Rotation.java

  1. /*
  2.  * Licensed to the Apache Software Foundation (ASF) under one or more
  3.  * contributor license agreements.  See the NOTICE file distributed with
  4.  * this work for additional information regarding copyright ownership.
  5.  * The ASF licenses this file to You under the Apache License, Version 2.0
  6.  * (the "License"); you may not use this file except in compliance with
  7.  * the License.  You may obtain a copy of the License at
  8.  *
  9.  *      http://www.apache.org/licenses/LICENSE-2.0
  10.  *
  11.  * Unless required by applicable law or agreed to in writing, software
  12.  * distributed under the License is distributed on an "AS IS" BASIS,
  13.  * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  14.  * See the License for the specific language governing permissions and
  15.  * limitations under the License.
  16.  */

  18. package org.apache.commons.math3.geometry.euclidean.threed;

  20. import java.io.Serializable;

  22. import org.apache.commons.math3.exception.MathArithmeticException;
  23. import org.apache.commons.math3.exception.MathIllegalArgumentException;
  24. import org.apache.commons.math3.exception.util.LocalizedFormats;
  25. import org.apache.commons.math3.util.FastMath;
  26. import org.apache.commons.math3.util.MathArrays;

  28. /**
  29.  * This class implements rotations in a three-dimensional space.
  30.  *
  31.  * <p>Rotations can be represented by several different mathematical
  32.  * entities (matrices, axe and angle, Cardan or Euler angles,
  33.  * quaternions). This class presents an higher level abstraction, more
  34.  * user-oriented and hiding this implementation details. Well, for the
  35.  * curious, we use quaternions for the internal representation. The
  36.  * user can build a rotation from any of these representations, and
  37.  * any of these representations can be retrieved from a
  38.  * <code>Rotation</code> instance (see the various constructors and
  39.  * getters). In addition, a rotation can also be built implicitly
  40.  * from a set of vectors and their image.</p>
  41.  * <p>This implies that this class can be used to convert from one
  42.  * representation to another one. For example, converting a rotation
  43.  * matrix into a set of Cardan angles from can be done using the
  44.  * following single line of code:</p>
  45.  * <pre>
  46.  * double[] angles = new Rotation(matrix, 1.0e-10).getAngles(RotationOrder.XYZ);
  47.  * </pre>
  48.  * <p>Focus is oriented on what a rotation <em>do</em> rather than on its
  49.  * underlying representation. Once it has been built, and regardless of its
  50.  * internal representation, a rotation is an <em>operator</em> which basically
  51.  * transforms three dimensional {@link Vector3D vectors} into other three
  52.  * dimensional {@link Vector3D vectors}. Depending on the application, the
  53.  * meaning of these vectors may vary and the semantics of the rotation also.</p>
  54.  * <p>For example in an spacecraft attitude simulation tool, users will often
  55.  * consider the vectors are fixed (say the Earth direction for example) and the
  56.  * frames change. The rotation transforms the coordinates of the vector in inertial
  57.  * frame into the coordinates of the same vector in satellite frame. In this
  58.  * case, the rotation implicitly defines the relation between the two frames.</p>
  59.  * <p>Another example could be a telescope control application, where the rotation
  60.  * would transform the sighting direction at rest into the desired observing
  61.  * direction when the telescope is pointed towards an object of interest. In this
  62.  * case the rotation transforms the direction at rest in a topocentric frame
  63.  * into the sighting direction in the same topocentric frame. This implies in this
  64.  * case the frame is fixed and the vector moves.</p>
  65.  * <p>In many case, both approaches will be combined. In our telescope example,
  66.  * we will probably also need to transform the observing direction in the topocentric
  67.  * frame into the observing direction in inertial frame taking into account the observatory
  68.  * location and the Earth rotation, which would essentially be an application of the
  69.  * first approach.</p>
  70.  *
  71.  * <p>These examples show that a rotation is what the user wants it to be. This
  72.  * class does not push the user towards one specific definition and hence does not
  73.  * provide methods like <code>projectVectorIntoDestinationFrame</code> or
  74.  * <code>computeTransformedDirection</code>. It provides simpler and more generic
  75.  * methods: {@link #applyTo(Vector3D) applyTo(Vector3D)} and {@link
  76.  * #applyInverseTo(Vector3D) applyInverseTo(Vector3D)}.</p>
  77.  *
  78.  * <p>Since a rotation is basically a vectorial operator, several rotations can be
  79.  * composed together and the composite operation <code>r = r<sub>1</sub> o
  80.  * r<sub>2</sub></code> (which means that for each vector <code>u</code>,
  81.  * <code>r(u) = r<sub>1</sub>(r<sub>2</sub>(u))</code>) is also a rotation. Hence
  82.  * we can consider that in addition to vectors, a rotation can be applied to other
  83.  * rotations as well (or to itself). With our previous notations, we would say we
  84.  * can apply <code>r<sub>1</sub></code> to <code>r<sub>2</sub></code> and the result
  85.  * we get is <code>r = r<sub>1</sub> o r<sub>2</sub></code>. For this purpose, the
  86.  * class provides the methods: {@link #applyTo(Rotation) applyTo(Rotation)} and
  87.  * {@link #applyInverseTo(Rotation) applyInverseTo(Rotation)}.</p>
  88.  *
  89.  * <p>Rotations are guaranteed to be immutable objects.</p>
  90.  *
  91.  * @see Vector3D
  92.  * @see RotationOrder
  93.  * @since 1.2
  94.  */

  96. public class Rotation implements Serializable {

  98.   /** Identity rotation. */
  99.   public static final Rotation IDENTITY = new Rotation(1.0, 0.0, 0.0, 0.0, false);

  101.   /** Serializable version identifier */
  102.   private static final long serialVersionUID = -2153622329907944313L;

  104.   /** Scalar coordinate of the quaternion. */
  105.   private final double q0;

  107.   /** First coordinate of the vectorial part of the quaternion. */
  108.   private final double q1;

  110.   /** Second coordinate of the vectorial part of the quaternion. */
  111.   private final double q2;

  113.   /** Third coordinate of the vectorial part of the quaternion. */
  114.   private final double q3;

  116.   /** Build a rotation from the quaternion coordinates.
  117.    * <p>A rotation can be built from a <em>normalized</em> quaternion,
  118.    * i.e. a quaternion for which q<sub>0</sub><sup>2</sup> +
  119.    * q<sub>1</sub><sup>2</sup> + q<sub>2</sub><sup>2</sup> +
  120.    * q<sub>3</sub><sup>2</sup> = 1. If the quaternion is not normalized,
  121.    * the constructor can normalize it in a preprocessing step.</p>
  122.    * <p>Note that some conventions put the scalar part of the quaternion
  123.    * as the 4<sup>th</sup> component and the vector part as the first three
  124.    * components. This is <em>not</em> our convention. We put the scalar part
  125.    * as the first component.</p>
  126.    * @param q0 scalar part of the quaternion
  127.    * @param q1 first coordinate of the vectorial part of the quaternion
  128.    * @param q2 second coordinate of the vectorial part of the quaternion
  129.    * @param q3 third coordinate of the vectorial part of the quaternion
  130.    * @param needsNormalization if true, the coordinates are considered
  131.    * not to be normalized, a normalization preprocessing step is performed
  132.    * before using them
  133.    */
  134.   public Rotation(double q0, double q1, double q2, double q3,
  135.                   boolean needsNormalization) {

  137.     if (needsNormalization) {
  138.       // normalization preprocessing
  139.       double inv = 1.0 / FastMath.sqrt(q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3);
  140.       q0 *= inv;
  141.       q1 *= inv;
  142.       q2 *= inv;
  143.       q3 *= inv;
  144.     }

  146.     this.q0 = q0;
  147.     this.q1 = q1;
  148.     this.q2 = q2;
  149.     this.q3 = q3;

  151.   }

  153.   /** Build a rotation from an axis and an angle.
  154.    * <p>We use the convention that angles are oriented according to
  155.    * the effect of the rotation on vectors around the axis. That means
  156.    * that if (i, j, k) is a direct frame and if we first provide +k as
  157.    * the axis and &pi;/2 as the angle to this constructor, and then
  158.    * {@link #applyTo(Vector3D) apply} the instance to +i, we will get
  159.    * +j.</p>
  160.    * <p>Another way to represent our convention is to say that a rotation
  161.    * of angle &theta; about the unit vector (x, y, z) is the same as the
  162.    * rotation build from quaternion components { cos(-&theta;/2),
  163.    * x * sin(-&theta;/2), y * sin(-&theta;/2), z * sin(-&theta;/2) }.
  164.    * Note the minus sign on the angle!</p>
  165.    * <p>On the one hand this convention is consistent with a vectorial
  166.    * perspective (moving vectors in fixed frames), on the other hand it
  167.    * is different from conventions with a frame perspective (fixed vectors
  168.    * viewed from different frames) like the ones used for example in spacecraft
  169.    * attitude community or in the graphics community.</p>
  170.    * @param axis axis around which to rotate
  171.    * @param angle rotation angle.
  172.    * @exception MathIllegalArgumentException if the axis norm is zero
  173.    */
  174.   public Rotation(Vector3D axis, double angle) throws MathIllegalArgumentException {

  176.     double norm = axis.getNorm();
  177.     if (norm == 0) {
  178.       throw new MathIllegalArgumentException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_AXIS);
  179.     }

  181.     double halfAngle = -0.5 * angle;
  182.     double coeff = FastMath.sin(halfAngle) / norm;

  184.     q0 = FastMath.cos (halfAngle);
  185.     q1 = coeff * axis.getX();
  186.     q2 = coeff * axis.getY();
  187.     q3 = coeff * axis.getZ();

  189.   }

  191.   /** Build a rotation from a 3X3 matrix.

  193.    * <p>Rotation matrices are orthogonal matrices, i.e. unit matrices
  194.    * (which are matrices for which m.m<sup>T</sup> = I) with real
  195.    * coefficients. The module of the determinant of unit matrices is
  196.    * 1, among the orthogonal 3X3 matrices, only the ones having a
  197.    * positive determinant (+1) are rotation matrices.</p>

  199.    * <p>When a rotation is defined by a matrix with truncated values
  200.    * (typically when it is extracted from a technical sheet where only
  201.    * four to five significant digits are available), the matrix is not
  202.    * orthogonal anymore. This constructor handles this case
  203.    * transparently by using a copy of the given matrix and applying a
  204.    * correction to the copy in order to perfect its orthogonality. If
  205.    * the Frobenius norm of the correction needed is above the given
  206.    * threshold, then the matrix is considered to be too far from a
  207.    * true rotation matrix and an exception is thrown.<p>

  209.    * @param m rotation matrix
  210.    * @param threshold convergence threshold for the iterative
  211.    * orthogonality correction (convergence is reached when the
  212.    * difference between two steps of the Frobenius norm of the
  213.    * correction is below this threshold)

  215.    * @exception NotARotationMatrixException if the matrix is not a 3X3
  216.    * matrix, or if it cannot be transformed into an orthogonal matrix
  217.    * with the given threshold, or if the determinant of the resulting
  218.    * orthogonal matrix is negative

  220.    */
  221.   public Rotation(double[][] m, double threshold)
  222.     throws NotARotationMatrixException {

  224.     // dimension check
  225.     if ((m.length != 3) || (m[0].length != 3) ||
  226.         (m[1].length != 3) || (m[2].length != 3)) {
  227.       throw new NotARotationMatrixException(
  228.               LocalizedFormats.ROTATION_MATRIX_DIMENSIONS,
  229.               m.length, m[0].length);
  230.     }

  232.     // compute a "close" orthogonal matrix
  233.     double[][] ort = orthogonalizeMatrix(m, threshold);

  235.     // check the sign of the determinant
  236.     double det = ort[0][0] * (ort[1][1] * ort[2][2] - ort[2][1] * ort[1][2]) -
  237.                  ort[1][0] * (ort[0][1] * ort[2][2] - ort[2][1] * ort[0][2]) +
  238.                  ort[2][0] * (ort[0][1] * ort[1][2] - ort[1][1] * ort[0][2]);
  239.     if (det < 0.0) {
  240.       throw new NotARotationMatrixException(
  241.               LocalizedFormats.CLOSEST_ORTHOGONAL_MATRIX_HAS_NEGATIVE_DETERMINANT,
  242.               det);
  243.     }

  245.     double[] quat = mat2quat(ort);
  246.     q0 = quat[0];
  247.     q1 = quat[1];
  248.     q2 = quat[2];
  249.     q3 = quat[3];

  251.   }

  253.   /** Build the rotation that transforms a pair of vector into another pair.

  255.    * <p>Except for possible scale factors, if the instance were applied to
  256.    * the pair (u<sub>1</sub>, u<sub>2</sub>) it will produce the pair
  257.    * (v<sub>1</sub>, v<sub>2</sub>).</p>

  259.    * <p>If the angular separation between u<sub>1</sub> and u<sub>2</sub> is
  260.    * not the same as the angular separation between v<sub>1</sub> and
  261.    * v<sub>2</sub>, then a corrected v'<sub>2</sub> will be used rather than
  262.    * v<sub>2</sub>, the corrected vector will be in the (v<sub>1</sub>,
  263.    * v<sub>2</sub>) plane.</p>

  265.    * @param u1 first vector of the origin pair
  266.    * @param u2 second vector of the origin pair
  267.    * @param v1 desired image of u1 by the rotation
  268.    * @param v2 desired image of u2 by the rotation
  269.    * @exception MathArithmeticException if the norm of one of the vectors is zero,
  270.    * or if one of the pair is degenerated (i.e. the vectors of the pair are colinear)
  271.    */
  272.   public Rotation(Vector3D u1, Vector3D u2, Vector3D v1, Vector3D v2)
  273.       throws MathArithmeticException {

  275.       // build orthonormalized base from u1, u2
  276.       // this fails when vectors are null or colinear, which is forbidden to define a rotation
  277.       final Vector3D u3 = u1.crossProduct(u2).normalize();
  278.       u2 = u3.crossProduct(u1).normalize();
  279.       u1 = u1.normalize();

  281.       // build an orthonormalized base from v1, v2
  282.       // this fails when vectors are null or colinear, which is forbidden to define a rotation
  283.       final Vector3D v3 = v1.crossProduct(v2).normalize();
  284.       v2 = v3.crossProduct(v1).normalize();
  285.       v1 = v1.normalize();

  287.       // buid a matrix transforming the first base into the second one
  288.       final double[][] m = new double[][] {
  289.           {
  290.               MathArrays.linearCombination(u1.getX(), v1.getX(), u2.getX(), v2.getX(), u3.getX(), v3.getX()),
  291.               MathArrays.linearCombination(u1.getY(), v1.getX(), u2.getY(), v2.getX(), u3.getY(), v3.getX()),
  292.               MathArrays.linearCombination(u1.getZ(), v1.getX(), u2.getZ(), v2.getX(), u3.getZ(), v3.getX())
  293.           },
  294.           {
  295.               MathArrays.linearCombination(u1.getX(), v1.getY(), u2.getX(), v2.getY(), u3.getX(), v3.getY()),
  296.               MathArrays.linearCombination(u1.getY(), v1.getY(), u2.getY(), v2.getY(), u3.getY(), v3.getY()),
  297.               MathArrays.linearCombination(u1.getZ(), v1.getY(), u2.getZ(), v2.getY(), u3.getZ(), v3.getY())
  298.           },
  299.           {
  300.               MathArrays.linearCombination(u1.getX(), v1.getZ(), u2.getX(), v2.getZ(), u3.getX(), v3.getZ()),
  301.               MathArrays.linearCombination(u1.getY(), v1.getZ(), u2.getY(), v2.getZ(), u3.getY(), v3.getZ()),
  302.               MathArrays.linearCombination(u1.getZ(), v1.getZ(), u2.getZ(), v2.getZ(), u3.getZ(), v3.getZ())
  303.           }
  304.       };

  306.       double[] quat = mat2quat(m);
  307.       q0 = quat[0];
  308.       q1 = quat[1];
  309.       q2 = quat[2];
  310.       q3 = quat[3];

  312.   }

  314.   /** Build one of the rotations that transform one vector into another one.

  316.    * <p>Except for a possible scale factor, if the instance were
  317.    * applied to the vector u it will produce the vector v. There is an
  318.    * infinite number of such rotations, this constructor choose the
  319.    * one with the smallest associated angle (i.e. the one whose axis
  320.    * is orthogonal to the (u, v) plane). If u and v are colinear, an
  321.    * arbitrary rotation axis is chosen.</p>

  323.    * @param u origin vector
  324.    * @param v desired image of u by the rotation
  325.    * @exception MathArithmeticException if the norm of one of the vectors is zero
  326.    */
  327.   public Rotation(Vector3D u, Vector3D v) throws MathArithmeticException {

  329.     double normProduct = u.getNorm() * v.getNorm();
  330.     if (normProduct == 0) {
  331.         throw new MathArithmeticException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_DEFINING_VECTOR);
  332.     }

  334.     double dot = u.dotProduct(v);

  336.     if (dot < ((2.0e-15 - 1.0) * normProduct)) {
  337.       // special case u = -v: we select a PI angle rotation around
  338.       // an arbitrary vector orthogonal to u
  339.       Vector3D w = u.orthogonal();
  340.       q0 = 0.0;
  341.       q1 = -w.getX();
  342.       q2 = -w.getY();
  343.       q3 = -w.getZ();
  344.     } else {
  345.       // general case: (u, v) defines a plane, we select
  346.       // the shortest possible rotation: axis orthogonal to this plane
  347.       q0 = FastMath.sqrt(0.5 * (1.0 + dot / normProduct));
  348.       double coeff = 1.0 / (2.0 * q0 * normProduct);
  349.       Vector3D q = v.crossProduct(u);
  350.       q1 = coeff * q.getX();
  351.       q2 = coeff * q.getY();
  352.       q3 = coeff * q.getZ();
  353.     }

  355.   }

  357.   /** Build a rotation from three Cardan or Euler elementary rotations.

  359.    * <p>Cardan rotations are three successive rotations around the
  360.    * canonical axes X, Y and Z, each axis being used once. There are
  361.    * 6 such sets of rotations (XYZ, XZY, YXZ, YZX, ZXY and ZYX). Euler
  362.    * rotations are three successive rotations around the canonical
  363.    * axes X, Y and Z, the first and last rotations being around the
  364.    * same axis. There are 6 such sets of rotations (XYX, XZX, YXY,
  365.    * YZY, ZXZ and ZYZ), the most popular one being ZXZ.</p>
  366.    * <p>Beware that many people routinely use the term Euler angles even
  367.    * for what really are Cardan angles (this confusion is especially
  368.    * widespread in the aerospace business where Roll, Pitch and Yaw angles
  369.    * are often wrongly tagged as Euler angles).</p>

  371.    * @param order order of rotations to use
  372.    * @param alpha1 angle of the first elementary rotation
  373.    * @param alpha2 angle of the second elementary rotation
  374.    * @param alpha3 angle of the third elementary rotation
  375.    */
  376.   public Rotation(RotationOrder order,
  377.                   double alpha1, double alpha2, double alpha3) {
  378.       Rotation r1 = new Rotation(order.getA1(), alpha1);
  379.       Rotation r2 = new Rotation(order.getA2(), alpha2);
  380.       Rotation r3 = new Rotation(order.getA3(), alpha3);
  381.       Rotation composed = r1.applyTo(r2.applyTo(r3));
  382.       q0 = composed.q0;
  383.       q1 = composed.q1;
  384.       q2 = composed.q2;
  385.       q3 = composed.q3;
  386.   }

  388.   /** Convert an orthogonal rotation matrix to a quaternion.
  389.    * @param ort orthogonal rotation matrix
  390.    * @return quaternion corresponding to the matrix
  391.    */
  392.   private static double[] mat2quat(final double[][] ort) {

  394.       final double[] quat = new double[4];

  396.       // There are different ways to compute the quaternions elements
  397.       // from the matrix. They all involve computing one element from
  398.       // the diagonal of the matrix, and computing the three other ones
  399.       // using a formula involving a division by the first element,
  400.       // which unfortunately can be zero. Since the norm of the
  401.       // quaternion is 1, we know at least one element has an absolute
  402.       // value greater or equal to 0.5, so it is always possible to
  403.       // select the right formula and avoid division by zero and even
  404.       // numerical inaccuracy. Checking the elements in turn and using
  405.       // the first one greater than 0.45 is safe (this leads to a simple
  406.       // test since qi = 0.45 implies 4 qi^2 - 1 = -0.19)
  407.       double s = ort[0][0] + ort[1][1] + ort[2][2];
  408.       if (s > -0.19) {
  409.           // compute q0 and deduce q1, q2 and q3
  410.           quat[0] = 0.5 * FastMath.sqrt(s + 1.0);
  411.           double inv = 0.25 / quat[0];
  412.           quat[1] = inv * (ort[1][2] - ort[2][1]);
  413.           quat[2] = inv * (ort[2][0] - ort[0][2]);
  414.           quat[3] = inv * (ort[0][1] - ort[1][0]);
  415.       } else {
  416.           s = ort[0][0] - ort[1][1] - ort[2][2];
  417.           if (s > -0.19) {
  418.               // compute q1 and deduce q0, q2 and q3
  419.               quat[1] = 0.5 * FastMath.sqrt(s + 1.0);
  420.               double inv = 0.25 / quat[1];
  421.               quat[0] = inv * (ort[1][2] - ort[2][1]);
  422.               quat[2] = inv * (ort[0][1] + ort[1][0]);
  423.               quat[3] = inv * (ort[0][2] + ort[2][0]);
  424.           } else {
  425.               s = ort[1][1] - ort[0][0] - ort[2][2];
  426.               if (s > -0.19) {
  427.                   // compute q2 and deduce q0, q1 and q3
  428.                   quat[2] = 0.5 * FastMath.sqrt(s + 1.0);
  429.                   double inv = 0.25 / quat[2];
  430.                   quat[0] = inv * (ort[2][0] - ort[0][2]);
  431.                   quat[1] = inv * (ort[0][1] + ort[1][0]);
  432.                   quat[3] = inv * (ort[2][1] + ort[1][2]);
  433.               } else {
  434.                   // compute q3 and deduce q0, q1 and q2
  435.                   s = ort[2][2] - ort[0][0] - ort[1][1];
  436.                   quat[3] = 0.5 * FastMath.sqrt(s + 1.0);
  437.                   double inv = 0.25 / quat[3];
  438.                   quat[0] = inv * (ort[0][1] - ort[1][0]);
  439.                   quat[1] = inv * (ort[0][2] + ort[2][0]);
  440.                   quat[2] = inv * (ort[2][1] + ort[1][2]);
  441.               }
  442.           }
  443.       }

  445.       return quat;

  447.   }

  449.   /** Revert a rotation.
  450.    * Build a rotation which reverse the effect of another
  451.    * rotation. This means that if r(u) = v, then r.revert(v) = u. The
  452.    * instance is not changed.
  453.    * @return a new rotation whose effect is the reverse of the effect
  454.    * of the instance
  455.    */
  456.   public Rotation revert() {
  457.     return new Rotation(-q0, q1, q2, q3, false);
  458.   }

  460.   /** Get the scalar coordinate of the quaternion.
  461.    * @return scalar coordinate of the quaternion
  462.    */
  463.   public double getQ0() {
  464.     return q0;
  465.   }

  467.   /** Get the first coordinate of the vectorial part of the quaternion.
  468.    * @return first coordinate of the vectorial part of the quaternion
  469.    */
  470.   public double getQ1() {
  471.     return q1;
  472.   }

  474.   /** Get the second coordinate of the vectorial part of the quaternion.
  475.    * @return second coordinate of the vectorial part of the quaternion
  476.    */
  477.   public double getQ2() {
  478.     return q2;
  479.   }

  481.   /** Get the third coordinate of the vectorial part of the quaternion.
  482.    * @return third coordinate of the vectorial part of the quaternion
  483.    */
  484.   public double getQ3() {
  485.     return q3;
  486.   }

  488.   /** Get the normalized axis of the rotation.
  489.    * @return normalized axis of the rotation
  490.    * @see #Rotation(Vector3D, double)
  491.    */
  492.   public Vector3D getAxis() {
  493.     double squaredSine = q1 * q1 + q2 * q2 + q3 * q3;
  494.     if (squaredSine == 0) {
  495.       return new Vector3D(1, 0, 0);
  496.     } else if (q0 < 0) {
  497.       double inverse = 1 / FastMath.sqrt(squaredSine);
  498.       return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
  499.     }
  500.     double inverse = -1 / FastMath.sqrt(squaredSine);
  501.     return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
  502.   }

  504.   /** Get the angle of the rotation.
  505.    * @return angle of the rotation (between 0 and &pi;)
  506.    * @see #Rotation(Vector3D, double)
  507.    */
  508.   public double getAngle() {
  509.     if ((q0 < -0.1) || (q0 > 0.1)) {
  510.       return 2 * FastMath.asin(FastMath.sqrt(q1 * q1 + q2 * q2 + q3 * q3));
  511.     } else if (q0 < 0) {
  512.       return 2 * FastMath.acos(-q0);
  513.     }
  514.     return 2 * FastMath.acos(q0);
  515.   }

  517.   /** Get the Cardan or Euler angles corresponding to the instance.

  519.    * <p>The equations show that each rotation can be defined by two
  520.    * different values of the Cardan or Euler angles set. For example
  521.    * if Cardan angles are used, the rotation defined by the angles
  522.    * a<sub>1</sub>, a<sub>2</sub> and a<sub>3</sub> is the same as
  523.    * the rotation defined by the angles &pi; + a<sub>1</sub>, &pi;
  524.    * - a<sub>2</sub> and &pi; + a<sub>3</sub>. This method implements
  525.    * the following arbitrary choices:</p>
  526.    * <ul>
  527.    *   <li>for Cardan angles, the chosen set is the one for which the
  528.    *   second angle is between -&pi;/2 and &pi;/2 (i.e its cosine is
  529.    *   positive),</li>
  530.    *   <li>for Euler angles, the chosen set is the one for which the
  531.    *   second angle is between 0 and &pi; (i.e its sine is positive).</li>
  532.    * </ul>

  534.    * <p>Cardan and Euler angle have a very disappointing drawback: all
  535.    * of them have singularities. This means that if the instance is
  536.    * too close to the singularities corresponding to the given
  537.    * rotation order, it will be impossible to retrieve the angles. For
  538.    * Cardan angles, this is often called gimbal lock. There is
  539.    * <em>nothing</em> to do to prevent this, it is an intrinsic problem
  540.    * with Cardan and Euler representation (but not a problem with the
  541.    * rotation itself, which is perfectly well defined). For Cardan
  542.    * angles, singularities occur when the second angle is close to
  543.    * -&pi;/2 or +&pi;/2, for Euler angle singularities occur when the
  544.    * second angle is close to 0 or &pi;, this implies that the identity
  545.    * rotation is always singular for Euler angles!</p>

  547.    * @param order rotation order to use
  548.    * @return an array of three angles, in the order specified by the set
  549.    * @exception CardanEulerSingularityException if the rotation is
  550.    * singular with respect to the angles set specified
  551.    */
  552.   public double[] getAngles(RotationOrder order)
  553.     throws CardanEulerSingularityException {

  555.     if (order == RotationOrder.XYZ) {

  557.       // r (Vector3D.plusK) coordinates are :
  558.       //  sin (theta), -cos (theta) sin (phi), cos (theta) cos (phi)
  559.       // (-r) (Vector3D.plusI) coordinates are :
  560.       // cos (psi) cos (theta), -sin (psi) cos (theta), sin (theta)
  561.       // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
  562.       Vector3D v1 = applyTo(Vector3D.PLUS_K);
  563.       Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
  564.       if  ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
  565.         throw new CardanEulerSingularityException(true);
  566.       }
  567.       return new double[] {
  568.         FastMath.atan2(-(v1.getY()), v1.getZ()),
  569.         FastMath.asin(v2.getZ()),
  570.         FastMath.atan2(-(v2.getY()), v2.getX())
  571.       };

  573.     } else if (order == RotationOrder.XZY) {

  575.       // r (Vector3D.plusJ) coordinates are :
  576.       // -sin (psi), cos (psi) cos (phi), cos (psi) sin (phi)
  577.       // (-r) (Vector3D.plusI) coordinates are :
  578.       // cos (theta) cos (psi), -sin (psi), sin (theta) cos (psi)
  579.       // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
  580.       Vector3D v1 = applyTo(Vector3D.PLUS_J);
  581.       Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
  582.       if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
  583.         throw new CardanEulerSingularityException(true);
  584.       }
  585.       return new double[] {
  586.         FastMath.atan2(v1.getZ(), v1.getY()),
  587.        -FastMath.asin(v2.getY()),
  588.         FastMath.atan2(v2.getZ(), v2.getX())
  589.       };

  591.     } else if (order == RotationOrder.YXZ) {

  593.       // r (Vector3D.plusK) coordinates are :
  594.       //  cos (phi) sin (theta), -sin (phi), cos (phi) cos (theta)
  595.       // (-r) (Vector3D.plusJ) coordinates are :
  596.       // sin (psi) cos (phi), cos (psi) cos (phi), -sin (phi)
  597.       // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
  598.       Vector3D v1 = applyTo(Vector3D.PLUS_K);
  599.       Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
  600.       if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
  601.         throw new CardanEulerSingularityException(true);
  602.       }
  603.       return new double[] {
  604.         FastMath.atan2(v1.getX(), v1.getZ()),
  605.        -FastMath.asin(v2.getZ()),
  606.         FastMath.atan2(v2.getX(), v2.getY())
  607.       };

  609.     } else if (order == RotationOrder.YZX) {

  611.       // r (Vector3D.plusI) coordinates are :
  612.       // cos (psi) cos (theta), sin (psi), -cos (psi) sin (theta)
  613.       // (-r) (Vector3D.plusJ) coordinates are :
  614.       // sin (psi), cos (phi) cos (psi), -sin (phi) cos (psi)
  615.       // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
  616.       Vector3D v1 = applyTo(Vector3D.PLUS_I);
  617.       Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
  618.       if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
  619.         throw new CardanEulerSingularityException(true);
  620.       }
  621.       return new double[] {
  622.         FastMath.atan2(-(v1.getZ()), v1.getX()),
  623.         FastMath.asin(v2.getX()),
  624.         FastMath.atan2(-(v2.getZ()), v2.getY())
  625.       };

  627.     } else if (order == RotationOrder.ZXY) {

  629.       // r (Vector3D.plusJ) coordinates are :
  630.       // -cos (phi) sin (psi), cos (phi) cos (psi), sin (phi)
  631.       // (-r) (Vector3D.plusK) coordinates are :
  632.       // -sin (theta) cos (phi), sin (phi), cos (theta) cos (phi)
  633.       // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
  634.       Vector3D v1 = applyTo(Vector3D.PLUS_J);
  635.       Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
  636.       if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
  637.         throw new CardanEulerSingularityException(true);
  638.       }
  639.       return new double[] {
  640.         FastMath.atan2(-(v1.getX()), v1.getY()),
  641.         FastMath.asin(v2.getY()),
  642.         FastMath.atan2(-(v2.getX()), v2.getZ())
  643.       };

  645.     } else if (order == RotationOrder.ZYX) {

  647.       // r (Vector3D.plusI) coordinates are :
  648.       //  cos (theta) cos (psi), cos (theta) sin (psi), -sin (theta)
  649.       // (-r) (Vector3D.plusK) coordinates are :
  650.       // -sin (theta), sin (phi) cos (theta), cos (phi) cos (theta)
  651.       // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
  652.       Vector3D v1 = applyTo(Vector3D.PLUS_I);
  653.       Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
  654.       if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
  655.         throw new CardanEulerSingularityException(true);
  656.       }
  657.       return new double[] {
  658.         FastMath.atan2(v1.getY(), v1.getX()),
  659.        -FastMath.asin(v2.getX()),
  660.         FastMath.atan2(v2.getY(), v2.getZ())
  661.       };

  663.     } else if (order == RotationOrder.XYX) {

  665.       // r (Vector3D.plusI) coordinates are :
  666.       //  cos (theta), sin (phi1) sin (theta), -cos (phi1) sin (theta)
  667.       // (-r) (Vector3D.plusI) coordinates are :
  668.       // cos (theta), sin (theta) sin (phi2), sin (theta) cos (phi2)
  669.       // and we can choose to have theta in the interval [0 ; PI]
  670.       Vector3D v1 = applyTo(Vector3D.PLUS_I);
  671.       Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
  672.       if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
  673.         throw new CardanEulerSingularityException(false);
  674.       }
  675.       return new double[] {
  676.         FastMath.atan2(v1.getY(), -v1.getZ()),
  677.         FastMath.acos(v2.getX()),
  678.         FastMath.atan2(v2.getY(), v2.getZ())
  679.       };

  681.     } else if (order == RotationOrder.XZX) {

  683.       // r (Vector3D.plusI) coordinates are :
  684.       //  cos (psi), cos (phi1) sin (psi), sin (phi1) sin (psi)
  685.       // (-r) (Vector3D.plusI) coordinates are :
  686.       // cos (psi), -sin (psi) cos (phi2), sin (psi) sin (phi2)
  687.       // and we can choose to have psi in the interval [0 ; PI]
  688.       Vector3D v1 = applyTo(Vector3D.PLUS_I);
  689.       Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
  690.       if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
  691.         throw new CardanEulerSingularityException(false);
  692.       }
  693.       return new double[] {
  694.         FastMath.atan2(v1.getZ(), v1.getY()),
  695.         FastMath.acos(v2.getX()),
  696.         FastMath.atan2(v2.getZ(), -v2.getY())
  697.       };

  699.     } else if (order == RotationOrder.YXY) {

  701.       // r (Vector3D.plusJ) coordinates are :
  702.       //  sin (theta1) sin (phi), cos (phi), cos (theta1) sin (phi)
  703.       // (-r) (Vector3D.plusJ) coordinates are :
  704.       // sin (phi) sin (theta2), cos (phi), -sin (phi) cos (theta2)
  705.       // and we can choose to have phi in the interval [0 ; PI]
  706.       Vector3D v1 = applyTo(Vector3D.PLUS_J);
  707.       Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
  708.       if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
  709.         throw new CardanEulerSingularityException(false);
  710.       }
  711.       return new double[] {
  712.         FastMath.atan2(v1.getX(), v1.getZ()),
  713.         FastMath.acos(v2.getY()),
  714.         FastMath.atan2(v2.getX(), -v2.getZ())
  715.       };

  717.     } else if (order == RotationOrder.YZY) {

  719.       // r (Vector3D.plusJ) coordinates are :
  720.       //  -cos (theta1) sin (psi), cos (psi), sin (theta1) sin (psi)
  721.       // (-r) (Vector3D.plusJ) coordinates are :
  722.       // sin (psi) cos (theta2), cos (psi), sin (psi) sin (theta2)
  723.       // and we can choose to have psi in the interval [0 ; PI]
  724.       Vector3D v1 = applyTo(Vector3D.PLUS_J);
  725.       Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
  726.       if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
  727.         throw new CardanEulerSingularityException(false);
  728.       }
  729.       return new double[] {
  730.         FastMath.atan2(v1.getZ(), -v1.getX()),
  731.         FastMath.acos(v2.getY()),
  732.         FastMath.atan2(v2.getZ(), v2.getX())
  733.       };

  735.     } else if (order == RotationOrder.ZXZ) {

  737.       // r (Vector3D.plusK) coordinates are :
  738.       //  sin (psi1) sin (phi), -cos (psi1) sin (phi), cos (phi)
  739.       // (-r) (Vector3D.plusK) coordinates are :
  740.       // sin (phi) sin (psi2), sin (phi) cos (psi2), cos (phi)
  741.       // and we can choose to have phi in the interval [0 ; PI]
  742.       Vector3D v1 = applyTo(Vector3D.PLUS_K);
  743.       Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
  744.       if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
  745.         throw new CardanEulerSingularityException(false);
  746.       }
  747.       return new double[] {
  748.         FastMath.atan2(v1.getX(), -v1.getY()),
  749.         FastMath.acos(v2.getZ()),
  750.         FastMath.atan2(v2.getX(), v2.getY())
  751.       };

  753.     } else { // last possibility is ZYZ

  755.       // r (Vector3D.plusK) coordinates are :
  756.       //  cos (psi1) sin (theta), sin (psi1) sin (theta), cos (theta)
  757.       // (-r) (Vector3D.plusK) coordinates are :
  758.       // -sin (theta) cos (psi2), sin (theta) sin (psi2), cos (theta)
  759.       // and we can choose to have theta in the interval [0 ; PI]
  760.       Vector3D v1 = applyTo(Vector3D.PLUS_K);
  761.       Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
  762.       if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
  763.         throw new CardanEulerSingularityException(false);
  764.       }
  765.       return new double[] {
  766.         FastMath.atan2(v1.getY(), v1.getX()),
  767.         FastMath.acos(v2.getZ()),
  768.         FastMath.atan2(v2.getY(), -v2.getX())
  769.       };

  771.     }

  773.   }

  775.   /** Get the 3X3 matrix corresponding to the instance
  776.    * @return the matrix corresponding to the instance
  777.    */
  778.   public double[][] getMatrix() {

  780.     // products
  781.     double q0q0  = q0 * q0;
  782.     double q0q1  = q0 * q1;
  783.     double q0q2  = q0 * q2;
  784.     double q0q3  = q0 * q3;
  785.     double q1q1  = q1 * q1;
  786.     double q1q2  = q1 * q2;
  787.     double q1q3  = q1 * q3;
  788.     double q2q2  = q2 * q2;
  789.     double q2q3  = q2 * q3;
  790.     double q3q3  = q3 * q3;

  792.     // create the matrix
  793.     double[][] m = new double[3][];
  794.     m[0] = new double[3];
  795.     m[1] = new double[3];
  796.     m[2] = new double[3];

  798.     m [0][0] = 2.0 * (q0q0 + q1q1) - 1.0;
  799.     m [1][0] = 2.0 * (q1q2 - q0q3);
  800.     m [2][0] = 2.0 * (q1q3 + q0q2);

  802.     m [0][1] = 2.0 * (q1q2 + q0q3);
  803.     m [1][1] = 2.0 * (q0q0 + q2q2) - 1.0;
  804.     m [2][1] = 2.0 * (q2q3 - q0q1);

  806.     m [0][2] = 2.0 * (q1q3 - q0q2);
  807.     m [1][2] = 2.0 * (q2q3 + q0q1);
  808.     m [2][2] = 2.0 * (q0q0 + q3q3) - 1.0;

  810.     return m;

  812.   }

  814.   /** Apply the rotation to a vector.
  815.    * @param u vector to apply the rotation to
  816.    * @return a new vector which is the image of u by the rotation
  817.    */
  818.   public Vector3D applyTo(Vector3D u) {

  820.     double x = u.getX();
  821.     double y = u.getY();
  822.     double z = u.getZ();

  824.     double s = q1 * x + q2 * y + q3 * z;

  826.     return new Vector3D(2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x,
  827.                         2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y,
  828.                         2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - z);

  830.   }

  832.   /** Apply the rotation to a vector stored in an array.
  833.    * @param in an array with three items which stores vector to rotate
  834.    * @param out an array with three items to put result to (it can be the same
  835.    * array as in)
  836.    */
  837.   public void applyTo(final double[] in, final double[] out) {

  839.       final double x = in[0];
  840.       final double y = in[1];
  841.       final double z = in[2];

  843.       final double s = q1 * x + q2 * y + q3 * z;

  845.       out[0] = 2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x;
  846.       out[1] = 2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y;
  847.       out[2] = 2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - z;

  849.   }

  851.   /** Apply the inverse of the rotation to a vector.
  852.    * @param u vector to apply the inverse of the rotation to
  853.    * @return a new vector which such that u is its image by the rotation
  854.    */
  855.   public Vector3D applyInverseTo(Vector3D u) {

  857.     double x = u.getX();
  858.     double y = u.getY();
  859.     double z = u.getZ();

  861.     double s = q1 * x + q2 * y + q3 * z;
  862.     double m0 = -q0;

  864.     return new Vector3D(2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x,
  865.                         2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y,
  866.                         2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - z);

  868.   }

  870.   /** Apply the inverse of the rotation to a vector stored in an array.
  871.    * @param in an array with three items which stores vector to rotate
  872.    * @param out an array with three items to put result to (it can be the same
  873.    * array as in)
  874.    */
  875.   public void applyInverseTo(final double[] in, final double[] out) {

  877.       final double x = in[0];
  878.       final double y = in[1];
  879.       final double z = in[2];

  881.       final double s = q1 * x + q2 * y + q3 * z;
  882.       final double m0 = -q0;

  884.       out[0] = 2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x;
  885.       out[1] = 2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y;
  886.       out[2] = 2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - z;

  888.   }

  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.   }

  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.   }

  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;

  953.     double[][] o = new double[3][3];
  954.     double[] o0 = o[0];
  955.     double[] o1 = o[1];
  956.     double[] o2 = o[2];

  958.     // iterative correction: Xn+1 = Xn - 0.5 * (Xn.Mt.Xn - M)
  959.     int i = 0;
  960.     while (++i < 11) {

  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;

  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]);

  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];

  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;

  1000.       // convergence test
  1001.       if (FastMath.abs(fn1 - fn) <= threshold) {
  1002.           return o;
  1003.       }

  1005.       // prepare next iteration
  1006.       x00 = o0[0];
  1007.       x01 = o0[1];
  1008.       x02 = o0[2];
  1009.       x10 = o1[0];
  1010.       x11 = o1[1];
  1011.       x12 = o1[2];
  1012.       x20 = o2[0];
  1013.       x21 = o2[1];
  1014.       x22 = o2[2];
  1015.       fn  = fn1;

  1017.     }

  1019.     // the algorithm did not converge after 10 iterations
  1020.     throw new NotARotationMatrixException(
  1021.             LocalizedFormats.UNABLE_TO_ORTHOGONOLIZE_MATRIX,
  1022.             i - 1);
  1023.   }

  1025.   /** Compute the <i>distance</i> between two rotations.
  1026.    * <p>The <i>distance</i> is intended here as a way to check if two
  1027.    * rotations are almost similar (i.e. they transform vectors the same way)
  1028.    * or very different. It is mathematically defined as the angle of
  1029.    * the rotation r that prepended to one of the rotations gives the other
  1030.    * one:</p>
  1031.    * <pre>
  1032.    *        r<sub>1</sub>(r) = r<sub>2</sub>
  1033.    * </pre>
  1034.    * <p>This distance is an angle between 0 and &pi;. Its value is the smallest
  1035.    * possible upper bound of the angle in radians between r<sub>1</sub>(v)
  1036.    * and r<sub>2</sub>(v) for all possible vectors v. This upper bound is
  1037.    * reached for some v. The distance is equal to 0 if and only if the two
  1038.    * rotations are identical.</p>
  1039.    * <p>Comparing two rotations should always be done using this value rather
  1040.    * than for example comparing the components of the quaternions. It is much
  1041.    * more stable, and has a geometric meaning. Also comparing quaternions
  1042.    * components is error prone since for example quaternions (0.36, 0.48, -0.48, -0.64)
  1043.    * and (-0.36, -0.48, 0.48, 0.64) represent exactly the same rotation despite
  1044.    * their components are different (they are exact opposites).</p>
  1045.    * @param r1 first rotation
  1046.    * @param r2 second rotation
  1047.    * @return <i>distance</i> between r1 and r2
  1048.    */
  1049.   public static double distance(Rotation r1, Rotation r2) {
  1050.       return r1.applyInverseTo(r2).getAngle();
  1051.   }

  1053. }
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