rotation.java

Apache的common math数学软件包

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 * Licensed to the Apache Software Foundation (ASF) under one or more
 * contributor license agreements.  See the NOTICE file distributed with
 * this work for additional information regarding copyright ownership.
 * The ASF licenses this file to You under the Apache License, Version 2.0
 * (the "License"); you may not use this file except in compliance with
 * the License.  You may obtain a copy of the License at
 *
 *      http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 */

package org.apache.commons.math.geometry;

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)
    throws NotARotationMatrixException {

    // dimension check
    if ((m.length != 3) || (m[0].length != 3) ||
        (m[1].length != 3) || (m[2].length != 3)) {
      throw new NotARotationMatrixException("a {0}x{1} matrix" +
                                            " cannot be a rotation matrix",
                                            new Object[] {
                                              Integer.toString(m.length),
                                              Integer.toString(m[0].length)
                                            });
    }

    // compute a "close" orthogonal matrix
    double[][] ort = orthogonalizeMatrix(m, threshold);

    // check the sign of the determinant
    double det = ort[0][0] * (ort[1][1] * ort[2][2] - ort[2][1] * ort[1][2]) -
                 ort[1][0] * (ort[0][1] * ort[2][2] - ort[2][1] * ort[0][2]) +
                 ort[2][0] * (ort[0][1] * ort[1][2] - ort[1][1] * ort[0][2]);
    if (det < 0.0) {
      throw new NotARotationMatrixException("the closest orthogonal matrix" +
                                            " has a negative determinant {0}",
                                            new Object[] {
                                              Double.toString(det)
                                            });
    }

    // There are different ways to compute the quaternions elements
    // from the matrix. They all involve computing one element from
    // the diagonal of the matrix, and computing the three other ones
    // using a formula involving a division by the first element,
    // which unfortunately can be zero. Since the norm of the
    // quaternion is 1, we know at least one element has an absolute
    // value greater or equal to 0.5, so it is always possible to
    // select the right formula and avoid division by zero and even
    // numerical inaccuracy. Checking the elements in turn and using
    // the first one greater than 0.45 is safe (this leads to a simple
    // test since qi = 0.45 implies 4 qi^2 - 1 = -0.19)
    double s = ort[0][0] + ort[1][1] + ort[2][2];
    if (s > -0.19) {
      // compute q0 and deduce q1, q2 and q3
      q0 = 0.5 * 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