    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    }