX(), v2.getY())
790          };
791    
792        } else { // last possibility is ZYZ
793    
794          // r (Vector3D.plusK) coordinates are :
795          //  cos (psi1) sin (theta), sin (psi1) sin (theta), cos (theta)
796          // (-r) (Vector3D.plusK) coordinates are :
797          // -sin (theta) cos (psi2), sin (theta) sin (psi2), cos (theta)
798          // and we can choose to have theta in the interval [0 ; PI]
799          Vector3D v1 = applyTo(Vector3D.PLUS_K);
800          Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
801          if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
802            throw new CardanEulerSingularityException(false);
803          }
804          return new double[] {
805            Math.atan2(v1.getY(), v1.getX()),
806            Math.acos(v2.getZ()),
807            Math.atan2(v2.getY(), -v2.getX())
808          };
809    
810        }
811    
812      }
813    
814      /** Get the 3X3 matrix corresponding to the instance
815       * @return the matrix corresponding to the instance
816       */
817      public double[][] getMatrix() {
818    
819        // products
820        double q0q0  = q0 * q0;
821        double q0q1  = q0 * q1;
822        double q0q2  = q0 * q2;
823        double q0q3  = q0 * q3;
824        double q1q1  = q1 * q1;
825        double q1q2  = q1 * q2;
826        double q1q3  = q1 * q3;
827        double q2q2  = q2 * q2;
828        double q2q3  = q2 * q3;
829        double q3q3  = q3 * q3;
830    
831        // create the matrix
832        double[][] m = new double[3][];
833        m[0] = new double[3];
834        m[1] = new double[3];
835        m[2] = new double[3];
836    
837        m [0][0] = 2.0 * (q0q0 + q1q1) - 1.0;
838        m [1][0] = 2.0 * (q1q2 - q0q3);
839        m [2][0] = 2.0 * (q1q3 + q0q2);
840    
841        m [0][1] = 2.0 * (q1q2 + q0q3);
842        m [1][1] = 2.0 * (q0q0 + q2q2) - 1.0;
843        m [2][1] = 2.0 * (q2q3 - q0q1);
844    
845        m [0][2] = 2.0 * (q1q3 - q0q2);
846        m [1][2] = 2.0 * (q2q3 + q0q1);
847        m [2][2] = 2.0 * (q0q0 + q3q3) - 1.0;
848    
849        return m;
850    
851      }
852    
853      /** Apply the rotation to a vector.
854       * @param u vector to apply the rotation to
855       * @return a new vector which is the image of u by the rotation
856       */
857      public Vector3D applyTo(Vector3D u) {
858    
859        double x = u.getX();
860        double y = u.getY();
861        double z = u.getZ();
862    
863        double s = q1 * x + q2 * y + q3 * z;
864    
865        return new Vector3D(2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x,
866                            2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y,
867                            2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - z);
868    
869      }
870    
871      /** Apply the inverse of the rotation to a vector.
872       * @param u vector to apply the inverse of the rotation to
873       * @return a new vector which such that u is its image by the rotation
874       */
875      public Vector3D applyInverseTo(Vector3D u) {
876    
877        double x = u.getX();
878        double y = u.getY();
879        double z = u.getZ();
880    
881        double s = q1 * x + q2 * y + q3 * z;
882        double m0 = -q0;
883    
884        return new Vector3D(2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x,
885                            2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y,
886                            2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - z);
887    
888      }
889    
890      /** Apply the instance to another rotation.
891       * Applying the instance to a rotation is computing the composition
892       * in an order compliant with the following rule : let u be any
893       * vector and v its image by r (i.e. r.applyTo(u) = v), let w be the image
894       * of v by the instance (i.e. applyTo(v) = w), then w = comp.applyTo(u),
895       * where comp = applyTo(r).
896       * @param r rotation to apply the rotation to
897       * @return a new rotation which is the composition of r by the instance
898       */
899      public Rotation applyTo(Rotation r) {
900        return new Rotation(r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),
901                            r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),
902                            r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),
903                            r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),
904                            false);
905      }
906    
907      /** Apply the inverse of the instance to another rotation.
908       * Applying the inverse of the instance to a rotation is computing
909       * the composition in an order compliant with the following rule :
910       * let u be any vector and v its image by r (i.e. r.applyTo(u) = v),
911       * let w be the inverse image of v by the instance
912       * (i.e. applyInverseTo(v) = w), then w = comp.applyTo(u), where
913       * comp = applyInverseTo(r).
914       * @param r rotation to apply the rotation to
915       * @return a new rotation which is the composition of r by the inverse
916       * of the instance
917       */
918      public Rotation applyInverseTo(Rotation r) {
919        return new Rotation(-r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),
920                            -r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),
921                            -r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),
922                            -r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),
923                            false);
924      }
925    
926      /** Perfect orthogonality on a 3X3 matrix.
927       * @param m initial matrix (not exactly orthogonal)
928       * @param threshold convergence threshold for the iterative
929       * orthogonality correction (convergence is reached when the
930       * difference between two steps of the Frobenius norm of the
931       * correction is below this threshold)
932       * @return an orthogonal matrix close to m
933       * @exception NotARotationMatrixException if the matrix cannot be
934       * orthogonalized with the given threshold after 10 iterations
935       */
936      private double[][] orthogonalizeMatrix(double[][] m, double threshold)
937        throws NotARotationMatrixException {
938        double[] m0 = m[0];
939        double[] m1 = m[1];
940        double[] m2 = m[2];
941        double x00 = m0[0];
942        double x01 = m0[1];
943        double x02 = m0[2];
944        double x10 = m1[0];
945        double x11 = m1[1];
946        double x12 = m1[2];
947        double x20 = m2[0];
948        double x21 = m2[1];
949        double x22 = m2[2];
950        double fn = 0;
951        double fn1;
952    
953        double[][] o = new double[3][3];
954        double[] o0 = o[0];
955        double[] o1 = o[1];
956        double[] o2 = o[2];
957    
958        // iterative correction: Xn+1 = Xn - 0.5 * (Xn.Mt.Xn - M)
959        int i = 0;
960        while (++i < 11) {
961    
962          // Mt.Xn
963          double mx00 = m0[0] * x00 + m1[0] * x10 + m2[0] * x20;
964          double mx10 = m0[1] * x00 + m1[1] * x10 + m2[1] * x20;
965          double mx20 = m0[2] * x00 + m1[2] * x10 + m2[2] * x20;
966          double mx01 = m0[0] * x01 + m1[0] * x11 + m2[0] * x21;
967          double mx11 = m0[1] * x01 + m1[1] * x11 + m2[1] * x21;
968          double mx21 = m0[2] * x01 + m1[2] * x11 + m2[2] * x21;
969          double mx02 = m0[0] * x02 + m1[0] * x12 + m2[0] * x22;
970          double mx12 = m0[1] * x02 + m1[1] * x12 + m2[1] * x22;
971          double mx22 = m0[2] * x02 + m1[2] * x12 + m2[2] * x22;
972    
973          // Xn+1
974          o0[0] = x00 - 0.5 * (x00 * mx00 + x01 * mx10 + x02 * mx20 - m0[0]);
975          o0[1] = x01 - 0.5 * (x00 * mx01 + x01 * mx11 + x02 * mx21 - m0[1]);
976          o0[2] = x02 - 0.5 * (x00 * mx02 + x01 * mx12 + x02 * mx22 - m0[2]);
977          o1[0] = x10 - 0.5 * (x10 * mx00 + x11 * mx10 + x12 * mx20 - m1[0]);
978          o1[1] = x11 - 0.5 * (x10 * mx01 + x11 * mx11 + x12 * mx21 - m1[1]);
979          o1[2] = x12 - 0.5 * (x10 * mx02 + x11 * mx12 + x12 * mx22 - m1[2]);
980          o2[0] = x20 - 0.5 * (x20 * mx00 + x21 * mx10 + x22 * mx20 - m2[0]);
981          o2[1] = x21 - 0.5 * (x20 * mx01 + x21 * mx11 + x22 * mx21 - m2[1]);
982          o2[2] = x22 - 0.5 * (x20 * mx02 + x21 * mx12 + x22 * mx22 - m2[2]);
983    
984          // correction on each elements
985          double corr00 = o0[0] - m0[0];
986          double corr01 = o0[1] - m0[1];
987          double corr02 = o0[2] - m0[2];
988          double corr10 = o1[0] - m1[0];
989          double corr11 = o1[1] - m1[1];
990          double corr12 = o1[2] - m1[2];
991          double corr20 = o2[0] - m2[0];
992          double corr21 = o2[1] - m2[1];
993          double corr22 = o2[2] - m2[2];
994    
995          // Frobenius norm of the correction
996          fn1 = corr00 * corr00 + corr01 * corr01 + corr02 * corr02 +
997                corr10 * corr10 + corr11 * corr11 + corr12 * corr12 +
998                corr20 * corr20 + corr21 * corr21 + corr22 * corr22;
999    
1000          // convergence test
1001          if (Math.abs(fn1 - fn) <= threshold)
1002            return o;
1003    
1004          // prepare next iteration
1005          x00 = o0[0];
1006          x01 = o0[1];
1007          x02 = o0[2];
1008          x10 = o1[0];
1009          x11 = o1[1];
1010          x12 = o1[2];
1011          x20 = o2[0];
1012          x21 = o2[1];
1013          x22 = o2[2];
1014          fn  = fn1;
1015    
1016        }
1017    
1018        // the algorithm did not converge after 10 iterations
1019        throw new NotARotationMatrixException(
1020                "unable to orthogonalize matrix in {0} iterations",
1021                i - 1);
1022      }
1023    
1024      /** Compute the <i>distance</i> between two rotations.
1025       * <p>The <i>distance</i> is intended here as a way to check if two
1026       * rotations are almost similar (i.e. they transform vectors the same way)
1027       * or very different. It is mathematically defined as the angle of
1028       * the rotation r that prepended to one of the rotations gives the other
1029       * one:</p>
1030       * <pre>
1031       *        r<sub>1</sub>(r) = r<sub>2</sub>
1032       * </pre>
1033       * <p>This distance is an angle between 0 and &pi;. Its value is the smallest
1034       * possible upper bound of the angle in radians between r<sub>1</sub>(v)
1035       * and r<sub>2</sub>(v) for all possible vectors v. This upper bound is
1036       * reached for some v. The distance is equal to 0 if and only if the two
1037       * rotations are identical.</p>
1038       * <p>Comparing two rotations should always be done using this value rather
1039       * than for example comparing the components of the quaternions. It is much
1040       * more stable, and has a geometric meaning. Also comparing quaternions
1041       * components is error prone since for example quaternions (0.36, 0.48, -0.48, -0.64)
1042       * and (-0.36, -0.48, 0.48, 0.64) represent exactly the same rotation despite
1043       * their components are different (they are exact opposites).</p>
1044       * @param r1 first rotation
1045       * @param r2 second rotation
1046       * @return <i>distance</i> between r1 and r2
1047       */
1048      public static double distance(Rotation r1, Rotation r2) {
1049          return r1.applyInverseTo(r2).getAngle();
1050      }
1051    
1052    }