org.apache.commons.math3.geometry.euclidean.threed.Rotation.java Source code

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/*
 * 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();
    }

}