Java/2D Graphics GUI/Matrix

Implementation of a 4x4 matrix suited for use in a 2D and 3D graphics rendering engine

 
/*
 * (C) 2004 - Geotechnical Software Services
 * 
 * This code is free software; you can redistribute it and/or
 * modify it under the terms of the GNU Lesser General Public 
 * License as published by the Free Software Foundation; either 
 * version 2.1 of the License, or (at your option) any later version.
 *
 * This code is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
 * GNU Lesser General Public License for more details.
 *
 * You should have received a copy of the GNU Lesser General Public 
 * License along with this program; if not, write to the Free 
 * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, 
 * MA  02111-1307, USA.
 */
//package no.geosoft.cc.geometry;

/**
 * Implementation of a 4x4 matrix suited for use in a 2D and 3D
 * graphics rendering engine.
 * 
 * @author 
 */   
//public 
class Vector4
{
  private double[] v_;

  
  private void initialize()
  {
    v_ = new double[4];
    for (int i = 0; i < 4; i++)
      v_[i] = 0.0;
  }

  /**
   * Create a default 4-element vector (all elements set to 0.0).
   */
  public Vector4()
  {
    initialize();
  }

  /**
   * Create a 4-element vector with the specified values.
   * 
   * @param v1  1st element.
   * @param v2  2nd element.
   * @param v3  3rd element.
   * @param v4  4th element
   */
  public Vector4 (double v1, double v2, double v3, double v4)
  {
    initialize();
    set (v1, v2, v3, v4);
  }

  /**
   * Construct a 4-element vector as a copy of the specified vector.
   * 
   * @param vector4
   */
  public Vector4 (Vector4 vector4)
  {
    initialize();
    set (vector4);
  }

  /**
   * Set the elements of this vector.
   * 
   * @param v1  1st element.
   * @param v2  2nd element.
   * @param v3  3rd element.
   * @param v4  4th element
   */
  public void set (double v1, double v2, double v3, double v4)
  {
    v_[0] = v1;
    v_[1] = v2;
    v_[2] = v3;
    v_[3] = v4;
  }

  /**
   * Set the elements of this vector according to the specified vector.
   * 
   * @param vector  Vector to copy.
   */
  public void set (Vector4 vector)
  {
    for (int i = 0; i < 4; i++)
      v_[0] = vector.v_[i];
  }
  
  /**
   * Check if this 4-element vector equals the specified object.
   * 
   * @return  TRue if the two equals, false otherwise.
   */
  public boolean equals (Object object)
  {
    Vector4 vector = (Vector4) object;
    
    return v_[0] == vector.v_[0] &&
           v_[1] == vector.v_[1] &&
           v_[2] == vector.v_[2] &&
           v_[3] == vector.v_[3];
  }

  /**
   * Return the i"th element of this vector.
   * 
   * @param i  Index of element to get (first is 0).
   * @return   i"th element of this vector.
   */
  public double getElement (int i)
  {
    return v_[i];
  }

  /**
   * Set the i"th element of this vector.
   * 
   * @param i  Index of element to set (first is 0).
   * @param    Value to set.
   */
  public void setElement (int i, double value)
  {
    v_[i] = value;
  }
  

  /**
   * Create a string representation of this vector.
   * 
   * @return  String representing this vector.
   */
  public String toString()
  {
    return ("Vector4: [" + 
            v_[0] + "," + v_[1] + "," + v_[2] + "," + v_[3] + "]");
  }
}

Rotations in a three-dimensional space

 
/*
 * 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.
 */

import java.io.Serializable;
/**
 * 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 implicitely
 * 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
 * followong 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
 * rotation transforms the coordinates coordinates of this vector in inertial
 * frame into the coordinates of the same vector in satellite frame. In this
 * case, the rotation implicitely defines the relation between the two frames.
 * 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 directionf at rest in a topocentric frame
 * into the sighting direction in the same topocentric frame. 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.</p>
 *
 * <p>These examples show that a rotation is what the user wants it to be, so 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: 627994 $ $Date: 2008-02-15 03:16:05 -0700 (Fri, 15 Feb 2008) $
 * @see Vector3D
 * @see RotationOrder
 * @since 1.2
 */
public class Rotation implements Serializable {
  /** Build the identity rotation.
   */
  public Rotation() {
    q0 = 1;
    q1 = 0;
    q2 = 0;
    q3 = 0;
  }
  /** 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>
   * @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 / Math.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>
   * @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 new ArithmeticException("zero norm for rotation axis");
    }
    double halfAngle = -0.5 * angle;
    double coeff = Math.sin(halfAngle) / norm;
    q0 = Math.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)
    {
    // dimension check
    if ((m.length != 3) || (m[0].length != 3) ||
        (m[1].length != 3) || (m[2].length != 3)) {
      System.out.println("a {0}x{1} matrix" +
                                            " cannot be a rotation matrix");
    }
    // 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) {
      System.out.println("the closest orthogonal matrix" +
                                            " has a negative determinant {0}");
    }
    // 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 * Math.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 * Math.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 * Math.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 * Math.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 new IllegalArgumentException("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 = Math.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   = Math.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 = Math.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 new IllegalArgumentException("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 = Math.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 beeing 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 beeing 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
   */
  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 / Math.sqrt(squaredSine);
      return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
    }
    double inverse = -1 / Math.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;)
   */
  public double getAngle() {
    if ((q0 < -0.1) || (q0 > 0.1)) {
      return 2 * Math.asin(Math.sqrt(q1 * q1 + q2 * q2 + q3 * q3));
    } else if (q0 < 0) {
      return 2 * Math.acos(-q0);
    }
    return 2 * Math.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)
    {
    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.plusK);
      Vector3D v2 = applyInverseTo(Vector3D.plusI);
      if  ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
        System.out.println("CardanEulerSingularityException");
      }
      return new double[] {
        Math.atan2(-(v1.getY()), v1.getZ()),
        Math.asin(v2.getZ()),
        Math.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.plusJ);
      Vector3D v2 = applyInverseTo(Vector3D.plusI);
      if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
        System.out.println("CardanEulerSingularityException");
      }
      return new double[] {
        Math.atan2(v1.getZ(), v1.getY()),
       -Math.asin(v2.getY()),
        Math.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.plusK);
      Vector3D v2 = applyInverseTo(Vector3D.plusJ);
      if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
        System.out.println("CardanEulerSingularityException");
      }
      return new double[] {
        Math.atan2(v1.getX(), v1.getZ()),
       -Math.asin(v2.getZ()),
        Math.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.plusI);
      Vector3D v2 = applyInverseTo(Vector3D.plusJ);
      if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
        System.out.println("CardanEulerSingularityException");
      }
      return new double[] {
        Math.atan2(-(v1.getZ()), v1.getX()),
        Math.asin(v2.getX()),
        Math.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.plusJ);
      Vector3D v2 = applyInverseTo(Vector3D.plusK);
      if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
        System.out.println("CardanEulerSingularityException");
      }
      return new double[] {
        Math.atan2(-(v1.getX()), v1.getY()),
        Math.asin(v2.getY()),
        Math.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.plusI);
      Vector3D v2 = applyInverseTo(Vector3D.plusK);
      if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
        System.out.println("CardanEulerSingularityException");
      }
      return new double[] {
        Math.atan2(v1.getY(), v1.getX()),
       -Math.asin(v2.getX()),
        Math.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.plusI);
      Vector3D v2 = applyInverseTo(Vector3D.plusI);
      if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
        System.out.println("CardanEulerSingularityException");
      }
      return new double[] {
        Math.atan2(v1.getY(), -v1.getZ()),
        Math.acos(v2.getX()),
        Math.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.plusI);
      Vector3D v2 = applyInverseTo(Vector3D.plusI);
      if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
        System.out.println("CardanEulerSingularityException");
      }
      return new double[] {
        Math.atan2(v1.getZ(), v1.getY()),
        Math.acos(v2.getX()),
        Math.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.plusJ);
      Vector3D v2 = applyInverseTo(Vector3D.plusJ);
      if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
        System.out.println("CardanEulerSingularityException");
      }
      return new double[] {
        Math.atan2(v1.getX(), v1.getZ()),
        Math.acos(v2.getY()),
        Math.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.plusJ);
      Vector3D v2 = applyInverseTo(Vector3D.plusJ);
      if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
        System.out.println("CardanEulerSingularityException");
      }
      return new double[] {
        Math.atan2(v1.getZ(), -v1.getX()),
        Math.acos(v2.getY()),
        Math.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.plusK);
      Vector3D v2 = applyInverseTo(Vector3D.plusK);
      if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
        System.out.println("CardanEulerSingularityException");
      }
      return new double[] {
        Math.atan2(v1.getX(), -v1.getY()),
        Math.acos(v2.getZ()),
        Math.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.plusK);
      Vector3D v2 = applyInverseTo(Vector3D.plusK);
      if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
        throw new RuntimeException("false");
      }
      return new double[] {
        Math.atan2(v1.getY(), v1.getX()),
        Math.acos(v2.getZ()),
        Math.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)
    {
    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 (Math.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;
    }
    return null;
    // the algorithm did not converge after 10 iterations
    //System.out.println("unable to orthogonalize matrix" +
                                          //" in {0} iterations");
  }
  /** 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;
  /** Serializable version identifier */
  private static final long serialVersionUID = 8225864499430109352L;
}

/**
 * This class is a utility representing a rotation order specification
 * for Cardan or Euler angles specification.
 *
 * This class cannot be instanciated by the user. He can only use one
 * of the twelve predefined supported orders as an argument to either
 * the {@link Rotation#Rotation(RotationOrder,double,double,double)}
 * constructor or the {@link Rotation#getAngles} method.
 *
 * @version $Revision: 620312 $ $Date: 2008-02-10 12:28:59 -0700 (Sun, 10 Feb 2008) $
 * @since 1.2
 */
final class RotationOrder {
  /** Private constructor.
   * This is a utility class that cannot be instantiated by the user,
   * so its only constructor is private.
   * @param name name of the rotation order
   * @param a1 axis of the first rotation
   * @param a2 axis of the second rotation
   * @param a3 axis of the third rotation
   */
  private RotationOrder(String name,
                        Vector3D a1, Vector3D a2, Vector3D a3) {
    this.name = name;
    this.a1   = a1;
    this.a2   = a2;
    this.a3   = a3;
  }
  /** Get a string representation of the instance.
   * @return a string representation of the instance (in fact, its name)
   */
  public String toString() {
    return name;
  }
  /** Get the axis of the first rotation.
   * @return axis of the first rotation
   */
  public Vector3D getA1() {
    return a1;
  }
  /** Get the axis of the second rotation.
   * @return axis of the second rotation
   */
  public Vector3D getA2() {
    return a2;
  }
  /** Get the axis of the second rotation.
   * @return axis of the second rotation
   */
  public Vector3D getA3() {
    return a3;
  }
  /** Set of Cardan angles.
   * this ordered set of rotations is around X, then around Y, then
   * around Z
   */
  public static final RotationOrder XYZ =
    new RotationOrder("XYZ", Vector3D.plusI, Vector3D.plusJ, Vector3D.plusK);
  /** Set of Cardan angles.
   * this ordered set of rotations is around X, then around Z, then
   * around Y
   */
  public static final RotationOrder XZY =
    new RotationOrder("XZY", Vector3D.plusI, Vector3D.plusK, Vector3D.plusJ);
  /** Set of Cardan angles.
   * this ordered set of rotations is around Y, then around X, then
   * around Z
   */
  public static final RotationOrder YXZ =
    new RotationOrder("YXZ", Vector3D.plusJ, Vector3D.plusI, Vector3D.plusK);
  /** Set of Cardan angles.
   * this ordered set of rotations is around Y, then around Z, then
   * around X
   */
  public static final RotationOrder YZX =
    new RotationOrder("YZX", Vector3D.plusJ, Vector3D.plusK, Vector3D.plusI);
  /** Set of Cardan angles.
   * this ordered set of rotations is around Z, then around X, then
   * around Y
   */
  public static final RotationOrder ZXY =
    new RotationOrder("ZXY", Vector3D.plusK, Vector3D.plusI, Vector3D.plusJ);
  /** Set of Cardan angles.
   * this ordered set of rotations is around Z, then around Y, then
   * around X
   */
  public static final RotationOrder ZYX =
    new RotationOrder("ZYX", Vector3D.plusK, Vector3D.plusJ, Vector3D.plusI);
  /** Set of Euler angles.
   * this ordered set of rotations is around X, then around Y, then
   * around X
   */
  public static final RotationOrder XYX =
    new RotationOrder("XYX", Vector3D.plusI, Vector3D.plusJ, Vector3D.plusI);
  /** Set of Euler angles.
   * this ordered set of rotations is around X, then around Z, then
   * around X
   */
  public static final RotationOrder XZX =
    new RotationOrder("XZX", Vector3D.plusI, Vector3D.plusK, Vector3D.plusI);
  /** Set of Euler angles.
   * this ordered set of rotations is around Y, then around X, then
   * around Y
   */
  public static final RotationOrder YXY =
    new RotationOrder("YXY", Vector3D.plusJ, Vector3D.plusI, Vector3D.plusJ);
  /** Set of Euler angles.
   * this ordered set of rotations is around Y, then around Z, then
   * around Y
   */
  public static final RotationOrder YZY =
    new RotationOrder("YZY", Vector3D.plusJ, Vector3D.plusK, Vector3D.plusJ);
  /** Set of Euler angles.
   * this ordered set of rotations is around Z, then around X, then
   * around Z
   */
  public static final RotationOrder ZXZ =
    new RotationOrder("ZXZ", Vector3D.plusK, Vector3D.plusI, Vector3D.plusK);
  /** Set of Euler angles.
   * this ordered set of rotations is around Z, then around Y, then
   * around Z
   */
  public static final RotationOrder ZYZ =
    new RotationOrder("ZYZ", Vector3D.plusK, Vector3D.plusJ, Vector3D.plusK);
  /** Name of the rotations order. */
  private final String name;
  /** Axis of the first rotation. */
  private final Vector3D a1;
  /** Axis of the second rotation. */
  private final Vector3D a2;
  /** Axis of the third rotation. */
  private final Vector3D a3;
}
/** 
 * This class implements vectors in a three-dimensional space.
 * <p>Instance of this class are guaranteed to be immutable.</p>
 * @version $Revision: 627998 $ $Date: 2008-02-15 03:24:50 -0700 (Fri, 15 Feb 2008) $
 * @since 1.2
 */
 class Vector3D
  implements Serializable {
  /** First canonical vector (coordinates: 1, 0, 0). */
  public static final Vector3D plusI = new Vector3D(1, 0, 0);
  /** Opposite of the first canonical vector (coordinates: -1, 0, 0). */
  public static final Vector3D minusI = new Vector3D(-1, 0, 0);
  /** Second canonical vector (coordinates: 0, 1, 0). */
  public static final Vector3D plusJ = new Vector3D(0, 1, 0);
  /** Opposite of the second canonical vector (coordinates: 0, -1, 0). */
  public static final Vector3D minusJ = new Vector3D(0, -1, 0);
  /** Third canonical vector (coordinates: 0, 0, 1). */
  public static final Vector3D plusK = new Vector3D(0, 0, 1);
  /** Opposite of the third canonical vector (coordinates: 0, 0, -1).  */
  public static final Vector3D minusK = new Vector3D(0, 0, -1);
  /** Null vector (coordinates: 0, 0, 0). */
  public static final Vector3D zero   = new Vector3D(0, 0, 0);
  /** Simple constructor.
   * Build a null vector.
   */
  public Vector3D() {
    x = 0;
    y = 0;
    z = 0;
  }
  /** Simple constructor.
   * Build a vector from its coordinates
   * @param x abscissa
   * @param y ordinate
   * @param z height
   * @see #getX()
   * @see #getY()
   * @see #getZ()
   */
  public Vector3D(double x, double y, double z) {
    this.x = x;
    this.y = y;
    this.z = z;
  }
  /** Simple constructor.
   * Build a vector from its azimuthal coordinates
   * @param alpha azimuth (&alpha;) around Z
   *              (0 is +X, &pi;/2 is +Y, &pi; is -X and 3&pi;/2 is -Y)
   * @param delta elevation (&delta;) above (XY) plane, from -&pi;/2 to +&pi;/2
   * @see #getAlpha()
   * @see #getDelta()
   */
  public Vector3D(double alpha, double delta) {
    double cosDelta = Math.cos(delta);
    this.x = Math.cos(alpha) * cosDelta;
    this.y = Math.sin(alpha) * cosDelta;
    this.z = Math.sin(delta);
  }
  /** Multiplicative constructor
   * Build a vector from another one and a scale factor. 
   * The vector built will be a * u
   * @param a scale factor
   * @param u base (unscaled) vector
   */
  public Vector3D(double a, Vector3D u) {
    this.x = a * u.x;
    this.y = a * u.y;
    this.z = a * u.z;
  }
  /** Linear constructor
   * Build a vector from two other ones and corresponding scale factors.
   * The vector built will be a1 * u1 + a2 * u2
   * @param a1 first scale factor
   * @param u1 first base (unscaled) vector
   * @param a2 second scale factor
   * @param u2 second base (unscaled) vector
   */
  public Vector3D(double a1, Vector3D u1, double a2, Vector3D u2) {
    this.x = a1 * u1.x + a2 * u2.x;
    this.y = a1 * u1.y + a2 * u2.y;
    this.z = a1 * u1.z + a2 * u2.z;
  }
  /** Linear constructor
   * Build a vector from three other ones and corresponding scale factors.
   * The vector built will be a1 * u1 + a2 * u2 + a3 * u3
   * @param a1 first scale factor
   * @param u1 first base (unscaled) vector
   * @param a2 second scale factor
   * @param u2 second base (unscaled) vector
   * @param a3 third scale factor
   * @param u3 third base (unscaled) vector
   */
  public Vector3D(double a1, Vector3D u1, double a2, Vector3D u2,
                  double a3, Vector3D u3) {
    this.x = a1 * u1.x + a2 * u2.x + a3 * u3.x;
    this.y = a1 * u1.y + a2 * u2.y + a3 * u3.y;
    this.z = a1 * u1.z + a2 * u2.z + a3 * u3.z;
  }
  /** Linear constructor
   * Build a vector from four other ones and corresponding scale factors.
   * The vector built will be a1 * u1 + a2 * u2 + a3 * u3 + a4 * u4
   * @param a1 first scale factor
   * @param u1 first base (unscaled) vector
   * @param a2 second scale factor
   * @param u2 second base (unscaled) vector
   * @param a3 third scale factor
   * @param u3 third base (unscaled) vector
   * @param a4 fourth scale factor
   * @param u4 fourth base (unscaled) vector
   */
  public Vector3D(double a1, Vector3D u1, double a2, Vector3D u2,
                  double a3, Vector3D u3, double a4, Vector3D u4) {
    this.x = a1 * u1.x + a2 * u2.x + a3 * u3.x + a4 * u4.x;
    this.y = a1 * u1.y + a2 * u2.y + a3 * u3.y + a4 * u4.y;
    this.z = a1 * u1.z + a2 * u2.z + a3 * u3.z + a4 * u4.z;
  }
  /** Get the abscissa of the vector.
   * @return abscissa of the vector
   * @see #Vector3D(double, double, double)
   */
  public double getX() {
    return x;
  }
  /** Get the ordinate of the vector.
   * @return ordinate of the vector
   * @see #Vector3D(double, double, double)
   */
  public double getY() {
    return y;
  }
  /** Get the height of the vector.
   * @return height of the vector
   * @see #Vector3D(double, double, double)
   */
  public double getZ() {
    return z;
  }
  /** Get the norm for the vector.
   * @return euclidian norm for the vector
   */
  public double getNorm() {
    return Math.sqrt (x * x + y * y + z * z);
  }
  /** Get the azimuth of the vector.
   * @return azimuth (&alpha;) of the vector, between -&pi; and +&pi;
   * @see #Vector3D(double, double)
   */
  public double getAlpha() {
    return Math.atan2(y, x);
  }
  /** Get the elevation of the vector.
   * @return elevation (&delta;) of the vector, between -&pi;/2 and +&pi;/2
   * @see #Vector3D(double, double)
   */
  public double getDelta() {
    return Math.asin(z / getNorm());
  }
  /** Add a vector to the instance.
   * @param v vector to add
   * @return a new vector
   */
  public Vector3D add(Vector3D v) {
    return new Vector3D(x + v.x, y + v.y, z + v.z);
  }
  /** Add a scaled vector to the instance.
   * @param factor scale factor to apply to v before adding it
   * @param v vector to add
   * @return a new vector
   */
  public Vector3D add(double factor, Vector3D v) {
    return new Vector3D(x + factor * v.x, y + factor * v.y, z + factor * v.z);
  }
  /** Subtract a vector from the instance.
   * @param v vector to subtract
   * @return a new vector
   */
  public Vector3D subtract(Vector3D v) {
    return new Vector3D(x - v.x, y - v.y, z - v.z);
  }
  /** Subtract a scaled vector from the instance.
   * @param factor scale factor to apply to v before subtracting it
   * @param v vector to subtract
   * @return a new vector
   */
  public Vector3D subtract(double factor, Vector3D v) {
    return new Vector3D(x - factor * v.x, y - factor * v.y, z - factor * v.z);
  }
  /** Get a normalized vector aligned with the instance.
   * @return a new normalized vector
   * @exception ArithmeticException if the norm is zero
   */
  public Vector3D normalize() {
    double s = getNorm();
    if (s == 0) {
      throw new ArithmeticException("cannot normalize a zero norm vector");
    }
    return scalarMultiply(1 / s);
  }
  /** Get a vector orthogonal to the instance.
   * <p>There are an infinite number of normalized vectors orthogonal
   * to the instance. This method picks up one of them almost
   * arbitrarily. It is useful when one needs to compute a reference
   * frame with one of the axes in a predefined direction. The
   * following example shows how to build a frame having the k axis
   * aligned with the known vector u :
   * <pre><code>
   *   Vector3D k = u.normalize();
   *   Vector3D i = k.orthogonal();
   *   Vector3D j = Vector3D.crossProduct(k, i);
   * </code></pre></p>
   * @return a new normalized vector orthogonal to the instance
   * @exception ArithmeticException if the norm of the instance is null
   */
  public Vector3D orthogonal() {
    double threshold = 0.6 * getNorm();
    if (threshold == 0) {
      throw new ArithmeticException("null norm");
    }
    if ((x >= -threshold) && (x <= threshold)) {
      double inverse  = 1 / Math.sqrt(y * y + z * z);
      return new Vector3D(0, inverse * z, -inverse * y);
    } else if ((y >= -threshold) && (y <= threshold)) {
      double inverse  = 1 / Math.sqrt(x * x + z * z);
      return new Vector3D(-inverse * z, 0, inverse * x);
    }
    double inverse  = 1 / Math.sqrt(x * x + y * y);
    return new Vector3D(inverse * y, -inverse * x, 0);
  }
  /** Compute the angular separation between two vectors.
   * <p>This method computes the angular separation between two
   * vectors using the dot product for well separated vectors and the
   * cross product for almost aligned vectors. This allow to have a
   * good accuracy in all cases, even for vectors very close to each
   * other.</p>
   * @param v1 first vector
   * @param v2 second vector
   * @return angular separation between v1 and v2
   * @exception ArithmeticException if either vector has a null norm
   */
  public static double angle(Vector3D v1, Vector3D v2) {
    double normProduct = v1.getNorm() * v2.getNorm();
    if (normProduct == 0) {
      throw new ArithmeticException("null norm");
    }
    double dot = dotProduct(v1, v2);
    double threshold = normProduct * 0.9999;
    if ((dot < -threshold) || (dot > threshold)) {
      // the vectors are almost aligned, compute using the sine
      Vector3D v3 = crossProduct(v1, v2);
      if (dot >= 0) {
        return Math.asin(v3.getNorm() / normProduct);
      }
      return Math.PI - Math.asin(v3.getNorm() / normProduct);
    }
    
    // the vectors are sufficiently separated to use the cosine
    return Math.acos(dot / normProduct);
  }
  /** Get the opposite of the instance.
   * @return a new vector which is opposite to the instance
   */
  public Vector3D negate() {
    return new Vector3D(-x, -y, -z);
  }
  /** Multiply the instance by a scalar
   * @param a scalar
   * @return a new vector
   */
  public Vector3D scalarMultiply(double a) {
    return new Vector3D(a * x, a * y, a * z);
  }
  /** Compute the dot-product of two vectors.
   * @param v1 first vector
   * @param v2 second vector
   * @return the dot product v1.v2
   */
  public static double dotProduct(Vector3D v1, Vector3D v2) {
    return v1.x * v2.x + v1.y * v2.y + v1.z * v2.z;
  }
  /** Compute the cross-product of two vectors.
   * @param v1 first vector
   * @param v2 second vector
   * @return the cross product v1 ^ v2 as a new Vector
   */
  public static Vector3D crossProduct(Vector3D v1, Vector3D v2) {
    return new Vector3D(v1.y * v2.z - v1.z * v2.y,
                        v1.z * v2.x - v1.x * v2.z,
                        v1.x * v2.y - v1.y * v2.x);
  }
  /** Abscissa. */
  private final double x;
  /** Ordinate. */
  private final double y;
  /** Height. */
  private final double z;
  /** Serializable version identifier */
  private static final long serialVersionUID = -5721105387745193385L;

}