Java/2D Graphics GUI/Matrix

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

<source lang="java"> /*

* (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] + "]");
 }

}


 </source>   



Rotations in a three-dimensional space

<source lang="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.
*/

import java.io.Serializable; /**

* This class implements rotations in a three-dimensional space.
*
*

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 * Rotation instance (see the various constructors and * getters). In addition, a rotation can also be built implicitely * from a set of vectors and their image.

*

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:

*
 * double[] angles = new Rotation(matrix, 1.0e-10).getAngles(RotationOrder.XYZ);
 * 
*

Focus is oriented on what a rotation do rather than on its * underlying representation. Once it has been built, and regardless of its * internal representation, a rotation is an operator 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.

*

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.

*
*

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 projectVectorIntoDestinationFrame or * computeTransformedDirection. It provides simpler and more generic * methods: {@link #applyTo(Vector3D) applyTo(Vector3D)} and {@link * #applyInverseTo(Vector3D) applyInverseTo(Vector3D)}.

*
*

Since a rotation is basically a vectorial operator, several rotations can be * composed together and the composite operation r = r1 o * r2 (which means that for each vector u, * r(u) = r1(r2(u))) 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 r1 to r2 and the result * we get is r = r1 o r2. For this purpose, the * class provides the methods: {@link #applyTo(Rotation) applyTo(Rotation)} and * {@link #applyInverseTo(Rotation) applyInverseTo(Rotation)}.

*
*

Rotations are guaranteed to be immutable objects.

*
* @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.
*

A rotation can be built from a normalized quaternion, * i.e. a quaternion for which q02 + * q12 + q22 + * q32 = 1. If the quaternion is not normalized, * the constructor can normalize it in a preprocessing step.

  * @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.
*

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.

  * @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.
*

Rotation matrices are orthogonal matrices, i.e. unit matrices * (which are matrices for which m.mT = 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.

*

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 (u1, u2) it will produce the pair * (v1, v2).

*

If the angular separation between u1 and u2 is * not the same as the angular separation between v1 and * v2, then a corrected v"2 will be used rather than * v2, the corrected vector will be in the (v1, * v2) plane.

  * @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.
*

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.

  * @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.
*

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.

*

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).

  * @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 π)
  */
 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.
*

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 * a1, a2 and a3 is the same as * the rotation defined by the angles π + a1, π * - a2 and π + a3. This method implements * the following arbitrary choices:

*
    *
  • for Cardan angles, the chosen set is the one for which the * second angle is between -π/2 and π/2 (i.e its cosine is * positive),
  • *
  • for Euler angles, the chosen set is the one for which the * second angle is between 0 and π (i.e its sine is positive).
  • *
*

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 * nothing 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 * -π/2 or +π/2, for Euler angle singularities occur when the * second angle is close to 0 or π, this implies that the identity * rotation is always singular for Euler angles!

  * @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.
*

Instance of this class are guaranteed to be immutable.

* @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 (α) around Z
  *              (0 is +X, π/2 is +Y, π is -X and 3π/2 is -Y)
  * @param delta elevation (δ) above (XY) plane, from -π/2 to +π/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 (α) of the vector, between -π and +π
  * @see #Vector3D(double, double)
  */
 public double getAlpha() {
   return Math.atan2(y, x);
 }
 /** Get the elevation of the vector.
  * @return elevation (δ) of the vector, between -π/2 and +π/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.
*

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 : *

<code>
   *   Vector3D k = u.normalize();
   *   Vector3D i = k.orthogonal();
   *   Vector3D j = Vector3D.crossProduct(k, i);
   * </code>

  * @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.
*

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.

  * @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;

}

 </source>