/*
    * Licensed to the Apache Software Foundation (ASF) under one or more
    * contributor license agreements.  See the NOTICE file distributed with
    * this work for additional information regarding copyright ownership.
    * The ASF licenses this file to You under the Apache License, Version 2.0
    * (the "License"); you may not use this file except in compliance with
    * the License.  You may obtain a copy of the License at
    *
    *      http://www.apache.org/licenses/LICENSE-2.0
   *
   * Unless required by applicable law or agreed to in writing, software
   * distributed under the License is distributed on an "AS IS" BASIS,
   * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
   * See the License for the specific language governing permissions and
   * limitations under the License.
   */
  
  package org.apache.commons.math3.geometry.euclidean.threed;
  
  import java.io.Serializable;
  
  import org.apache.commons.math3.exception.MathArithmeticException;
  import org.apache.commons.math3.exception.MathIllegalArgumentException;
  import org.apache.commons.math3.exception.util.LocalizedFormats;
  import org.apache.commons.math3.util.FastMath;
  import org.apache.commons.math3.util.MathArrays;
  
  /**
   * This class implements rotations in a three-dimensional space.
   *
   * <p>Rotations can be represented by several different mathematical
   * entities (matrices, axe and angle, Cardan or Euler angles,
   * quaternions). This class presents an higher level abstraction, more
   * user-oriented and hiding this implementation details. Well, for the
   * curious, we use quaternions for the internal representation. The
   * user can build a rotation from any of these representations, and
   * any of these representations can be retrieved from a
   * <code>Rotation</code> instance (see the various constructors and
   * getters). In addition, a rotation can also be built implicitly
   * from a set of vectors and their image.</p>
   * <p>This implies that this class can be used to convert from one
   * representation to another one. For example, converting a rotation
   * matrix into a set of Cardan angles from can be done using the
   * following single line of code:</p>
   * <pre>
   * double[] angles = new Rotation(matrix, 1.0e-10).getAngles(RotationOrder.XYZ);
   * </pre>
   * <p>Focus is oriented on what a rotation <em>do</em> rather than on its
   * underlying representation. Once it has been built, and regardless of its
   * internal representation, a rotation is an <em>operator</em> which basically
   * transforms three dimensional {@link Vector3D vectors} into other three
   * dimensional {@link Vector3D vectors}. Depending on the application, the
   * meaning of these vectors may vary and the semantics of the rotation also.</p>
   * <p>For example in an spacecraft attitude simulation tool, users will often
   * consider the vectors are fixed (say the Earth direction for example) and the
   * frames change. The rotation transforms the coordinates of the vector in inertial
   * frame into the coordinates of the same vector in satellite frame. In this
   * case, the rotation implicitly defines the relation between the two frames.</p>
   * <p>Another example could be a telescope control application, where the rotation
   * would transform the sighting direction at rest into the desired observing
   * direction when the telescope is pointed towards an object of interest. In this
   * case the rotation transforms the direction at rest in a topocentric frame
   * into the sighting direction in the same topocentric frame. This implies in this
   * case the frame is fixed and the vector moves.</p>
   * <p>In many case, both approaches will be combined. In our telescope example,
   * we will probably also need to transform the observing direction in the topocentric
   * frame into the observing direction in inertial frame taking into account the observatory
   * location and the Earth rotation, which would essentially be an application of the
   * first approach.</p>
   *
   * <p>These examples show that a rotation is what the user wants it to be. This
   * class does not push the user towards one specific definition and hence does not
   * provide methods like <code>projectVectorIntoDestinationFrame</code> or
   * <code>computeTransformedDirection</code>. It provides simpler and more generic
   * methods: {@link #applyTo(Vector3D) applyTo(Vector3D)} and {@link
   * #applyInverseTo(Vector3D) applyInverseTo(Vector3D)}.</p>
   *
   * <p>Since a rotation is basically a vectorial operator, several rotations can be
   * composed together and the composite operation <code>r = r<sub>1</sub> o
   * r<sub>2</sub></code> (which means that for each vector <code>u</code>,
   * <code>r(u) = r<sub>1</sub>(r<sub>2</sub>(u))</code>) is also a rotation. Hence
   * we can consider that in addition to vectors, a rotation can be applied to other
   * rotations as well (or to itself). With our previous notations, we would say we
   * can apply <code>r<sub>1</sub></code> to <code>r<sub>2</sub></code> and the result
   * we get is <code>r = r<sub>1</sub> o r<sub>2</sub></code>. For this purpose, the
   * class provides the methods: {@link #applyTo(Rotation) applyTo(Rotation)} and
   * {@link #applyInverseTo(Rotation) applyInverseTo(Rotation)}.</p>
   *
   * <p>Rotations are guaranteed to be immutable objects.</p>
   *
   * @version $Id: Rotation.java 1416643 2012-12-03 19:37:14Z tn $
   * @see Vector3D
   * @see RotationOrder
   * @since 1.2
   */
  
  public class Rotation implements Serializable {
  
    /** Identity rotation. */
   public static final Rotation IDENTITY = new Rotation(1.0, 0.0, 0.0, 0.0, false);
 
   /** Serializable version identifier */
   private static final long serialVersionUID = -2153622329907944313L;
 
   /** Scalar coordinate of the quaternion. */
   private final double q0;
 
   /** First coordinate of the vectorial part of the quaternion. */
   private final double q1;
 
   /** Second coordinate of the vectorial part of the quaternion. */
   private final double q2;
 
   /** Third coordinate of the vectorial part of the quaternion. */
   private final double q3;
 
   /** Build a rotation from the quaternion coordinates.
    * <p>A rotation can be built from a <em>normalized</em> quaternion,
    * i.e. a quaternion for which q<sub>0</sub><sup>2</sup> +
    * q<sub>1</sub><sup>2</sup> + q<sub>2</sub><sup>2</sup> +
    * q<sub>3</sub><sup>2</sup> = 1. If the quaternion is not normalized,
    * the constructor can normalize it in a preprocessing step.</p>
    * <p>Note that some conventions put the scalar part of the quaternion
    * as the 4<sup>th</sup> component and the vector part as the first three
    * components. This is <em>not</em> our convention. We put the scalar part
    * as the first component.</p>
    * @param q0 scalar part of the quaternion
    * @param q1 first coordinate of the vectorial part of the quaternion
    * @param q2 second coordinate of the vectorial part of the quaternion
    * @param q3 third coordinate of the vectorial part of the quaternion
    * @param needsNormalization if true, the coordinates are considered
    * not to be normalized, a normalization preprocessing step is performed
    * before using them
    */
   public Rotation(double q0, double q1, double q2, double q3,
                   boolean needsNormalization) {
 
     if (needsNormalization) {
       // normalization preprocessing
       double inv = 1.0 / FastMath.sqrt(q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3);
       q0 *= inv;
       q1 *= inv;
       q2 *= inv;
       q3 *= inv;
     }
 
     this.q0 = q0;
     this.q1 = q1;
     this.q2 = q2;
     this.q3 = q3;
 
   }
 
   /** Build a rotation from an axis and an angle.
    * <p>We use the convention that angles are oriented according to
    * the effect of the rotation on vectors around the axis. That means
    * that if (i, j, k) is a direct frame and if we first provide +k as
    * the axis and &pi;/2 as the angle to this constructor, and then
    * {@link #applyTo(Vector3D) apply} the instance to +i, we will get
    * +j.</p>
    * <p>Another way to represent our convention is to say that a rotation
    * of angle &theta; about the unit vector (x, y, z) is the same as the
    * rotation build from quaternion components { cos(-&theta;/2),
    * x * sin(-&theta;/2), y * sin(-&theta;/2), z * sin(-&theta;/2) }.
    * Note the minus sign on the angle!</p>
    * <p>On the one hand this convention is consistent with a vectorial
    * perspective (moving vectors in fixed frames), on the other hand it
    * is different from conventions with a frame perspective (fixed vectors
    * viewed from different frames) like the ones used for example in spacecraft
    * attitude community or in the graphics community.</p>
    * @param axis axis around which to rotate
    * @param angle rotation angle.
    * @exception MathIllegalArgumentException if the axis norm is zero
    */
   public Rotation(Vector3D axis, double angle) throws MathIllegalArgumentException {
 
     double norm = axis.getNorm();
     if (norm == 0) {
       throw new MathIllegalArgumentException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_AXIS);
     }
 
     double halfAngle = -0.5 * angle;
     double coeff = FastMath.sin(halfAngle) / norm;
 
     q0 = FastMath.cos (halfAngle);
     q1 = coeff * axis.getX();
     q2 = coeff * axis.getY();
     q3 = coeff * axis.getZ();
 
   }
 
   /** Build a rotation from a 3X3 matrix.
 
    * <p>Rotation matrices are orthogonal matrices, i.e. unit matrices
    * (which are matrices for which m.m<sup>T</sup> = I) with real
    * coefficients. The module of the determinant of unit matrices is
    * 1, among the orthogonal 3X3 matrices, only the ones having a
    * positive determinant (+1) are rotation matrices.</p>
 
    * <p>When a rotation is defined by a matrix with truncated values
    * (typically when it is extracted from a technical sheet where only
    * four to five significant digits are available), the matrix is not
    * orthogonal anymore. This constructor handles this case
    * transparently by using a copy of the given matrix and applying a
    * correction to the copy in order to perfect its orthogonality. If
    * the Frobenius norm of the correction needed is above the given
    * threshold, then the matrix is considered to be too far from a
    * true rotation matrix and an exception is thrown.<p>
 
    * @param m rotation matrix
    * @param threshold convergence threshold for the iterative
    * orthogonality correction (convergence is reached when the
    * difference between two steps of the Frobenius norm of the
    * correction is below this threshold)
 
    * @exception NotARotationMatrixException if the matrix is not a 3X3
    * matrix, or if it cannot be transformed into an orthogonal matrix
    * with the given threshold, or if the determinant of the resulting
    * orthogonal matrix is negative
 
    */
   public Rotation(double[][] m, double threshold)
     throws NotARotationMatrixException {
 
     // dimension check
     if ((m.length != 3) || (m[0].length != 3) ||
         (m[1].length != 3) || (m[2].length != 3)) {
       throw new NotARotationMatrixException(
               LocalizedFormats.ROTATION_MATRIX_DIMENSIONS,
               m.length, m[0].length);
     }
 
     // compute a "close" orthogonal matrix
     double[][] ort = orthogonalizeMatrix(m, threshold);
 
     // check the sign of the determinant
     double det = ort[0][0] * (ort[1][1] * ort[2][2] - ort[2][1] * ort[1][2]) -
                  ort[1][0] * (ort[0][1] * ort[2][2] - ort[2][1] * ort[0][2]) +
                  ort[2][0] * (ort[0][1] * ort[1][2] - ort[1][1] * ort[0][2]);
     if (det < 0.0) {
       throw new NotARotationMatrixException(
               LocalizedFormats.CLOSEST_ORTHOGONAL_MATRIX_HAS_NEGATIVE_DETERMINANT,
               det);
     }
 
     double[] quat = mat2quat(ort);
     q0 = quat[0];
     q1 = quat[1];
     q2 = quat[2];
     q3 = quat[3];
 
   }
 
   /** Build the rotation that transforms a pair of vector into another pair.
 
    * <p>Except for possible scale factors, if the instance were applied to
    * the pair (u<sub>1</sub>, u<sub>2</sub>) it will produce the pair
    * (v<sub>1</sub>, v<sub>2</sub>).</p>
 
    * <p>If the angular separation between u<sub>1</sub> and u<sub>2</sub> is
    * not the same as the angular separation between v<sub>1</sub> and
    * v<sub>2</sub>, then a corrected v'<sub>2</sub> will be used rather than
    * v<sub>2</sub>, the corrected vector will be in the (v<sub>1</sub>,
    * v<sub>2</sub>) plane.</p>
 
    * @param u1 first vector of the origin pair
    * @param u2 second vector of the origin pair
    * @param v1 desired image of u1 by the rotation
    * @param v2 desired image of u2 by the rotation
    * @exception MathArithmeticException if the norm of one of the vectors is zero,
    * or if one of the pair is degenerated (i.e. the vectors of the pair are colinear)
    */
   public Rotation(Vector3D u1, Vector3D u2, Vector3D v1, Vector3D v2)
       throws MathArithmeticException {
 
       // build orthonormalized base from u1, u2
       // this fails when vectors are null or colinear, which is forbidden to define a rotation
       final Vector3D u3 = u1.crossProduct(u2).normalize();
       u2 = u3.crossProduct(u1).normalize();
       u1 = u1.normalize();
 
       // build an orthonormalized base from v1, v2
       // this fails when vectors are null or colinear, which is forbidden to define a rotation
       final Vector3D v3 = v1.crossProduct(v2).normalize();
       v2 = v3.crossProduct(v1).normalize();
       v1 = v1.normalize();
 
       // buid a matrix transforming the first base into the second one
       final double[][] m = new double[][] {
           {
               MathArrays.linearCombination(u1.getX(), v1.getX(), u2.getX(), v2.getX(), u3.getX(), v3.getX()),
               MathArrays.linearCombination(u1.getY(), v1.getX(), u2.getY(), v2.getX(), u3.getY(), v3.getX()),
               MathArrays.linearCombination(u1.getZ(), v1.getX(), u2.getZ(), v2.getX(), u3.getZ(), v3.getX())
           },
           {
               MathArrays.linearCombination(u1.getX(), v1.getY(), u2.getX(), v2.getY(), u3.getX(), v3.getY()),
               MathArrays.linearCombination(u1.getY(), v1.getY(), u2.getY(), v2.getY(), u3.getY(), v3.getY()),
               MathArrays.linearCombination(u1.getZ(), v1.getY(), u2.getZ(), v2.getY(), u3.getZ(), v3.getY())
           },
           {
               MathArrays.linearCombination(u1.getX(), v1.getZ(), u2.getX(), v2.getZ(), u3.getX(), v3.getZ()),
               MathArrays.linearCombination(u1.getY(), v1.getZ(), u2.getY(), v2.getZ(), u3.getY(), v3.getZ()),
               MathArrays.linearCombination(u1.getZ(), v1.getZ(), u2.getZ(), v2.getZ(), u3.getZ(), v3.getZ())
           }
       };
 
       double[] quat = mat2quat(m);
       q0 = quat[0];
       q1 = quat[1];
       q2 = quat[2];
       q3 = quat[3];
 
   }
 
   /** Build one of the rotations that transform one vector into another one.
 
    * <p>Except for a possible scale factor, if the instance were
    * applied to the vector u it will produce the vector v. There is an
    * infinite number of such rotations, this constructor choose the
    * one with the smallest associated angle (i.e. the one whose axis
    * is orthogonal to the (u, v) plane). If u and v are colinear, an
    * arbitrary rotation axis is chosen.</p>
 
    * @param u origin vector
    * @param v desired image of u by the rotation
    * @exception MathArithmeticException if the norm of one of the vectors is zero
    */
   public Rotation(Vector3D u, Vector3D v) throws MathArithmeticException {
 
     double normProduct = u.getNorm() * v.getNorm();
     if (normProduct == 0) {
         throw new MathArithmeticException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_DEFINING_VECTOR);
     }
 
     double dot = u.dotProduct(v);
 
     if (dot < ((2.0e-15 - 1.0) * normProduct)) {
       // special case u = -v: we select a PI angle rotation around
       // an arbitrary vector orthogonal to u
       Vector3D w = u.orthogonal();
       q0 = 0.0;
       q1 = -w.getX();
       q2 = -w.getY();
       q3 = -w.getZ();
     } else {
       // general case: (u, v) defines a plane, we select
       // the shortest possible rotation: axis orthogonal to this plane
       q0 = FastMath.sqrt(0.5 * (1.0 + dot / normProduct));
       double coeff = 1.0 / (2.0 * q0 * normProduct);
       Vector3D q = v.crossProduct(u);
       q1 = coeff * q.getX();
       q2 = coeff * q.getY();
       q3 = coeff * q.getZ();
     }
 
   }
 
   /** Build a rotation from three Cardan or Euler elementary rotations.
 
    * <p>Cardan rotations are three successive rotations around the
    * canonical axes X, Y and Z, each axis being used once. There are
    * 6 such sets of rotations (XYZ, XZY, YXZ, YZX, ZXY and ZYX). Euler
    * rotations are three successive rotations around the canonical
    * axes X, Y and Z, the first and last rotations being around the
    * same axis. There are 6 such sets of rotations (XYX, XZX, YXY,
    * YZY, ZXZ and ZYZ), the most popular one being ZXZ.</p>
    * <p>Beware that many people routinely use the term Euler angles even
    * for what really are Cardan angles (this confusion is especially
    * widespread in the aerospace business where Roll, Pitch and Yaw angles
    * are often wrongly tagged as Euler angles).</p>
 
    * @param order order of rotations to use
    * @param alpha1 angle of the first elementary rotation
    * @param alpha2 angle of the second elementary rotation
    * @param alpha3 angle of the third elementary rotation
    */
   public Rotation(RotationOrder order,
                   double alpha1, double alpha2, double alpha3) {
       Rotation r1 = new Rotation(order.getA1(), alpha1);
       Rotation r2 = new Rotation(order.getA2(), alpha2);
       Rotation r3 = new Rotation(order.getA3(), alpha3);
       Rotation composed = r1.applyTo(r2.applyTo(r3));
       q0 = composed.q0;
       q1 = composed.q1;
       q2 = composed.q2;
       q3 = composed.q3;
   }
 
   /** Convert an orthogonal rotation matrix to a quaternion.
    * @param ort orthogonal rotation matrix
    * @return quaternion corresponding to the matrix
    */
   private static double[] mat2quat(final double[][] ort) {
 
       final double[] quat = new double[4];
 
       // There are different ways to compute the quaternions elements
       // from the matrix. They all involve computing one element from
       // the diagonal of the matrix, and computing the three other ones
       // using a formula involving a division by the first element,
       // which unfortunately can be zero. Since the norm of the
       // quaternion is 1, we know at least one element has an absolute
       // value greater or equal to 0.5, so it is always possible to
       // select the right formula and avoid division by zero and even
       // numerical inaccuracy. Checking the elements in turn and using
       // the first one greater than 0.45 is safe (this leads to a simple
       // test since qi = 0.45 implies 4 qi^2 - 1 = -0.19)
       double s = ort[0][0] + ort[1][1] + ort[2][2];
       if (s > -0.19) {
           // compute q0 and deduce q1, q2 and q3
           quat[0] = 0.5 * FastMath.sqrt(s + 1.0);
           double inv = 0.25 / quat[0];
           quat[1] = inv * (ort[1][2] - ort[2][1]);
           quat[2] = inv * (ort[2][0] - ort[0][2]);
           quat[3] = inv * (ort[0][1] - ort[1][0]);
       } else {
           s = ort[0][0] - ort[1][1] - ort[2][2];
           if (s > -0.19) {
               // compute q1 and deduce q0, q2 and q3
               quat[1] = 0.5 * FastMath.sqrt(s + 1.0);
               double inv = 0.25 / quat[1];
               quat[0] = inv * (ort[1][2] - ort[2][1]);
               quat[2] = inv * (ort[0][1] + ort[1][0]);
               quat[3] = inv * (ort[0][2] + ort[2][0]);
           } else {
               s = ort[1][1] - ort[0][0] - ort[2][2];
               if (s > -0.19) {
                   // compute q2 and deduce q0, q1 and q3
                   quat[2] = 0.5 * FastMath.sqrt(s + 1.0);
                   double inv = 0.25 / quat[2];
                   quat[0] = inv * (ort[2][0] - ort[0][2]);
                   quat[1] = inv * (ort[0][1] + ort[1][0]);
                   quat[3] = inv * (ort[2][1] + ort[1][2]);
               } else {
                   // compute q3 and deduce q0, q1 and q2
                   s = ort[2][2] - ort[0][0] - ort[1][1];
                   quat[3] = 0.5 * FastMath.sqrt(s + 1.0);
                   double inv = 0.25 / quat[3];
                   quat[0] = inv * (ort[0][1] - ort[1][0]);
                   quat[1] = inv * (ort[0][2] + ort[2][0]);
                   quat[2] = inv * (ort[2][1] + ort[1][2]);
               }
           }
       }
 
       return quat;
 
   }
 
   /** Revert a rotation.
    * Build a rotation which reverse the effect of another
    * rotation. This means that if r(u) = v, then r.revert(v) = u. The
    * instance is not changed.
    * @return a new rotation whose effect is the reverse of the effect
    * of the instance
    */
   public Rotation revert() {
     return new Rotation(-q0, q1, q2, q3, false);
   }
 
   /** Get the scalar coordinate of the quaternion.
    * @return scalar coordinate of the quaternion
    */
   public double getQ0() {
     return q0;
   }
 
   /** Get the first coordinate of the vectorial part of the quaternion.
    * @return first coordinate of the vectorial part of the quaternion
    */
   public double getQ1() {
     return q1;
   }
 
   /** Get the second coordinate of the vectorial part of the quaternion.
    * @return second coordinate of the vectorial part of the quaternion
    */
   public double getQ2() {
     return q2;
   }
 
   /** Get the third coordinate of the vectorial part of the quaternion.
    * @return third coordinate of the vectorial part of the quaternion
    */
   public double getQ3() {
     return q3;
   }
 
   /** Get the normalized axis of the rotation.
    * @return normalized axis of the rotation
    * @see #Rotation(Vector3D, double)
    */
   public Vector3D getAxis() {
     double squaredSine = q1 * q1 + q2 * q2 + q3 * q3;
     if (squaredSine == 0) {
       return new Vector3D(1, 0, 0);
     } else if (q0 < 0) {
       double inverse = 1 / FastMath.sqrt(squaredSine);
       return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
     }
     double inverse = -1 / FastMath.sqrt(squaredSine);
     return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
   }
 
   /** Get the angle of the rotation.
    * @return angle of the rotation (between 0 and &pi;)
    * @see #Rotation(Vector3D, double)
    */
   public double getAngle() {
     if ((q0 < -0.1) || (q0 > 0.1)) {
       return 2 * FastMath.asin(FastMath.sqrt(q1 * q1 + q2 * q2 + q3 * q3));
     } else if (q0 < 0) {
       return 2 * FastMath.acos(-q0);
     }
     return 2 * FastMath.acos(q0);
   }
 
   /** Get the Cardan or Euler angles corresponding to the instance.
 
    * <p>The equations show that each rotation can be defined by two
    * different values of the Cardan or Euler angles set. For example
    * if Cardan angles are used, the rotation defined by the angles
    * a<sub>1</sub>, a<sub>2</sub> and a<sub>3</sub> is the same as
    * the rotation defined by the angles &pi; + a<sub>1</sub>, &pi;
    * - a<sub>2</sub> and &pi; + a<sub>3</sub>. This method implements
    * the following arbitrary choices:</p>
    * <ul>
    *   <li>for Cardan angles, the chosen set is the one for which the
    *   second angle is between -&pi;/2 and &pi;/2 (i.e its cosine is
    *   positive),</li>
    *   <li>for Euler angles, the chosen set is the one for which the
    *   second angle is between 0 and &pi; (i.e its sine is positive).</li>
    * </ul>
 
    * <p>Cardan and Euler angle have a very disappointing drawback: all
    * of them have singularities. This means that if the instance is
    * too close to the singularities corresponding to the given
    * rotation order, it will be impossible to retrieve the angles. For
    * Cardan angles, this is often called gimbal lock. There is
    * <em>nothing</em> to do to prevent this, it is an intrinsic problem
    * with Cardan and Euler representation (but not a problem with the
    * rotation itself, which is perfectly well defined). For Cardan
    * angles, singularities occur when the second angle is close to
    * -&pi;/2 or +&pi;/2, for Euler angle singularities occur when the
    * second angle is close to 0 or &pi;, this implies that the identity
    * rotation is always singular for Euler angles!</p>
 
    * @param order rotation order to use
    * @return an array of three angles, in the order specified by the set
    * @exception CardanEulerSingularityException if the rotation is
    * singular with respect to the angles set specified
    */
   public double[] getAngles(RotationOrder order)
     throws CardanEulerSingularityException {
 
     if (order == RotationOrder.XYZ) {
 
       // r (Vector3D.plusK) coordinates are :
       //  sin (theta), -cos (theta) sin (phi), cos (theta) cos (phi)
       // (-r) (Vector3D.plusI) coordinates are :
       // cos (psi) cos (theta), -sin (psi) cos (theta), sin (theta)
       // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
       Vector3D v1 = applyTo(Vector3D.PLUS_K);
       Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
       if  ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
         throw new CardanEulerSingularityException(true);
       }
       return new double[] {
         FastMath.atan2(-(v1.getY()), v1.getZ()),
         FastMath.asin(v2.getZ()),
         FastMath.atan2(-(v2.getY()), v2.getX())
       };
 
     } else if (order == RotationOrder.XZY) {
 
       // r (Vector3D.plusJ) coordinates are :
       // -sin (psi), cos (psi) cos (phi), cos (psi) sin (phi)
       // (-r) (Vector3D.plusI) coordinates are :
       // cos (theta) cos (psi), -sin (psi), sin (theta) cos (psi)
       // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
       Vector3D v1 = applyTo(Vector3D.PLUS_J);
       Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
       if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
         throw new CardanEulerSingularityException(true);
       }
       return new double[] {
         FastMath.atan2(v1.getZ(), v1.getY()),
        -FastMath.asin(v2.getY()),
         FastMath.atan2(v2.getZ(), v2.getX())
       };
 
     } else if (order == RotationOrder.YXZ) {
 
       // r (Vector3D.plusK) coordinates are :
       //  cos (phi) sin (theta), -sin (phi), cos (phi) cos (theta)
       // (-r) (Vector3D.plusJ) coordinates are :
       // sin (psi) cos (phi), cos (psi) cos (phi), -sin (phi)
       // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
       Vector3D v1 = applyTo(Vector3D.PLUS_K);
       Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
       if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
         throw new CardanEulerSingularityException(true);
       }
       return new double[] {
         FastMath.atan2(v1.getX(), v1.getZ()),
        -FastMath.asin(v2.getZ()),
         FastMath.atan2(v2.getX(), v2.getY())
       };
 
     } else if (order == RotationOrder.YZX) {
 
       // r (Vector3D.plusI) coordinates are :
       // cos (psi) cos (theta), sin (psi), -cos (psi) sin (theta)
       // (-r) (Vector3D.plusJ) coordinates are :
       // sin (psi), cos (phi) cos (psi), -sin (phi) cos (psi)
       // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
       Vector3D v1 = applyTo(Vector3D.PLUS_I);
       Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
       if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
         throw new CardanEulerSingularityException(true);
       }
       return new double[] {
         FastMath.atan2(-(v1.getZ()), v1.getX()),
         FastMath.asin(v2.getX()),
         FastMath.atan2(-(v2.getZ()), v2.getY())
       };
 
     } else if (order == RotationOrder.ZXY) {
 
       // r (Vector3D.plusJ) coordinates are :
       // -cos (phi) sin (psi), cos (phi) cos (psi), sin (phi)
       // (-r) (Vector3D.plusK) coordinates are :
       // -sin (theta) cos (phi), sin (phi), cos (theta) cos (phi)
       // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
       Vector3D v1 = applyTo(Vector3D.PLUS_J);
       Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
       if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
         throw new CardanEulerSingularityException(true);
       }
       return new double[] {
         FastMath.atan2(-(v1.getX()), v1.getY()),
         FastMath.asin(v2.getY()),
         FastMath.atan2(-(v2.getX()), v2.getZ())
       };
 
     } else if (order == RotationOrder.ZYX) {
 
       // r (Vector3D.plusI) coordinates are :
       //  cos (theta) cos (psi), cos (theta) sin (psi), -sin (theta)
       // (-r) (Vector3D.plusK) coordinates are :
       // -sin (theta), sin (phi) cos (theta), cos (phi) cos (theta)
       // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
       Vector3D v1 = applyTo(Vector3D.PLUS_I);
       Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
       if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
         throw new CardanEulerSingularityException(true);
       }
       return new double[] {
         FastMath.atan2(v1.getY(), v1.getX()),
        -FastMath.asin(v2.getX()),
         FastMath.atan2(v2.getY(), v2.getZ())
       };
 
     } else if (order == RotationOrder.XYX) {
 
       // r (Vector3D.plusI) coordinates are :
       //  cos (theta), sin (phi1) sin (theta), -cos (phi1) sin (theta)
       // (-r) (Vector3D.plusI) coordinates are :
       // cos (theta), sin (theta) sin (phi2), sin (theta) cos (phi2)
       // and we can choose to have theta in the interval [0 ; PI]
       Vector3D v1 = applyTo(Vector3D.PLUS_I);
       Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
       if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
         throw new CardanEulerSingularityException(false);
       }
       return new double[] {
         FastMath.atan2(v1.getY(), -v1.getZ()),
         FastMath.acos(v2.getX()),
         FastMath.atan2(v2.getY(), v2.getZ())
       };
 
     } else if (order == RotationOrder.XZX) {
 
       // r (Vector3D.plusI) coordinates are :
       //  cos (psi), cos (phi1) sin (psi), sin (phi1) sin (psi)
       // (-r) (Vector3D.plusI) coordinates are :
       // cos (psi), -sin (psi) cos (phi2), sin (psi) sin (phi2)
       // and we can choose to have psi in the interval [0 ; PI]
       Vector3D v1 = applyTo(Vector3D.PLUS_I);
       Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
       if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
         throw new CardanEulerSingularityException(false);
       }
       return new double[] {
         FastMath.atan2(v1.getZ(), v1.getY()),
         FastMath.acos(v2.getX()),
         FastMath.atan2(v2.getZ(), -v2.getY())
       };
 
     } else if (order == RotationOrder.YXY) {
 
       // r (Vector3D.plusJ) coordinates are :
       //  sin (theta1) sin (phi), cos (phi), cos (theta1) sin (phi)
       // (-r) (Vector3D.plusJ) coordinates are :
       // sin (phi) sin (theta2), cos (phi), -sin (phi) cos (theta2)
       // and we can choose to have phi in the interval [0 ; PI]
       Vector3D v1 = applyTo(Vector3D.PLUS_J);
       Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
       if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
         throw new CardanEulerSingularityException(false);
       }
       return new double[] {
         FastMath.atan2(v1.getX(), v1.getZ()),
         FastMath.acos(v2.getY()),
         FastMath.atan2(v2.getX(), -v2.getZ())
       };
 
     } else if (order == RotationOrder.YZY) {
 
       // r (Vector3D.plusJ) coordinates are :
       //  -cos (theta1) sin (psi), cos (psi), sin (theta1) sin (psi)
       // (-r) (Vector3D.plusJ) coordinates are :
       // sin (psi) cos (theta2), cos (psi), sin (psi) sin (theta2)
       // and we can choose to have psi in the interval [0 ; PI]
       Vector3D v1 = applyTo(Vector3D.PLUS_J);
       Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
       if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
         throw new CardanEulerSingularityException(false);
       }
       return new double[] {
         FastMath.atan2(v1.getZ(), -v1.getX()),
         FastMath.acos(v2.getY()),
         FastMath.atan2(v2.getZ(), v2.getX())
       };
 
     } else if (order == RotationOrder.ZXZ) {
 
       // r (Vector3D.plusK) coordinates are :
       //  sin (psi1) sin (phi), -cos (psi1) sin (phi), cos (phi)
       // (-r) (Vector3D.plusK) coordinates are :
       // sin (phi) sin (psi2), sin (phi) cos (psi2), cos (phi)
       // and we can choose to have phi in the interval [0 ; PI]
       Vector3D v1 = applyTo(Vector3D.PLUS_K);
       Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
       if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
         throw new CardanEulerSingularityException(false);
       }
       return new double[] {
         FastMath.atan2(v1.getX(), -v1.getY()),
         FastMath.acos(v2.getZ()),
         FastMath.atan2(v2.getX(), v2.getY())
       };
 
     } else { // last possibility is ZYZ
 
       // r (Vector3D.plusK) coordinates are :
       //  cos (psi1) sin (theta), sin (psi1) sin (theta), cos (theta)
       // (-r) (Vector3D.plusK) coordinates are :
       // -sin (theta) cos (psi2), sin (theta) sin (psi2), cos (theta)
       // and we can choose to have theta in the interval [0 ; PI]
       Vector3D v1 = applyTo(Vector3D.PLUS_K);
       Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
       if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
         throw new CardanEulerSingularityException(false);
       }
       return new double[] {
         FastMath.atan2(v1.getY(), v1.getX()),
         FastMath.acos(v2.getZ()),
         FastMath.atan2(v2.getY(), -v2.getX())
       };
 
     }
 
   }
 
   /** Get the 3X3 matrix corresponding to the instance
    * @return the matrix corresponding to the instance
    */
   public double[][] getMatrix() {
 
     // products
     double q0q0  = q0 * q0;
     double q0q1  = q0 * q1;
     double q0q2  = q0 * q2;
     double q0q3  = q0 * q3;
     double q1q1  = q1 * q1;
     double q1q2  = q1 * q2;
     double q1q3  = q1 * q3;
     double q2q2  = q2 * q2;
     double q2q3  = q2 * q3;
     double q3q3  = q3 * q3;
 
     // create the matrix
     double[][] m = new double[3][];
     m[0] = new double[3];
     m[1] = new double[3];
     m[2] = new double[3];
 
     m [0][0] = 2.0 * (q0q0 + q1q1) - 1.0;
     m [1][0] = 2.0 * (q1q2 - q0q3);
     m [2][0] = 2.0 * (q1q3 + q0q2);
 
     m [0][1] = 2.0 * (q1q2 + q0q3);
     m [1][1] = 2.0 * (q0q0 + q2q2) - 1.0;
     m [2][1] = 2.0 * (q2q3 - q0q1);
 
     m [0][2] = 2.0 * (q1q3 - q0q2);
     m [1][2] = 2.0 * (q2q3 + q0q1);
     m [2][2] = 2.0 * (q0q0 + q3q3) - 1.0;
 
     return m;
 
   }
 
   /** Apply the rotation to a vector.
    * @param u vector to apply the rotation to
    * @return a new vector which is the image of u by the rotation
    */
   public Vector3D applyTo(Vector3D u) {
 
     double x = u.getX();
     double y = u.getY();
     double z = u.getZ();
 
     double s = q1 * x + q2 * y + q3 * z;
 
     return new Vector3D(2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x,
                         2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y,
                         2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - z);
 
   }
 
   /** Apply the rotation to a vector stored in an array.
    * @param in an array with three items which stores vector to rotate
    * @param out an array with three items to put result to (it can be the same
    * array as in)
    */
   public void applyTo(final double[] in, final double[] out) {
 
       final double x = in[0];
       final double y = in[1];
       final double z = in[2];
 
       final double s = q1 * x + q2 * y + q3 * z;
 
       out[0] = 2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x;
       out[1] = 2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y;
       out[2] = 2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - z;
 
   }
 
   /** Apply the inverse of the rotation to a vector.
    * @param u vector to apply the inverse of the rotation to
    * @return a new vector which such that u is its image by the rotation
    */
   public Vector3D applyInverseTo(Vector3D u) {
 
     double x = u.getX();
     double y = u.getY();
     double z = u.getZ();
 
     double s = q1 * x + q2 * y + q3 * z;
     double m0 = -q0;
 
     return new Vector3D(2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x,
                         2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y,
                         2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - z);
 
   }
 
   /** Apply the inverse of the rotation to a vector stored in an array.
    * @param in an array with three items which stores vector to rotate
    * @param out an array with three items to put result to (it can be the same
    * array as in)
    */
   public void applyInverseTo(final double[] in, final double[] out) {
 
       final double x = in[0];
       final double y = in[1];
       final double z = in[2];
 
       final double s = q1 * x + q2 * y + q3 * z;
       final double m0 = -q0;
 
       out[0] = 2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x;
       out[1] = 2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y;
       out[2] = 2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - z;
 
   }
 
   /** Apply the instance to another rotation.
    * Applying the instance to a rotation is computing the composition
    * in an order compliant with the following rule : let u be any
    * vector and v its image by r (i.e. r.applyTo(u) = v), let w be the image
    * of v by the instance (i.e. applyTo(v) = w), then w = comp.applyTo(u),
    * where comp = applyTo(r).
    * @param r rotation to apply the rotation to
    * @return a new rotation which is the composition of r by the instance
    */
   public Rotation applyTo(Rotation r) {
     return new Rotation(r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),
                         r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),
                         r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),
                         r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),
                         false);
   }
 
   /** Apply the inverse of the instance to another rotation.
    * Applying the inverse of the instance to a rotation is computing
    * the composition in an order compliant with the following rule :
    * let u be any vector and v its image by r (i.e. r.applyTo(u) = v),
    * let w be the inverse image of v by the instance
    * (i.e. applyInverseTo(v) = w), then w = comp.applyTo(u), where
    * comp = applyInverseTo(r).
    * @param r rotation to apply the rotation to
    * @return a new rotation which is the composition of r by the inverse
    * of the instance
    */
   public Rotation applyInverseTo(Rotation r) {
     return new Rotation(-r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),
                         -r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),
                         -r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),
                         -r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),
                         false);
   }
 
   /** Perfect orthogonality on a 3X3 matrix.
    * @param m initial matrix (not exactly orthogonal)
    * @param threshold convergence threshold for the iterative
    * orthogonality correction (convergence is reached when the
    * difference between two steps of the Frobenius norm of the
    * correction is below this threshold)
    * @return an orthogonal matrix close to m
    * @exception NotARotationMatrixException if the matrix cannot be
    * orthogonalized with the given threshold after 10 iterations
    */
   private double[][] orthogonalizeMatrix(double[][] m, double threshold)
     throws NotARotationMatrixException {
     double[] m0 = m[0];
     double[] m1 = m[1];
     double[] m2 = m[2];
     double x00 = m0[0];
     double x01 = m0[1];
     double x02 = m0[2];
     double x10 = m1[0];
     double x11 = m1[1];
     double x12 = m1[2];
     double x20 = m2[0];
     double x21 = m2[1];
     double x22 = m2[2];
     double fn = 0;
     double fn1;
 
     double[][] o = new double[3][3];
     double[] o0 = o[0];
     double[] o1 = o[1];
     double[] o2 = o[2];
 
     // iterative correction: Xn+1 = Xn - 0.5 * (Xn.Mt.Xn - M)
     int i = 0;
     while (++i < 11) {
 
       // Mt.Xn
       double mx00 = m0[0] * x00 + m1[0] * x10 + m2[0] * x20;
       double mx10 = m0[1] * x00 + m1[1] * x10 + m2[1] * x20;
       double mx20 = m0[2] * x00 + m1[2] * x10 + m2[2] * x20;
       double mx01 = m0[0] * x01 + m1[0] * x11 + m2[0] * x21;
       double mx11 = m0[1] * x01 + m1[1] * x11 + m2[1] * x21;
       double mx21 = m0[2] * x01 + m1[2] * x11 + m2[2] * x21;
       double mx02 = m0[0] * x02 + m1[0] * x12 + m2[0] * x22;
       double mx12 = m0[1] * x02 + m1[1] * x12 + m2[1] * x22;
       double mx22 = m0[2] * x02 + m1[2] * x12 + m2[2] * x22;
 
       // Xn+1
       o0[0] = x00 - 0.5 * (x00 * mx00 + x01 * mx10 + x02 * mx20 - m0[0]);
       o0[1] = x01 - 0.5 * (x00 * mx01 + x01 * mx11 + x02 * mx21 - m0[1]);
       o0[2] = x02 - 0.5 * (x00 * mx02 + x01 * mx12 + x02 * mx22 - m0[2]);
       o1[0] = x10 - 0.5 * (x10 * mx00 + x11 * mx10 + x12 * mx20 - m1[0]);
       o1[1] = x11 - 0.5 * (x10 * mx01 + x11 * mx11 + x12 * mx21 - m1[1]);
       o1[2] = x12 - 0.5 * (x10 * mx02 + x11 * mx12 + x12 * mx22 - m1[2]);
       o2[0] = x20 - 0.5 * (x20 * mx00 + x21 * mx10 + x22 * mx20 - m2[0]);
       o2[1] = x21 - 0.5 * (x20 * mx01 + x21 * mx11 + x22 * mx21 - m2[1]);
       o2[2] = x22 - 0.5 * (x20 * mx02 + x21 * mx12 + x22 * mx22 - m2[2]);
 
       // correction on each elements
       double corr00 = o0[0] - m0[0];
       double corr01 = o0[1] - m0[1];
       double corr02 = o0[2] - m0[2];
       double corr10 = o1[0] - m1[0];
       double corr11 = o1[1] - m1[1];
       double corr12 = o1[2] - m1[2];
       double corr20 = o2[0] - m2[0];
       double corr21 = o2[1] - m2[1];
       double corr22 = o2[2] - m2[2];
 
       // Frobenius norm of the correction
       fn1 = corr00 * corr00 + corr01 * corr01 + corr02 * corr02 +
             corr10 * corr10 + corr11 * corr11 + corr12 * corr12 +
             corr20 * corr20 + corr21 * corr21 + corr22 * corr22;
 
       // convergence test
       if (FastMath.abs(fn1 - fn) <= threshold) {
           return o;
       }
 
       // prepare next iteration
       x00 = o0[0];
       x01 = o0[1];
       x02 = o0[2];
       x10 = o1[0];
       x11 = o1[1];
       x12 = o1[2];
       x20 = o2[0];
       x21 = o2[1];
       x22 = o2[2];
       fn  = fn1;
 
     }
 
     // the algorithm did not converge after 10 iterations
     throw new NotARotationMatrixException(
             LocalizedFormats.UNABLE_TO_ORTHOGONOLIZE_MATRIX,
             i - 1);
   }
 
   /** Compute the <i>distance</i> between two rotations.
    * <p>The <i>distance</i> is intended here as a way to check if two
    * rotations are almost similar (i.e. they transform vectors the same way)
    * or very different. It is mathematically defined as the angle of
    * the rotation r that prepended to one of the rotations gives the other
    * one:</p>
    * <pre>
    *        r<sub>1</sub>(r) = r<sub>2</sub>
    * </pre>
    * <p>This distance is an angle between 0 and &pi;. Its value is the smallest
    * possible upper bound of the angle in radians between r<sub>1</sub>(v)
    * and r<sub>2</sub>(v) for all possible vectors v. This upper bound is
    * reached for some v. The distance is equal to 0 if and only if the two
    * rotations are identical.</p>
    * <p>Comparing two rotations should always be done using this value rather
    * than for example comparing the components of the quaternions. It is much
    * more stable, and has a geometric meaning. Also comparing quaternions
    * components is error prone since for example quaternions (0.36, 0.48, -0.48, -0.64)
    * and (-0.36, -0.48, 0.48, 0.64) represent exactly the same rotation despite
    * their components are different (they are exact opposites).</p>
    * @param r1 first rotation
    * @param r2 second rotation
    * @return <i>distance</i> between r1 and r2
    */
   public static double distance(Rotation r1, Rotation r2) {
       return r1.applyInverseTo(r2).getAngle();
   }
 
 }