rotation.java

Apache的common math数学软件包

        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)) {
        throw new CardanEulerSingularityException(false);
      }
      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)) {
        throw new CardanEulerSingularityException(false);
      }
      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)) {
        throw new CardanEulerSingularityException(false);
      }
      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)) {
        throw new CardanEulerSingularityException(false);
      }
      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)) {
        throw new CardanEulerSingularityException(false);
      }
      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 CardanEulerSingularityException(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)
    throws NotARotationMatrixException {
    double[] m0 = m[0];
    double[] m1 = m[1];
    double[] m2 = m[2];
    double x00 = m0[0];
    double x01 = m0[1];
    double x02 = m0[2];
    double x10 = m1[0];
    double x11 = m1[1];
    double x12 = m1[2];
    double x20 = m2[0];
    double x21 = m2[1];
    double x22 = m2[2];
    double fn = 0;
    double fn1;

    double[][] o = new double[3][3];
    double[] o0 = o[0];
    double[] o1 = o[1];
    double[] o2 = o[2];

    // iterative correction: Xn+1 = Xn - 0.5 * (Xn.Mt.Xn - M)
    int i = 0;
    while (++i < 11) {

      // Mt.Xn
      double mx00 = m0[0] * x00 + m1[0] * x10 + m2[0] * x20;
      double mx10 = m0[1] * x00 + m1[1] * x10 + m2[1] * x20;
      double mx20 = m0[2] * x00 + m1[2] * x10 + m2[2] * x20;
      double mx01 = m0[0] * x01 + m1[0] * x11 + m2[0] * x21;
      double mx11 = m0[1] * x01 + m1[1] * x11 + m2[1] * x21;
      double mx21 = m0[2] * x01 + m1[2] * x11 + m2[2] * x21;
      double mx02 = m0[0] * x02 + m1[0] * x12 + m2[0] * x22;
      double mx12 = m0[1] * x02 + m1[1] * x12 + m2[1] * x22;
      double mx22 = m0[2] * x02 + m1[2] * x12 + m2[2] * x22;

      // Xn+1
      o0[0] = x00 - 0.5 * (x00 * mx00 + x01 * mx10 + x02 * mx20 - m0[0]);
      o0[1] = x01 - 0.5 * (x00 * mx01 + x01 * mx11 + x02 * mx21 - m0[1]);
      o0[2] = x02 - 0.5 * (x00 * mx02 + x01 * mx12 + x02 * mx22 - m0[2]);
      o1[0] = x10 - 0.5 * (x10 * mx00 + x11 * mx10 + x12 * mx20 - m1[0]);
      o1[1] = x11 - 0.5 * (x10 * mx01 + x11 * mx11 + x12 * mx21 - m1[1]);
      o1[2] = x12 - 0.5 * (x10 * mx02 + x11 * mx12 + x12 * mx22 - m1[2]);
      o2[0] = x20 - 0.5 * (x20 * mx00 + x21 * mx10 + x22 * mx20 - m2[0]);
      o2[1] = x21 - 0.5 * (x20 * mx01 + x21 * mx11 + x22 * mx21 - m2[1]);
      o2[2] = x22 - 0.5 * (x20 * mx02 + x21 * mx12 + x22 * mx22 - m2[2]);

      // correction on each elements
      double corr00 = o0[0] - m0[0];
      double corr01 = o0[1] - m0[1];
      double corr02 = o0[2] - m0[2];
      double corr10 = o1[0] - m1[0];
      double corr11 = o1[1] - m1[1];
      double corr12 = o1[2] - m1[2];
      double corr20 = o2[0] - m2[0];
      double corr21 = o2[1] - m2[1];
      double corr22 = o2[2] - m2[2];

      // Frobenius norm of the correction
      fn1 = corr00 * corr00 + corr01 * corr01 + corr02 * corr02 +
            corr10 * corr10 + corr11 * corr11 + corr12 * corr12 +
            corr20 * corr20 + corr21 * corr21 + corr22 * corr22;

      // convergence test
      if (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;

    }

    // the algorithm did not converge after 10 iterations
    throw new NotARotationMatrixException("unable to orthogonalize matrix" +
                                          " in {0} iterations",
                                          new Object[] {
                                            Integer.toString(i - 1)
                                          });
  }

  /** 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;

}