rotation.java

Apache的common math数学软件包

    // 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)
    throws CardanEulerSingularityException {

    if (order == RotationOrder.XYZ) {

      // r (Vector3D.plusK) coordinates are :
      //  sin (theta), -cos (theta) sin (phi), cos (theta) cos (phi)
      // (-r) (Vector3D.plusI) coordinates are :
      // cos (psi) cos (theta), -sin (psi) cos (theta), sin (theta)
      // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
      Vector3D v1 = applyTo(Vector3D.plusK);
      Vector3D v2 = applyInverseTo(Vector3D.plusI);
      if  ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
        throw new CardanEulerSingularityException(true);
      }
      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)) {
        throw new CardanEulerSingularityException(true);
      }
      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)) {
        throw new CardanEulerSingularityException(true);
      }
      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)) {
        throw new CardanEulerSingularityException(true);
      }
      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)) {
        throw new CardanEulerSingularityException(true);
      }
      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)) {
        throw new CardanEulerSingularityException(true);
      }
      return new double[] {
        Math.atan2(v1.getY(), v1.getX()),
       -Math.asin(v2.getX()),