math/geometry/Rotation.java

/*
 * 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.math.geometry;

import java.io.Serializable;

import org.apache.commons.math.MathRuntimeException;
import org.apache.commons.math.exception.util.LocalizedFormats;
import org.apache.commons.math.util.FastMath;

/**
 * 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 $Revision: 1067500 $ $Date: 2011-02-05 21:11:30 +0100 (sam. 05 févr. 2011) $
 * @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 ArithmeticException if the axis norm is zero
   */
  public Rotation(Vector3D axis, double angle) {

    double norm = axis.getNorm();
    if (norm == 0) {
      throw MathRuntimeException.createArithmeticException(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);
    }

    // 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
      q0 = 0.5 * FastMath.sqrt(s + 1.0);
      double inv = 0.25 / q0;
      q1 = inv * (ort[1][2] - ort[2][1]);
      q2 = inv * (ort[2][0] - ort[0][2]);
      q3 = 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
        q1 = 0.5 * FastMath.sqrt(s + 1.0);
        double inv = 0.25 / q1;
        q0 = inv * (ort[1][2] - ort[2][1]);
        q2 = inv * (ort[0][1] + ort[1][0]);
        q3 = 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
          q2 = 0.5 * FastMath.sqrt(s + 1.0);
          double inv = 0.25 / q2;
          q0 = inv * (ort[2][0] - ort[0][2]);
          q1 = inv * (ort[0][1] + ort[1][0]);
          q3 = 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];
          q3 = 0.5 * FastMath.sqrt(s + 1.0);
          double inv = 0.25 / q3;
          q0 = inv * (ort[0][1] - ort[1][0]);
          q1 = inv * (ort[0][2] + ort[2][0]);
          q2 = inv * (ort[2][1] + ort[1][2]);
        }
      }
    }

  }

  /** 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 IllegalArgumentException if the norm of one of the vectors is zero
   */
  public Rotation(Vector3D u1, Vector3D u2, Vector3D v1, Vector3D v2) {

  // norms computation
  double u1u1 = Vector3D.dotProduct(u1, u1);
  double u2u2 = Vector3D.dotProduct(u2, u2);
  double v1v1 = Vector3D.dotProduct(v1, v1);
  double v2v2 = Vector3D.dotProduct(v2, v2);
  if ((u1u1 == 0) || (u2u2 == 0) || (v1v1 == 0) || (v2v2 == 0)) {
    throw MathRuntimeException.createIllegalArgumentException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_DEFINING_VECTOR);
  }

  double u1x = u1.getX();
  double u1y = u1.getY();
  double u1z = u1.getZ();

  double u2x = u2.getX();
  double u2y = u2.getY();
  double u2z = u2.getZ();

  // normalize v1 in order to have (v1'|v1') = (u1|u1)
  double coeff = FastMath.sqrt (u1u1 / v1v1);
  double v1x   = coeff * v1.getX();
  double v1y   = coeff * v1.getY();
  double v1z   = coeff * v1.getZ();
  v1 = new Vector3D(v1x, v1y, v1z);

  // adjust v2 in order to have (u1|u2) = (v1|v2) and (v2'|v2') = (u2|u2)
  double u1u2   = Vector3D.dotProduct(u1, u2);
  double v1v2   = Vector3D.dotProduct(v1, v2);
  double coeffU = u1u2 / u1u1;
  double coeffV = v1v2 / u1u1;
  double beta   = FastMath.sqrt((u2u2 - u1u2 * coeffU) / (v2v2 - v1v2 * coeffV));
  double alpha  = coeffU - beta * coeffV;
  double v2x    = alpha * v1x + beta * v2.getX();
  double v2y    = alpha * v1y + beta * v2.getY();
  double v2z    = alpha * v1z + beta * v2.getZ();
  v2 = new Vector3D(v2x, v2y, v2z);

  // preliminary computation (we use explicit formulation instead
  // of relying on the Vector3D class in order to avoid building lots
  // of temporary objects)
  Vector3D uRef = u1;
  Vector3D vRef = v1;
  double dx1 = v1x - u1.getX();
  double dy1 = v1y - u1.getY();
  double dz1 = v1z - u1.getZ();
  double dx2 = v2x - u2.getX();
  double dy2 = v2y - u2.getY();
  double dz2 = v2z - u2.getZ();
  Vector3D k = new Vector3D(dy1 * dz2 - dz1 * dy2,
                            dz1 * dx2 - dx1 * dz2,
                            dx1 * dy2 - dy1 * dx2);
  double c = k.getX() * (u1y * u2z - u1z * u2y) +
             k.getY() * (u1z * u2x - u1x * u2z) +
             k.getZ() * (u1x * u2y - u1y * u2x);

  if (c == 0) {
    // the (q1, q2, q3) vector is in the (u1, u2) plane
    // we try other vectors
    Vector3D u3 = Vector3D.crossProduct(u1, u2);
    Vector3D v3 = Vector3D.crossProduct(v1, v2);
    double u3x  = u3.getX();
    double u3y  = u3.getY();
    double u3z  = u3.getZ();
    double v3x  = v3.getX();
    double v3y  = v3.getY();
    double v3z  = v3.getZ();

    double dx3 = v3x - u3x;
    double dy3 = v3y - u3y;
    double dz3 = v3z - u3z;
    k = new Vector3D(dy1 * dz3 - dz1 * dy3,
                     dz1 * dx3 - dx1 * dz3,
                     dx1 * dy3 - dy1 * dx3);
    c = k.getX() * (u1y * u3z - u1z * u3y) +
        k.getY() * (u1z * u3x - u1x * u3z) +
        k.getZ() * (u1x * u3y - u1y * u3x);

    if (c == 0) {
      // the (q1, q2, q3) vector is aligned with u1:
      // we try (u2, u3) and (v2, v3)
      k = new Vector3D(dy2 * dz3 - dz2 * dy3,
                       dz2 * dx3 - dx2 * dz3,
                       dx2 * dy3 - dy2 * dx3);
      c = k.getX() * (u2y * u3z - u2z * u3y) +
          k.getY() * (u2z * u3x - u2x * u3z) +
          k.getZ() * (u2x * u3y - u2y * u3x);

      if (c == 0) {
        // the (q1, q2, q3) vector is aligned with everything
        // this is really the identity rotation
        q0 = 1.0;
        q1 = 0.0;
        q2 = 0.0;
        q3 = 0.0;
        return;
      }

      // we will have to use u2 and v2 to compute the scalar part
      uRef = u2;
      vRef = v2;

    }

  }

  // compute the vectorial part
  c = FastMath.sqrt(c);
  double inv = 1.0 / (c + c);
  q1 = inv * k.getX();
  q2 = inv * k.getY();
  q3 = inv * k.getZ();

  // compute the scalar part
   k = new Vector3D(uRef.getY() * q3 - uRef.getZ() * q2,
                    uRef.getZ() * q1 - uRef.getX() * q3,
                    uRef.getX() * q2 - uRef.getY() * q1);
   c = Vector3D.dotProduct(k, k);
  q0 = Vector3D.dotProduct(vRef, k) / (c + c);

  }

  /** 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 IllegalArgumentException if the norm of one of the vectors is zero
   */
  public Rotation(Vector3D u, Vector3D v) {

    double normProduct = u.getNorm() * v.getNorm();
    if (normProduct == 0) {
        throw MathRuntimeException.createIllegalArgumentException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_DEFINING_VECTOR);
    }

    double dot = Vector3D.dotProduct(u, 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);
      q1 = coeff * (v.getY() * u.getZ() - v.getZ() * u.getY());
      q2 = coeff * (v.getZ() * u.getX() - v.getX() * u.getZ());
      q3 = coeff * (v.getX() * u.getY() - v.getY() * u.getX());
    }

  }

  /** 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;
  }

  /** 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 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 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();
  }

}