FieldRotation.java

  1. /*
  2.  * Licensed to the Apache Software Foundation (ASF) under one or more
  3.  * contributor license agreements.  See the NOTICE file distributed with
  4.  * this work for additional information regarding copyright ownership.
  5.  * The ASF licenses this file to You under the Apache License, Version 2.0
  6.  * (the "License"); you may not use this file except in compliance with
  7.  * the License.  You may obtain a copy of the License at
  8.  *
  9.  *      http://www.apache.org/licenses/LICENSE-2.0
  10.  *
  11.  * Unless required by applicable law or agreed to in writing, software
  12.  * distributed under the License is distributed on an "AS IS" BASIS,
  13.  * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  14.  * See the License for the specific language governing permissions and
  15.  * limitations under the License.
  16.  */

  18. package org.apache.commons.math3.geometry.euclidean.threed;

  20. import java.io.Serializable;

  22. import org.apache.commons.math3.RealFieldElement;
  23. import org.apache.commons.math3.Field;
  24. import org.apache.commons.math3.exception.MathArithmeticException;
  25. import org.apache.commons.math3.exception.MathIllegalArgumentException;
  26. import org.apache.commons.math3.exception.util.LocalizedFormats;
  27. import org.apache.commons.math3.util.FastMath;
  28. import org.apache.commons.math3.util.MathArrays;

  30. /**
  31.  * This class is a re-implementation of {@link Rotation} using {@link RealFieldElement}.
  32.  * <p>Instance of this class are guaranteed to be immutable.</p>
  33.  *
  34.  * @param <T> the type of the field elements
  35.  * @see FieldVector3D
  36.  * @see RotationOrder
  37.  * @since 3.2
  38.  */

  40. public class FieldRotation<T extends RealFieldElement<T>> implements Serializable {

  42.     /** Serializable version identifier */
  43.     private static final long serialVersionUID = 20130224l;

  45.     /** Scalar coordinate of the quaternion. */
  46.     private final T q0;

  48.     /** First coordinate of the vectorial part of the quaternion. */
  49.     private final T q1;

  51.     /** Second coordinate of the vectorial part of the quaternion. */
  52.     private final T q2;

  54.     /** Third coordinate of the vectorial part of the quaternion. */
  55.     private final T q3;

  57.     /** Build a rotation from the quaternion coordinates.
  58.      * <p>A rotation can be built from a <em>normalized</em> quaternion,
  59.      * i.e. a quaternion for which q<sub>0</sub><sup>2</sup> +
  60.      * q<sub>1</sub><sup>2</sup> + q<sub>2</sub><sup>2</sup> +
  61.      * q<sub>3</sub><sup>2</sup> = 1. If the quaternion is not normalized,
  62.      * the constructor can normalize it in a preprocessing step.</p>
  63.      * <p>Note that some conventions put the scalar part of the quaternion
  64.      * as the 4<sup>th</sup> component and the vector part as the first three
  65.      * components. This is <em>not</em> our convention. We put the scalar part
  66.      * as the first component.</p>
  67.      * @param q0 scalar part of the quaternion
  68.      * @param q1 first coordinate of the vectorial part of the quaternion
  69.      * @param q2 second coordinate of the vectorial part of the quaternion
  70.      * @param q3 third coordinate of the vectorial part of the quaternion
  71.      * @param needsNormalization if true, the coordinates are considered
  72.      * not to be normalized, a normalization preprocessing step is performed
  73.      * before using them
  74.      */
  75.     public FieldRotation(final T q0, final T q1, final T q2, final T q3, final boolean needsNormalization) {

  77.         if (needsNormalization) {
  78.             // normalization preprocessing
  79.             final T inv =
  80.                     q0.multiply(q0).add(q1.multiply(q1)).add(q2.multiply(q2)).add(q3.multiply(q3)).sqrt().reciprocal();
  81.             this.q0 = inv.multiply(q0);
  82.             this.q1 = inv.multiply(q1);
  83.             this.q2 = inv.multiply(q2);
  84.             this.q3 = inv.multiply(q3);
  85.         } else {
  86.             this.q0 = q0;
  87.             this.q1 = q1;
  88.             this.q2 = q2;
  89.             this.q3 = q3;
  90.         }

  92.     }

  94.     /** Build a rotation from an axis and an angle.
  95.      * <p>We use the convention that angles are oriented according to
  96.      * the effect of the rotation on vectors around the axis. That means
  97.      * that if (i, j, k) is a direct frame and if we first provide +k as
  98.      * the axis and &pi;/2 as the angle to this constructor, and then
  99.      * {@link #applyTo(FieldVector3D) apply} the instance to +i, we will get
  100.      * +j.</p>
  101.      * <p>Another way to represent our convention is to say that a rotation
  102.      * of angle &theta; about the unit vector (x, y, z) is the same as the
  103.      * rotation build from quaternion components { cos(-&theta;/2),
  104.      * x * sin(-&theta;/2), y * sin(-&theta;/2), z * sin(-&theta;/2) }.
  105.      * Note the minus sign on the angle!</p>
  106.      * <p>On the one hand this convention is consistent with a vectorial
  107.      * perspective (moving vectors in fixed frames), on the other hand it
  108.      * is different from conventions with a frame perspective (fixed vectors
  109.      * viewed from different frames) like the ones used for example in spacecraft
  110.      * attitude community or in the graphics community.</p>
  111.      * @param axis axis around which to rotate
  112.      * @param angle rotation angle.
  113.      * @exception MathIllegalArgumentException if the axis norm is zero
  114.      */
  115.     public FieldRotation(final FieldVector3D<T> axis, final T angle)
  116.         throws MathIllegalArgumentException {

  118.         final T norm = axis.getNorm();
  119.         if (norm.getReal() == 0) {
  120.             throw new MathIllegalArgumentException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_AXIS);
  121.         }

  123.         final T halfAngle = angle.multiply(-0.5);
  124.         final T coeff = halfAngle.sin().divide(norm);

  126.         q0 = halfAngle.cos();
  127.         q1 = coeff.multiply(axis.getX());
  128.         q2 = coeff.multiply(axis.getY());
  129.         q3 = coeff.multiply(axis.getZ());

  131.     }

  133.     /** Build a rotation from a 3X3 matrix.

  135.      * <p>Rotation matrices are orthogonal matrices, i.e. unit matrices
  136.      * (which are matrices for which m.m<sup>T</sup> = I) with real
  137.      * coefficients. The module of the determinant of unit matrices is
  138.      * 1, among the orthogonal 3X3 matrices, only the ones having a
  139.      * positive determinant (+1) are rotation matrices.</p>

  141.      * <p>When a rotation is defined by a matrix with truncated values
  142.      * (typically when it is extracted from a technical sheet where only
  143.      * four to five significant digits are available), the matrix is not
  144.      * orthogonal anymore. This constructor handles this case
  145.      * transparently by using a copy of the given matrix and applying a
  146.      * correction to the copy in order to perfect its orthogonality. If
  147.      * the Frobenius norm of the correction needed is above the given
  148.      * threshold, then the matrix is considered to be too far from a
  149.      * true rotation matrix and an exception is thrown.<p>

  151.      * @param m rotation matrix
  152.      * @param threshold convergence threshold for the iterative
  153.      * orthogonality correction (convergence is reached when the
  154.      * difference between two steps of the Frobenius norm of the
  155.      * correction is below this threshold)

  157.      * @exception NotARotationMatrixException if the matrix is not a 3X3
  158.      * matrix, or if it cannot be transformed into an orthogonal matrix
  159.      * with the given threshold, or if the determinant of the resulting
  160.      * orthogonal matrix is negative

  162.      */
  163.     public FieldRotation(final T[][] m, final double threshold)
  164.         throws NotARotationMatrixException {

  166.         // dimension check
  167.         if ((m.length != 3) || (m[0].length != 3) ||
  168.                 (m[1].length != 3) || (m[2].length != 3)) {
  169.             throw new NotARotationMatrixException(
  170.                                                   LocalizedFormats.ROTATION_MATRIX_DIMENSIONS,
  171.                                                   m.length, m[0].length);
  172.         }

  174.         // compute a "close" orthogonal matrix
  175.         final T[][] ort = orthogonalizeMatrix(m, threshold);

  177.         // check the sign of the determinant
  178.         final T d0 = ort[1][1].multiply(ort[2][2]).subtract(ort[2][1].multiply(ort[1][2]));
  179.         final T d1 = ort[0][1].multiply(ort[2][2]).subtract(ort[2][1].multiply(ort[0][2]));
  180.         final T d2 = ort[0][1].multiply(ort[1][2]).subtract(ort[1][1].multiply(ort[0][2]));
  181.         final T det =
  182.                 ort[0][0].multiply(d0).subtract(ort[1][0].multiply(d1)).add(ort[2][0].multiply(d2));
  183.         if (det.getReal() < 0.0) {
  184.             throw new NotARotationMatrixException(
  185.                                                   LocalizedFormats.CLOSEST_ORTHOGONAL_MATRIX_HAS_NEGATIVE_DETERMINANT,
  186.                                                   det);
  187.         }

  189.         final T[] quat = mat2quat(ort);
  190.         q0 = quat[0];
  191.         q1 = quat[1];
  192.         q2 = quat[2];
  193.         q3 = quat[3];

  195.     }

  197.     /** Build the rotation that transforms a pair of vector into another pair.

  199.      * <p>Except for possible scale factors, if the instance were applied to
  200.      * the pair (u<sub>1</sub>, u<sub>2</sub>) it will produce the pair
  201.      * (v<sub>1</sub>, v<sub>2</sub>).</p>

  203.      * <p>If the angular separation between u<sub>1</sub> and u<sub>2</sub> is
  204.      * not the same as the angular separation between v<sub>1</sub> and
  205.      * v<sub>2</sub>, then a corrected v'<sub>2</sub> will be used rather than
  206.      * v<sub>2</sub>, the corrected vector will be in the (v<sub>1</sub>,
  207.      * v<sub>2</sub>) plane.</p>

  209.      * @param u1 first vector of the origin pair
  210.      * @param u2 second vector of the origin pair
  211.      * @param v1 desired image of u1 by the rotation
  212.      * @param v2 desired image of u2 by the rotation
  213.      * @exception MathArithmeticException if the norm of one of the vectors is zero,
  214.      * or if one of the pair is degenerated (i.e. the vectors of the pair are colinear)
  215.      */
  216.     public FieldRotation(FieldVector3D<T> u1, FieldVector3D<T> u2, FieldVector3D<T> v1, FieldVector3D<T> v2)
  217.         throws MathArithmeticException {

  219.         // build orthonormalized base from u1, u2
  220.         // this fails when vectors are null or colinear, which is forbidden to define a rotation
  221.         final FieldVector3D<T> u3 = FieldVector3D.crossProduct(u1, u2).normalize();
  222.         u2 = FieldVector3D.crossProduct(u3, u1).normalize();
  223.         u1 = u1.normalize();

  225.         // build an orthonormalized base from v1, v2
  226.         // this fails when vectors are null or colinear, which is forbidden to define a rotation
  227.         final FieldVector3D<T> v3 = FieldVector3D.crossProduct(v1, v2).normalize();
  228.         v2 = FieldVector3D.crossProduct(v3, v1).normalize();
  229.         v1 = v1.normalize();

  231.         // buid a matrix transforming the first base into the second one
  232.         final T[][] array = MathArrays.buildArray(u1.getX().getField(), 3, 3);
  233.         array[0][0] = u1.getX().multiply(v1.getX()).add(u2.getX().multiply(v2.getX())).add(u3.getX().multiply(v3.getX()));
  234.         array[0][1] = u1.getY().multiply(v1.getX()).add(u2.getY().multiply(v2.getX())).add(u3.getY().multiply(v3.getX()));
  235.         array[0][2] = u1.getZ().multiply(v1.getX()).add(u2.getZ().multiply(v2.getX())).add(u3.getZ().multiply(v3.getX()));
  236.         array[1][0] = u1.getX().multiply(v1.getY()).add(u2.getX().multiply(v2.getY())).add(u3.getX().multiply(v3.getY()));
  237.         array[1][1] = u1.getY().multiply(v1.getY()).add(u2.getY().multiply(v2.getY())).add(u3.getY().multiply(v3.getY()));
  238.         array[1][2] = u1.getZ().multiply(v1.getY()).add(u2.getZ().multiply(v2.getY())).add(u3.getZ().multiply(v3.getY()));
  239.         array[2][0] = u1.getX().multiply(v1.getZ()).add(u2.getX().multiply(v2.getZ())).add(u3.getX().multiply(v3.getZ()));
  240.         array[2][1] = u1.getY().multiply(v1.getZ()).add(u2.getY().multiply(v2.getZ())).add(u3.getY().multiply(v3.getZ()));
  241.         array[2][2] = u1.getZ().multiply(v1.getZ()).add(u2.getZ().multiply(v2.getZ())).add(u3.getZ().multiply(v3.getZ()));

  243.         T[] quat = mat2quat(array);
  244.         q0 = quat[0];
  245.         q1 = quat[1];
  246.         q2 = quat[2];
  247.         q3 = quat[3];

  249.     }

  251.     /** Build one of the rotations that transform one vector into another one.

  253.      * <p>Except for a possible scale factor, if the instance were
  254.      * applied to the vector u it will produce the vector v. There is an
  255.      * infinite number of such rotations, this constructor choose the
  256.      * one with the smallest associated angle (i.e. the one whose axis
  257.      * is orthogonal to the (u, v) plane). If u and v are colinear, an
  258.      * arbitrary rotation axis is chosen.</p>

  260.      * @param u origin vector
  261.      * @param v desired image of u by the rotation
  262.      * @exception MathArithmeticException if the norm of one of the vectors is zero
  263.      */
  264.     public FieldRotation(final FieldVector3D<T> u, final FieldVector3D<T> v) throws MathArithmeticException {

  266.         final T normProduct = u.getNorm().multiply(v.getNorm());
  267.         if (normProduct.getReal() == 0) {
  268.             throw new MathArithmeticException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_DEFINING_VECTOR);
  269.         }

  271.         final T dot = FieldVector3D.dotProduct(u, v);

  273.         if (dot.getReal() < ((2.0e-15 - 1.0) * normProduct.getReal())) {
  274.             // special case u = -v: we select a PI angle rotation around
  275.             // an arbitrary vector orthogonal to u
  276.             final FieldVector3D<T> w = u.orthogonal();
  277.             q0 = normProduct.getField().getZero();
  278.             q1 = w.getX().negate();
  279.             q2 = w.getY().negate();
  280.             q3 = w.getZ().negate();
  281.         } else {
  282.             // general case: (u, v) defines a plane, we select
  283.             // the shortest possible rotation: axis orthogonal to this plane
  284.             q0 = dot.divide(normProduct).add(1.0).multiply(0.5).sqrt();
  285.             final T coeff = q0.multiply(normProduct).multiply(2.0).reciprocal();
  286.             final FieldVector3D<T> q = FieldVector3D.crossProduct(v, u);
  287.             q1 = coeff.multiply(q.getX());
  288.             q2 = coeff.multiply(q.getY());
  289.             q3 = coeff.multiply(q.getZ());
  290.         }

  292.     }

  294.     /** Build a rotation from three Cardan or Euler elementary rotations.

  296.      * <p>Cardan rotations are three successive rotations around the
  297.      * canonical axes X, Y and Z, each axis being used once. There are
  298.      * 6 such sets of rotations (XYZ, XZY, YXZ, YZX, ZXY and ZYX). Euler
  299.      * rotations are three successive rotations around the canonical
  300.      * axes X, Y and Z, the first and last rotations being around the
  301.      * same axis. There are 6 such sets of rotations (XYX, XZX, YXY,
  302.      * YZY, ZXZ and ZYZ), the most popular one being ZXZ.</p>
  303.      * <p>Beware that many people routinely use the term Euler angles even
  304.      * for what really are Cardan angles (this confusion is especially
  305.      * widespread in the aerospace business where Roll, Pitch and Yaw angles
  306.      * are often wrongly tagged as Euler angles).</p>

  308.      * @param order order of rotations to use
  309.      * @param alpha1 angle of the first elementary rotation
  310.      * @param alpha2 angle of the second elementary rotation
  311.      * @param alpha3 angle of the third elementary rotation
  312.      */
  313.     public FieldRotation(final RotationOrder order, final T alpha1, final T alpha2, final T alpha3) {
  314.         final T one = alpha1.getField().getOne();
  315.         final FieldRotation<T> r1 = new FieldRotation<T>(new FieldVector3D<T>(one, order.getA1()), alpha1);
  316.         final FieldRotation<T> r2 = new FieldRotation<T>(new FieldVector3D<T>(one, order.getA2()), alpha2);
  317.         final FieldRotation<T> r3 = new FieldRotation<T>(new FieldVector3D<T>(one, order.getA3()), alpha3);
  318.         final FieldRotation<T> composed = r1.applyTo(r2.applyTo(r3));
  319.         q0 = composed.q0;
  320.         q1 = composed.q1;
  321.         q2 = composed.q2;
  322.         q3 = composed.q3;
  323.     }

  325.     /** Convert an orthogonal rotation matrix to a quaternion.
  326.      * @param ort orthogonal rotation matrix
  327.      * @return quaternion corresponding to the matrix
  328.      */
  329.     private T[] mat2quat(final T[][] ort) {

  331.         final T[] quat = MathArrays.buildArray(ort[0][0].getField(), 4);

  333.         // There are different ways to compute the quaternions elements
  334.         // from the matrix. They all involve computing one element from
  335.         // the diagonal of the matrix, and computing the three other ones
  336.         // using a formula involving a division by the first element,
  337.         // which unfortunately can be zero. Since the norm of the
  338.         // quaternion is 1, we know at least one element has an absolute
  339.         // value greater or equal to 0.5, so it is always possible to
  340.         // select the right formula and avoid division by zero and even
  341.         // numerical inaccuracy. Checking the elements in turn and using
  342.         // the first one greater than 0.45 is safe (this leads to a simple
  343.         // test since qi = 0.45 implies 4 qi^2 - 1 = -0.19)
  344.         T s = ort[0][0].add(ort[1][1]).add(ort[2][2]);
  345.         if (s.getReal() > -0.19) {
  346.             // compute q0 and deduce q1, q2 and q3
  347.             quat[0] = s.add(1.0).sqrt().multiply(0.5);
  348.             T inv = quat[0].reciprocal().multiply(0.25);
  349.             quat[1] = inv.multiply(ort[1][2].subtract(ort[2][1]));
  350.             quat[2] = inv.multiply(ort[2][0].subtract(ort[0][2]));
  351.             quat[3] = inv.multiply(ort[0][1].subtract(ort[1][0]));
  352.         } else {
  353.             s = ort[0][0].subtract(ort[1][1]).subtract(ort[2][2]);
  354.             if (s.getReal() > -0.19) {
  355.                 // compute q1 and deduce q0, q2 and q3
  356.                 quat[1] = s.add(1.0).sqrt().multiply(0.5);
  357.                 T inv = quat[1].reciprocal().multiply(0.25);
  358.                 quat[0] = inv.multiply(ort[1][2].subtract(ort[2][1]));
  359.                 quat[2] = inv.multiply(ort[0][1].add(ort[1][0]));
  360.                 quat[3] = inv.multiply(ort[0][2].add(ort[2][0]));
  361.             } else {
  362.                 s = ort[1][1].subtract(ort[0][0]).subtract(ort[2][2]);
  363.                 if (s.getReal() > -0.19) {
  364.                     // compute q2 and deduce q0, q1 and q3
  365.                     quat[2] = s.add(1.0).sqrt().multiply(0.5);
  366.                     T inv = quat[2].reciprocal().multiply(0.25);
  367.                     quat[0] = inv.multiply(ort[2][0].subtract(ort[0][2]));
  368.                     quat[1] = inv.multiply(ort[0][1].add(ort[1][0]));
  369.                     quat[3] = inv.multiply(ort[2][1].add(ort[1][2]));
  370.                 } else {
  371.                     // compute q3 and deduce q0, q1 and q2
  372.                     s = ort[2][2].subtract(ort[0][0]).subtract(ort[1][1]);
  373.                     quat[3] = s.add(1.0).sqrt().multiply(0.5);
  374.                     T inv = quat[3].reciprocal().multiply(0.25);
  375.                     quat[0] = inv.multiply(ort[0][1].subtract(ort[1][0]));
  376.                     quat[1] = inv.multiply(ort[0][2].add(ort[2][0]));
  377.                     quat[2] = inv.multiply(ort[2][1].add(ort[1][2]));
  378.                 }
  379.             }
  380.         }

  382.         return quat;

  384.     }

  386.     /** Revert a rotation.
  387.      * Build a rotation which reverse the effect of another
  388.      * rotation. This means that if r(u) = v, then r.revert(v) = u. The
  389.      * instance is not changed.
  390.      * @return a new rotation whose effect is the reverse of the effect
  391.      * of the instance
  392.      */
  393.     public FieldRotation<T> revert() {
  394.         return new FieldRotation<T>(q0.negate(), q1, q2, q3, false);
  395.     }

  397.     /** Get the scalar coordinate of the quaternion.
  398.      * @return scalar coordinate of the quaternion
  399.      */
  400.     public T getQ0() {
  401.         return q0;
  402.     }

  404.     /** Get the first coordinate of the vectorial part of the quaternion.
  405.      * @return first coordinate of the vectorial part of the quaternion
  406.      */
  407.     public T getQ1() {
  408.         return q1;
  409.     }

  411.     /** Get the second coordinate of the vectorial part of the quaternion.
  412.      * @return second coordinate of the vectorial part of the quaternion
  413.      */
  414.     public T getQ2() {
  415.         return q2;
  416.     }

  418.     /** Get the third coordinate of the vectorial part of the quaternion.
  419.      * @return third coordinate of the vectorial part of the quaternion
  420.      */
  421.     public T getQ3() {
  422.         return q3;
  423.     }

  425.     /** Get the normalized axis of the rotation.
  426.      * @return normalized axis of the rotation
  427.      * @see #FieldRotation(FieldVector3D, RealFieldElement)
  428.      */
  429.     public FieldVector3D<T> getAxis() {
  430.         final T squaredSine = q1.multiply(q1).add(q2.multiply(q2)).add(q3.multiply(q3));
  431.         if (squaredSine.getReal() == 0) {
  432.             final Field<T> field = squaredSine.getField();
  433.             return new FieldVector3D<T>(field.getOne(), field.getZero(), field.getZero());
  434.         } else if (q0.getReal() < 0) {
  435.             T inverse = squaredSine.sqrt().reciprocal();
  436.             return new FieldVector3D<T>(q1.multiply(inverse), q2.multiply(inverse), q3.multiply(inverse));
  437.         }
  438.         final T inverse = squaredSine.sqrt().reciprocal().negate();
  439.         return new FieldVector3D<T>(q1.multiply(inverse), q2.multiply(inverse), q3.multiply(inverse));
  440.     }

  442.     /** Get the angle of the rotation.
  443.      * @return angle of the rotation (between 0 and &pi;)
  444.      * @see #FieldRotation(FieldVector3D, RealFieldElement)
  445.      */
  446.     public T getAngle() {
  447.         if ((q0.getReal() < -0.1) || (q0.getReal() > 0.1)) {
  448.             return q1.multiply(q1).add(q2.multiply(q2)).add(q3.multiply(q3)).sqrt().asin().multiply(2);
  449.         } else if (q0.getReal() < 0) {
  450.             return q0.negate().acos().multiply(2);
  451.         }
  452.         return q0.acos().multiply(2);
  453.     }

  455.     /** Get the Cardan or Euler angles corresponding to the instance.

  457.      * <p>The equations show that each rotation can be defined by two
  458.      * different values of the Cardan or Euler angles set. For example
  459.      * if Cardan angles are used, the rotation defined by the angles
  460.      * a<sub>1</sub>, a<sub>2</sub> and a<sub>3</sub> is the same as
  461.      * the rotation defined by the angles &pi; + a<sub>1</sub>, &pi;
  462.      * - a<sub>2</sub> and &pi; + a<sub>3</sub>. This method implements
  463.      * the following arbitrary choices:</p>
  464.      * <ul>
  465.      *   <li>for Cardan angles, the chosen set is the one for which the
  466.      *   second angle is between -&pi;/2 and &pi;/2 (i.e its cosine is
  467.      *   positive),</li>
  468.      *   <li>for Euler angles, the chosen set is the one for which the
  469.      *   second angle is between 0 and &pi; (i.e its sine is positive).</li>
  470.      * </ul>

  472.      * <p>Cardan and Euler angle have a very disappointing drawback: all
  473.      * of them have singularities. This means that if the instance is
  474.      * too close to the singularities corresponding to the given
  475.      * rotation order, it will be impossible to retrieve the angles. For
  476.      * Cardan angles, this is often called gimbal lock. There is
  477.      * <em>nothing</em> to do to prevent this, it is an intrinsic problem
  478.      * with Cardan and Euler representation (but not a problem with the
  479.      * rotation itself, which is perfectly well defined). For Cardan
  480.      * angles, singularities occur when the second angle is close to
  481.      * -&pi;/2 or +&pi;/2, for Euler angle singularities occur when the
  482.      * second angle is close to 0 or &pi;, this implies that the identity
  483.      * rotation is always singular for Euler angles!</p>

  485.      * @param order rotation order to use
  486.      * @return an array of three angles, in the order specified by the set
  487.      * @exception CardanEulerSingularityException if the rotation is
  488.      * singular with respect to the angles set specified
  489.      */
  490.     public T[] getAngles(final RotationOrder order)
  491.         throws CardanEulerSingularityException {

  493.         if (order == RotationOrder.XYZ) {

  495.             // r (+K) coordinates are :
  496.             //  sin (theta), -cos (theta) sin (phi), cos (theta) cos (phi)
  497.             // (-r) (+I) coordinates are :
  498.             // cos (psi) cos (theta), -sin (psi) cos (theta), sin (theta)
  499.             final // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
  500.             FieldVector3D<T> v1 = applyTo(vector(0, 0, 1));
  501.             final FieldVector3D<T> v2 = applyInverseTo(vector(1, 0, 0));
  502.             if  ((v2.getZ().getReal() < -0.9999999999) || (v2.getZ().getReal() > 0.9999999999)) {
  503.                 throw new CardanEulerSingularityException(true);
  504.             }
  505.             return buildArray(v1.getY().negate().atan2(v1.getZ()),
  506.                               v2.getZ().asin(),
  507.                               v2.getY().negate().atan2(v2.getX()));

  509.         } else if (order == RotationOrder.XZY) {

  511.             // r (+J) coordinates are :
  512.             // -sin (psi), cos (psi) cos (phi), cos (psi) sin (phi)
  513.             // (-r) (+I) coordinates are :
  514.             // cos (theta) cos (psi), -sin (psi), sin (theta) cos (psi)
  515.             // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
  516.             final FieldVector3D<T> v1 = applyTo(vector(0, 1, 0));
  517.             final FieldVector3D<T> v2 = applyInverseTo(vector(1, 0, 0));
  518.             if ((v2.getY().getReal() < -0.9999999999) || (v2.getY().getReal() > 0.9999999999)) {
  519.                 throw new CardanEulerSingularityException(true);
  520.             }
  521.             return buildArray(v1.getZ().atan2(v1.getY()),
  522.                               v2.getY().asin().negate(),
  523.                               v2.getZ().atan2(v2.getX()));

  525.         } else if (order == RotationOrder.YXZ) {

  527.             // r (+K) coordinates are :
  528.             //  cos (phi) sin (theta), -sin (phi), cos (phi) cos (theta)
  529.             // (-r) (+J) coordinates are :
  530.             // sin (psi) cos (phi), cos (psi) cos (phi), -sin (phi)
  531.             // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
  532.             final FieldVector3D<T> v1 = applyTo(vector(0, 0, 1));
  533.             final FieldVector3D<T> v2 = applyInverseTo(vector(0, 1, 0));
  534.             if ((v2.getZ().getReal() < -0.9999999999) || (v2.getZ().getReal() > 0.9999999999)) {
  535.                 throw new CardanEulerSingularityException(true);
  536.             }
  537.             return buildArray(v1.getX().atan2(v1.getZ()),
  538.                               v2.getZ().asin().negate(),
  539.                               v2.getX().atan2(v2.getY()));

  541.         } else if (order == RotationOrder.YZX) {

  543.             // r (+I) coordinates are :
  544.             // cos (psi) cos (theta), sin (psi), -cos (psi) sin (theta)
  545.             // (-r) (+J) coordinates are :
  546.             // sin (psi), cos (phi) cos (psi), -sin (phi) cos (psi)
  547.             // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
  548.             final FieldVector3D<T> v1 = applyTo(vector(1, 0, 0));
  549.             final FieldVector3D<T> v2 = applyInverseTo(vector(0, 1, 0));
  550.             if ((v2.getX().getReal() < -0.9999999999) || (v2.getX().getReal() > 0.9999999999)) {
  551.                 throw new CardanEulerSingularityException(true);
  552.             }
  553.             return buildArray(v1.getZ().negate().atan2(v1.getX()),
  554.                               v2.getX().asin(),
  555.                               v2.getZ().negate().atan2(v2.getY()));

  557.         } else if (order == RotationOrder.ZXY) {

  559.             // r (+J) coordinates are :
  560.             // -cos (phi) sin (psi), cos (phi) cos (psi), sin (phi)
  561.             // (-r) (+K) coordinates are :
  562.             // -sin (theta) cos (phi), sin (phi), cos (theta) cos (phi)
  563.             // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
  564.             final FieldVector3D<T> v1 = applyTo(vector(0, 1, 0));
  565.             final FieldVector3D<T> v2 = applyInverseTo(vector(0, 0, 1));
  566.             if ((v2.getY().getReal() < -0.9999999999) || (v2.getY().getReal() > 0.9999999999)) {
  567.                 throw new CardanEulerSingularityException(true);
  568.             }
  569.             return buildArray(v1.getX().negate().atan2(v1.getY()),
  570.                               v2.getY().asin(),
  571.                               v2.getX().negate().atan2(v2.getZ()));

  573.         } else if (order == RotationOrder.ZYX) {

  575.             // r (+I) coordinates are :
  576.             //  cos (theta) cos (psi), cos (theta) sin (psi), -sin (theta)
  577.             // (-r) (+K) coordinates are :
  578.             // -sin (theta), sin (phi) cos (theta), cos (phi) cos (theta)
  579.             // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
  580.             final FieldVector3D<T> v1 = applyTo(vector(1, 0, 0));
  581.             final FieldVector3D<T> v2 = applyInverseTo(vector(0, 0, 1));
  582.             if ((v2.getX().getReal() < -0.9999999999) || (v2.getX().getReal() > 0.9999999999)) {
  583.                 throw new CardanEulerSingularityException(true);
  584.             }
  585.             return buildArray(v1.getY().atan2(v1.getX()),
  586.                               v2.getX().asin().negate(),
  587.                               v2.getY().atan2(v2.getZ()));

  589.         } else if (order == RotationOrder.XYX) {

  591.             // r (+I) coordinates are :
  592.             //  cos (theta), sin (phi1) sin (theta), -cos (phi1) sin (theta)
  593.             // (-r) (+I) coordinates are :
  594.             // cos (theta), sin (theta) sin (phi2), sin (theta) cos (phi2)
  595.             // and we can choose to have theta in the interval [0 ; PI]
  596.             final FieldVector3D<T> v1 = applyTo(vector(1, 0, 0));
  597.             final FieldVector3D<T> v2 = applyInverseTo(vector(1, 0, 0));
  598.             if ((v2.getX().getReal() < -0.9999999999) || (v2.getX().getReal() > 0.9999999999)) {
  599.                 throw new CardanEulerSingularityException(false);
  600.             }
  601.             return buildArray(v1.getY().atan2(v1.getZ().negate()),
  602.                               v2.getX().acos(),
  603.                               v2.getY().atan2(v2.getZ()));

  605.         } else if (order == RotationOrder.XZX) {

  607.             // r (+I) coordinates are :
  608.             //  cos (psi), cos (phi1) sin (psi), sin (phi1) sin (psi)
  609.             // (-r) (+I) coordinates are :
  610.             // cos (psi), -sin (psi) cos (phi2), sin (psi) sin (phi2)
  611.             // and we can choose to have psi in the interval [0 ; PI]
  612.             final FieldVector3D<T> v1 = applyTo(vector(1, 0, 0));
  613.             final FieldVector3D<T> v2 = applyInverseTo(vector(1, 0, 0));
  614.             if ((v2.getX().getReal() < -0.9999999999) || (v2.getX().getReal() > 0.9999999999)) {
  615.                 throw new CardanEulerSingularityException(false);
  616.             }
  617.             return buildArray(v1.getZ().atan2(v1.getY()),
  618.                               v2.getX().acos(),
  619.                               v2.getZ().atan2(v2.getY().negate()));

  621.         } else if (order == RotationOrder.YXY) {

  623.             // r (+J) coordinates are :
  624.             //  sin (theta1) sin (phi), cos (phi), cos (theta1) sin (phi)
  625.             // (-r) (+J) coordinates are :
  626.             // sin (phi) sin (theta2), cos (phi), -sin (phi) cos (theta2)
  627.             // and we can choose to have phi in the interval [0 ; PI]
  628.             final FieldVector3D<T> v1 = applyTo(vector(0, 1, 0));
  629.             final FieldVector3D<T> v2 = applyInverseTo(vector(0, 1, 0));
  630.             if ((v2.getY().getReal() < -0.9999999999) || (v2.getY().getReal() > 0.9999999999)) {
  631.                 throw new CardanEulerSingularityException(false);
  632.             }
  633.             return buildArray(v1.getX().atan2(v1.getZ()),
  634.                               v2.getY().acos(),
  635.                               v2.getX().atan2(v2.getZ().negate()));

  637.         } else if (order == RotationOrder.YZY) {

  639.             // r (+J) coordinates are :
  640.             //  -cos (theta1) sin (psi), cos (psi), sin (theta1) sin (psi)
  641.             // (-r) (+J) coordinates are :
  642.             // sin (psi) cos (theta2), cos (psi), sin (psi) sin (theta2)
  643.             // and we can choose to have psi in the interval [0 ; PI]
  644.             final FieldVector3D<T> v1 = applyTo(vector(0, 1, 0));
  645.             final FieldVector3D<T> v2 = applyInverseTo(vector(0, 1, 0));
  646.             if ((v2.getY().getReal() < -0.9999999999) || (v2.getY().getReal() > 0.9999999999)) {
  647.                 throw new CardanEulerSingularityException(false);
  648.             }
  649.             return buildArray(v1.getZ().atan2(v1.getX().negate()),
  650.                               v2.getY().acos(),
  651.                               v2.getZ().atan2(v2.getX()));

  653.         } else if (order == RotationOrder.ZXZ) {

  655.             // r (+K) coordinates are :
  656.             //  sin (psi1) sin (phi), -cos (psi1) sin (phi), cos (phi)
  657.             // (-r) (+K) coordinates are :
  658.             // sin (phi) sin (psi2), sin (phi) cos (psi2), cos (phi)
  659.             // and we can choose to have phi in the interval [0 ; PI]
  660.             final FieldVector3D<T> v1 = applyTo(vector(0, 0, 1));
  661.             final FieldVector3D<T> v2 = applyInverseTo(vector(0, 0, 1));
  662.             if ((v2.getZ().getReal() < -0.9999999999) || (v2.getZ().getReal() > 0.9999999999)) {
  663.                 throw new CardanEulerSingularityException(false);
  664.             }
  665.             return buildArray(v1.getX().atan2(v1.getY().negate()),
  666.                               v2.getZ().acos(),
  667.                               v2.getX().atan2(v2.getY()));

  669.         } else { // last possibility is ZYZ

  671.             // r (+K) coordinates are :
  672.             //  cos (psi1) sin (theta), sin (psi1) sin (theta), cos (theta)
  673.             // (-r) (+K) coordinates are :
  674.             // -sin (theta) cos (psi2), sin (theta) sin (psi2), cos (theta)
  675.             // and we can choose to have theta in the interval [0 ; PI]
  676.             final FieldVector3D<T> v1 = applyTo(vector(0, 0, 1));
  677.             final FieldVector3D<T> v2 = applyInverseTo(vector(0, 0, 1));
  678.             if ((v2.getZ().getReal() < -0.9999999999) || (v2.getZ().getReal() > 0.9999999999)) {
  679.                 throw new CardanEulerSingularityException(false);
  680.             }
  681.             return buildArray(v1.getY().atan2(v1.getX()),
  682.                               v2.getZ().acos(),
  683.                               v2.getY().atan2(v2.getX().negate()));

  685.         }

  687.     }

  689.     /** Create a dimension 3 array.
  690.      * @param a0 first array element
  691.      * @param a1 second array element
  692.      * @param a2 third array element
  693.      * @return new array
  694.      */
  695.     private T[] buildArray(final T a0, final T a1, final T a2) {
  696.         final T[] array = MathArrays.buildArray(a0.getField(), 3);
  697.         array[0] = a0;
  698.         array[1] = a1;
  699.         array[2] = a2;
  700.         return array;
  701.     }

  703.     /** Create a constant vector.
  704.      * @param x abscissa
  705.      * @param y ordinate
  706.      * @param z height
  707.      * @return a constant vector
  708.      */
  709.     private FieldVector3D<T> vector(final double x, final double y, final double z) {
  710.         final T zero = q0.getField().getZero();
  711.         return new FieldVector3D<T>(zero.add(x), zero.add(y), zero.add(z));
  712.     }

  714.     /** Get the 3X3 matrix corresponding to the instance
  715.      * @return the matrix corresponding to the instance
  716.      */
  717.     public T[][] getMatrix() {

  719.         // products
  720.         final T q0q0  = q0.multiply(q0);
  721.         final T q0q1  = q0.multiply(q1);
  722.         final T q0q2  = q0.multiply(q2);
  723.         final T q0q3  = q0.multiply(q3);
  724.         final T q1q1  = q1.multiply(q1);
  725.         final T q1q2  = q1.multiply(q2);
  726.         final T q1q3  = q1.multiply(q3);
  727.         final T q2q2  = q2.multiply(q2);
  728.         final T q2q3  = q2.multiply(q3);
  729.         final T q3q3  = q3.multiply(q3);

  731.         // create the matrix
  732.         final T[][] m = MathArrays.buildArray(q0.getField(), 3, 3);

  734.         m [0][0] = q0q0.add(q1q1).multiply(2).subtract(1);
  735.         m [1][0] = q1q2.subtract(q0q3).multiply(2);
  736.         m [2][0] = q1q3.add(q0q2).multiply(2);

  738.         m [0][1] = q1q2.add(q0q3).multiply(2);
  739.         m [1][1] = q0q0.add(q2q2).multiply(2).subtract(1);
  740.         m [2][1] = q2q3.subtract(q0q1).multiply(2);

  742.         m [0][2] = q1q3.subtract(q0q2).multiply(2);
  743.         m [1][2] = q2q3.add(q0q1).multiply(2);
  744.         m [2][2] = q0q0.add(q3q3).multiply(2).subtract(1);

  746.         return m;

  748.     }

  750.     /** Convert to a constant vector without derivatives.
  751.      * @return a constant vector
  752.      */
  753.     public Rotation toRotation() {
  754.         return new Rotation(q0.getReal(), q1.getReal(), q2.getReal(), q3.getReal(), false);
  755.     }

  757.     /** Apply the rotation to a vector.
  758.      * @param u vector to apply the rotation to
  759.      * @return a new vector which is the image of u by the rotation
  760.      */
  761.     public FieldVector3D<T> applyTo(final FieldVector3D<T> u) {

  763.         final T x = u.getX();
  764.         final T y = u.getY();
  765.         final T z = u.getZ();

  767.         final T s = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z));

  769.         return new FieldVector3D<T>(q0.multiply(x.multiply(q0).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x),
  770.                                     q0.multiply(y.multiply(q0).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y),
  771.                                     q0.multiply(z.multiply(q0).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z));

  773.     }

  775.     /** Apply the rotation to a vector.
  776.      * @param u vector to apply the rotation to
  777.      * @return a new vector which is the image of u by the rotation
  778.      */
  779.     public FieldVector3D<T> applyTo(final Vector3D u) {

  781.         final double x = u.getX();
  782.         final double y = u.getY();
  783.         final double z = u.getZ();

  785.         final T s = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z));

  787.         return new FieldVector3D<T>(q0.multiply(q0.multiply(x).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x),
  788.                                     q0.multiply(q0.multiply(y).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y),
  789.                                     q0.multiply(q0.multiply(z).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z));

  791.     }

  793.     /** Apply the rotation to a vector stored in an array.
  794.      * @param in an array with three items which stores vector to rotate
  795.      * @param out an array with three items to put result to (it can be the same
  796.      * array as in)
  797.      */
  798.     public void applyTo(final T[] in, final T[] out) {

  800.         final T x = in[0];
  801.         final T y = in[1];
  802.         final T z = in[2];

  804.         final T s = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z));

  806.         out[0] = q0.multiply(x.multiply(q0).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x);
  807.         out[1] = q0.multiply(y.multiply(q0).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y);
  808.         out[2] = q0.multiply(z.multiply(q0).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z);

  810.     }

  812.     /** Apply the rotation to a vector stored in an array.
  813.      * @param in an array with three items which stores vector to rotate
  814.      * @param out an array with three items to put result to
  815.      */
  816.     public void applyTo(final double[] in, final T[] out) {

  818.         final double x = in[0];
  819.         final double y = in[1];
  820.         final double z = in[2];

  822.         final T s = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z));

  824.         out[0] = q0.multiply(q0.multiply(x).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x);
  825.         out[1] = q0.multiply(q0.multiply(y).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y);
  826.         out[2] = q0.multiply(q0.multiply(z).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z);

  828.     }

  830.     /** Apply a rotation to a vector.
  831.      * @param r rotation to apply
  832.      * @param u vector to apply the rotation to
  833.      * @param <T> the type of the field elements
  834.      * @return a new vector which is the image of u by the rotation
  835.      */
  836.     public static <T extends RealFieldElement<T>> FieldVector3D<T> applyTo(final Rotation r, final FieldVector3D<T> u) {

  838.         final T x = u.getX();
  839.         final T y = u.getY();
  840.         final T z = u.getZ();

  842.         final T s = x.multiply(r.getQ1()).add(y.multiply(r.getQ2())).add(z.multiply(r.getQ3()));

  844.         return new FieldVector3D<T>(x.multiply(r.getQ0()).subtract(z.multiply(r.getQ2()).subtract(y.multiply(r.getQ3()))).multiply(r.getQ0()).add(s.multiply(r.getQ1())).multiply(2).subtract(x),
  845.                                     y.multiply(r.getQ0()).subtract(x.multiply(r.getQ3()).subtract(z.multiply(r.getQ1()))).multiply(r.getQ0()).add(s.multiply(r.getQ2())).multiply(2).subtract(y),
  846.                                     z.multiply(r.getQ0()).subtract(y.multiply(r.getQ1()).subtract(x.multiply(r.getQ2()))).multiply(r.getQ0()).add(s.multiply(r.getQ3())).multiply(2).subtract(z));

  848.     }

  850.     /** Apply the inverse of the rotation to a vector.
  851.      * @param u vector to apply the inverse of the rotation to
  852.      * @return a new vector which such that u is its image by the rotation
  853.      */
  854.     public FieldVector3D<T> applyInverseTo(final FieldVector3D<T> u) {

  856.         final T x = u.getX();
  857.         final T y = u.getY();
  858.         final T z = u.getZ();

  860.         final T s  = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z));
  861.         final T m0 = q0.negate();

  863.         return new FieldVector3D<T>(m0.multiply(x.multiply(m0).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x),
  864.                                     m0.multiply(y.multiply(m0).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y),
  865.                                     m0.multiply(z.multiply(m0).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z));

  867.     }

  869.     /** Apply the inverse of the rotation to a vector.
  870.      * @param u vector to apply the inverse of the rotation to
  871.      * @return a new vector which such that u is its image by the rotation
  872.      */
  873.     public FieldVector3D<T> applyInverseTo(final Vector3D u) {

  875.         final double x = u.getX();
  876.         final double y = u.getY();
  877.         final double z = u.getZ();

  879.         final T s  = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z));
  880.         final T m0 = q0.negate();

  882.         return new FieldVector3D<T>(m0.multiply(m0.multiply(x).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x),
  883.                                     m0.multiply(m0.multiply(y).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y),
  884.                                     m0.multiply(m0.multiply(z).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z));

  886.     }

  888.     /** Apply the inverse of the rotation to a vector stored in an array.
  889.      * @param in an array with three items which stores vector to rotate
  890.      * @param out an array with three items to put result to (it can be the same
  891.      * array as in)
  892.      */
  893.     public void applyInverseTo(final T[] in, final T[] out) {

  895.         final T x = in[0];
  896.         final T y = in[1];
  897.         final T z = in[2];

  899.         final T s = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z));
  900.         final T m0 = q0.negate();

  902.         out[0] = m0.multiply(x.multiply(m0).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x);
  903.         out[1] = m0.multiply(y.multiply(m0).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y);
  904.         out[2] = m0.multiply(z.multiply(m0).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z);

  906.     }

  908.     /** Apply the inverse of the rotation to a vector stored in an array.
  909.      * @param in an array with three items which stores vector to rotate
  910.      * @param out an array with three items to put result to
  911.      */
  912.     public void applyInverseTo(final double[] in, final T[] out) {

  914.         final double x = in[0];
  915.         final double y = in[1];
  916.         final double z = in[2];

  918.         final T s = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z));
  919.         final T m0 = q0.negate();

  921.         out[0] = m0.multiply(m0.multiply(x).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x);
  922.         out[1] = m0.multiply(m0.multiply(y).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y);
  923.         out[2] = m0.multiply(m0.multiply(z).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z);

  925.     }

  927.     /** Apply the inverse of a rotation to a vector.
  928.      * @param r rotation to apply
  929.      * @param u vector to apply the inverse of the rotation to
  930.      * @param <T> the type of the field elements
  931.      * @return a new vector which such that u is its image by the rotation
  932.      */
  933.     public static <T extends RealFieldElement<T>> FieldVector3D<T> applyInverseTo(final Rotation r, final FieldVector3D<T> u) {

  935.         final T x = u.getX();
  936.         final T y = u.getY();
  937.         final T z = u.getZ();

  939.         final T s  = x.multiply(r.getQ1()).add(y.multiply(r.getQ2())).add(z.multiply(r.getQ3()));
  940.         final double m0 = -r.getQ0();

  942.         return new FieldVector3D<T>(x.multiply(m0).subtract(z.multiply(r.getQ2()).subtract(y.multiply(r.getQ3()))).multiply(m0).add(s.multiply(r.getQ1())).multiply(2).subtract(x),
  943.                                     y.multiply(m0).subtract(x.multiply(r.getQ3()).subtract(z.multiply(r.getQ1()))).multiply(m0).add(s.multiply(r.getQ2())).multiply(2).subtract(y),
  944.                                     z.multiply(m0).subtract(y.multiply(r.getQ1()).subtract(x.multiply(r.getQ2()))).multiply(m0).add(s.multiply(r.getQ3())).multiply(2).subtract(z));

  946.     }

  948.     /** Apply the instance to another rotation.
  949.      * Applying the instance to a rotation is computing the composition
  950.      * in an order compliant with the following rule : let u be any
  951.      * vector and v its image by r (i.e. r.applyTo(u) = v), let w be the image
  952.      * of v by the instance (i.e. applyTo(v) = w), then w = comp.applyTo(u),
  953.      * where comp = applyTo(r).
  954.      * @param r rotation to apply the rotation to
  955.      * @return a new rotation which is the composition of r by the instance
  956.      */
  957.     public FieldRotation<T> applyTo(final FieldRotation<T> r) {
  958.         return new FieldRotation<T>(r.q0.multiply(q0).subtract(r.q1.multiply(q1).add(r.q2.multiply(q2)).add(r.q3.multiply(q3))),
  959.                                     r.q1.multiply(q0).add(r.q0.multiply(q1)).add(r.q2.multiply(q3).subtract(r.q3.multiply(q2))),
  960.                                     r.q2.multiply(q0).add(r.q0.multiply(q2)).add(r.q3.multiply(q1).subtract(r.q1.multiply(q3))),
  961.                                     r.q3.multiply(q0).add(r.q0.multiply(q3)).add(r.q1.multiply(q2).subtract(r.q2.multiply(q1))),
  962.                                     false);
  963.     }

  965.     /** Apply the instance to another rotation.
  966.      * Applying the instance to a rotation is computing the composition
  967.      * in an order compliant with the following rule : let u be any
  968.      * vector and v its image by r (i.e. r.applyTo(u) = v), let w be the image
  969.      * of v by the instance (i.e. applyTo(v) = w), then w = comp.applyTo(u),
  970.      * where comp = applyTo(r).
  971.      * @param r rotation to apply the rotation to
  972.      * @return a new rotation which is the composition of r by the instance
  973.      */
  974.     public FieldRotation<T> applyTo(final Rotation r) {
  975.         return new FieldRotation<T>(q0.multiply(r.getQ0()).subtract(q1.multiply(r.getQ1()).add(q2.multiply(r.getQ2())).add(q3.multiply(r.getQ3()))),
  976.                                     q0.multiply(r.getQ1()).add(q1.multiply(r.getQ0())).add(q3.multiply(r.getQ2()).subtract(q2.multiply(r.getQ3()))),
  977.                                     q0.multiply(r.getQ2()).add(q2.multiply(r.getQ0())).add(q1.multiply(r.getQ3()).subtract(q3.multiply(r.getQ1()))),
  978.                                     q0.multiply(r.getQ3()).add(q3.multiply(r.getQ0())).add(q2.multiply(r.getQ1()).subtract(q1.multiply(r.getQ2()))),
  979.                                     false);
  980.     }

  982.     /** Apply a rotation to another rotation.
  983.      * Applying a rotation to another rotation is computing the composition
  984.      * in an order compliant with the following rule : let u be any
  985.      * vector and v its image by rInner (i.e. rInner.applyTo(u) = v), let w be the image
  986.      * of v by rOuter (i.e. rOuter.applyTo(v) = w), then w = comp.applyTo(u),
  987.      * where comp = applyTo(rOuter, rInner).
  988.      * @param r1 rotation to apply
  989.      * @param rInner rotation to apply the rotation to
  990.      * @param <T> the type of the field elements
  991.      * @return a new rotation which is the composition of r by the instance
  992.      */
  993.     public static <T extends RealFieldElement<T>> FieldRotation<T> applyTo(final Rotation r1, final FieldRotation<T> rInner) {
  994.         return new FieldRotation<T>(rInner.q0.multiply(r1.getQ0()).subtract(rInner.q1.multiply(r1.getQ1()).add(rInner.q2.multiply(r1.getQ2())).add(rInner.q3.multiply(r1.getQ3()))),
  995.                                     rInner.q1.multiply(r1.getQ0()).add(rInner.q0.multiply(r1.getQ1())).add(rInner.q2.multiply(r1.getQ3()).subtract(rInner.q3.multiply(r1.getQ2()))),
  996.                                     rInner.q2.multiply(r1.getQ0()).add(rInner.q0.multiply(r1.getQ2())).add(rInner.q3.multiply(r1.getQ1()).subtract(rInner.q1.multiply(r1.getQ3()))),
  997.                                     rInner.q3.multiply(r1.getQ0()).add(rInner.q0.multiply(r1.getQ3())).add(rInner.q1.multiply(r1.getQ2()).subtract(rInner.q2.multiply(r1.getQ1()))),
  998.                                     false);
  999.     }

  1001.     /** Apply the inverse of the instance to another rotation.
  1002.      * Applying the inverse of the instance to a rotation is computing
  1003.      * the composition in an order compliant with the following rule :
  1004.      * let u be any vector and v its image by r (i.e. r.applyTo(u) = v),
  1005.      * let w be the inverse image of v by the instance
  1006.      * (i.e. applyInverseTo(v) = w), then w = comp.applyTo(u), where
  1007.      * comp = applyInverseTo(r).
  1008.      * @param r rotation to apply the rotation to
  1009.      * @return a new rotation which is the composition of r by the inverse
  1010.      * of the instance
  1011.      */
  1012.     public FieldRotation<T> applyInverseTo(final FieldRotation<T> r) {
  1013.         return new FieldRotation<T>(r.q0.multiply(q0).add(r.q1.multiply(q1).add(r.q2.multiply(q2)).add(r.q3.multiply(q3))).negate(),
  1014.                                     r.q0.multiply(q1).add(r.q2.multiply(q3).subtract(r.q3.multiply(q2))).subtract(r.q1.multiply(q0)),
  1015.                                     r.q0.multiply(q2).add(r.q3.multiply(q1).subtract(r.q1.multiply(q3))).subtract(r.q2.multiply(q0)),
  1016.                                     r.q0.multiply(q3).add(r.q1.multiply(q2).subtract(r.q2.multiply(q1))).subtract(r.q3.multiply(q0)),
  1017.                                     false);
  1018.     }

  1020.     /** Apply the inverse of the instance to another rotation.
  1021.      * Applying the inverse of the instance to a rotation is computing
  1022.      * the composition in an order compliant with the following rule :
  1023.      * let u be any vector and v its image by r (i.e. r.applyTo(u) = v),
  1024.      * let w be the inverse image of v by the instance
  1025.      * (i.e. applyInverseTo(v) = w), then w = comp.applyTo(u), where
  1026.      * comp = applyInverseTo(r).
  1027.      * @param r rotation to apply the rotation to
  1028.      * @return a new rotation which is the composition of r by the inverse
  1029.      * of the instance
  1030.      */
  1031.     public FieldRotation<T> applyInverseTo(final Rotation r) {
  1032.         return new FieldRotation<T>(q0.multiply(r.getQ0()).add(q1.multiply(r.getQ1()).add(q2.multiply(r.getQ2())).add(q3.multiply(r.getQ3()))).negate(),
  1033.                                     q1.multiply(r.getQ0()).add(q3.multiply(r.getQ2()).subtract(q2.multiply(r.getQ3()))).subtract(q0.multiply(r.getQ1())),
  1034.                                     q2.multiply(r.getQ0()).add(q1.multiply(r.getQ3()).subtract(q3.multiply(r.getQ1()))).subtract(q0.multiply(r.getQ2())),
  1035.                                     q3.multiply(r.getQ0()).add(q2.multiply(r.getQ1()).subtract(q1.multiply(r.getQ2()))).subtract(q0.multiply(r.getQ3())),
  1036.                                     false);
  1037.     }

  1039.     /** Apply the inverse of a rotation to another rotation.
  1040.      * Applying the inverse of a rotation to another rotation is computing
  1041.      * the composition in an order compliant with the following rule :
  1042.      * let u be any vector and v its image by rInner (i.e. rInner.applyTo(u) = v),
  1043.      * let w be the inverse image of v by rOuter
  1044.      * (i.e. rOuter.applyInverseTo(v) = w), then w = comp.applyTo(u), where
  1045.      * comp = applyInverseTo(rOuter, rInner).
  1046.      * @param rOuter rotation to apply the rotation to
  1047.      * @param rInner rotation to apply the rotation to
  1048.      * @param <T> the type of the field elements
  1049.      * @return a new rotation which is the composition of r by the inverse
  1050.      * of the instance
  1051.      */
  1052.     public static <T extends RealFieldElement<T>> FieldRotation<T> applyInverseTo(final Rotation rOuter, final FieldRotation<T> rInner) {
  1053.         return new FieldRotation<T>(rInner.q0.multiply(rOuter.getQ0()).add(rInner.q1.multiply(rOuter.getQ1()).add(rInner.q2.multiply(rOuter.getQ2())).add(rInner.q3.multiply(rOuter.getQ3()))).negate(),
  1054.                                     rInner.q0.multiply(rOuter.getQ1()).add(rInner.q2.multiply(rOuter.getQ3()).subtract(rInner.q3.multiply(rOuter.getQ2()))).subtract(rInner.q1.multiply(rOuter.getQ0())),
  1055.                                     rInner.q0.multiply(rOuter.getQ2()).add(rInner.q3.multiply(rOuter.getQ1()).subtract(rInner.q1.multiply(rOuter.getQ3()))).subtract(rInner.q2.multiply(rOuter.getQ0())),
  1056.                                     rInner.q0.multiply(rOuter.getQ3()).add(rInner.q1.multiply(rOuter.getQ2()).subtract(rInner.q2.multiply(rOuter.getQ1()))).subtract(rInner.q3.multiply(rOuter.getQ0())),
  1057.                                     false);
  1058.     }

  1060.     /** Perfect orthogonality on a 3X3 matrix.
  1061.      * @param m initial matrix (not exactly orthogonal)
  1062.      * @param threshold convergence threshold for the iterative
  1063.      * orthogonality correction (convergence is reached when the
  1064.      * difference between two steps of the Frobenius norm of the
  1065.      * correction is below this threshold)
  1066.      * @return an orthogonal matrix close to m
  1067.      * @exception NotARotationMatrixException if the matrix cannot be
  1068.      * orthogonalized with the given threshold after 10 iterations
  1069.      */
  1070.     private T[][] orthogonalizeMatrix(final T[][] m, final double threshold)
  1071.         throws NotARotationMatrixException {

  1073.         T x00 = m[0][0];
  1074.         T x01 = m[0][1];
  1075.         T x02 = m[0][2];
  1076.         T x10 = m[1][0];
  1077.         T x11 = m[1][1];
  1078.         T x12 = m[1][2];
  1079.         T x20 = m[2][0];
  1080.         T x21 = m[2][1];
  1081.         T x22 = m[2][2];
  1082.         double fn = 0;
  1083.         double fn1;

  1085.         final T[][] o = MathArrays.buildArray(m[0][0].getField(), 3, 3);

  1087.         // iterative correction: Xn+1 = Xn - 0.5 * (Xn.Mt.Xn - M)
  1088.         int i = 0;
  1089.         while (++i < 11) {

  1091.             // Mt.Xn
  1092.             final T mx00 = m[0][0].multiply(x00).add(m[1][0].multiply(x10)).add(m[2][0].multiply(x20));
  1093.             final T mx10 = m[0][1].multiply(x00).add(m[1][1].multiply(x10)).add(m[2][1].multiply(x20));
  1094.             final T mx20 = m[0][2].multiply(x00).add(m[1][2].multiply(x10)).add(m[2][2].multiply(x20));
  1095.             final T mx01 = m[0][0].multiply(x01).add(m[1][0].multiply(x11)).add(m[2][0].multiply(x21));
  1096.             final T mx11 = m[0][1].multiply(x01).add(m[1][1].multiply(x11)).add(m[2][1].multiply(x21));
  1097.             final T mx21 = m[0][2].multiply(x01).add(m[1][2].multiply(x11)).add(m[2][2].multiply(x21));
  1098.             final T mx02 = m[0][0].multiply(x02).add(m[1][0].multiply(x12)).add(m[2][0].multiply(x22));
  1099.             final T mx12 = m[0][1].multiply(x02).add(m[1][1].multiply(x12)).add(m[2][1].multiply(x22));
  1100.             final T mx22 = m[0][2].multiply(x02).add(m[1][2].multiply(x12)).add(m[2][2].multiply(x22));

  1102.             // Xn+1
  1103.             o[0][0] = x00.subtract(x00.multiply(mx00).add(x01.multiply(mx10)).add(x02.multiply(mx20)).subtract(m[0][0]).multiply(0.5));
  1104.             o[0][1] = x01.subtract(x00.multiply(mx01).add(x01.multiply(mx11)).add(x02.multiply(mx21)).subtract(m[0][1]).multiply(0.5));
  1105.             o[0][2] = x02.subtract(x00.multiply(mx02).add(x01.multiply(mx12)).add(x02.multiply(mx22)).subtract(m[0][2]).multiply(0.5));
  1106.             o[1][0] = x10.subtract(x10.multiply(mx00).add(x11.multiply(mx10)).add(x12.multiply(mx20)).subtract(m[1][0]).multiply(0.5));
  1107.             o[1][1] = x11.subtract(x10.multiply(mx01).add(x11.multiply(mx11)).add(x12.multiply(mx21)).subtract(m[1][1]).multiply(0.5));
  1108.             o[1][2] = x12.subtract(x10.multiply(mx02).add(x11.multiply(mx12)).add(x12.multiply(mx22)).subtract(m[1][2]).multiply(0.5));
  1109.             o[2][0] = x20.subtract(x20.multiply(mx00).add(x21.multiply(mx10)).add(x22.multiply(mx20)).subtract(m[2][0]).multiply(0.5));
  1110.             o[2][1] = x21.subtract(x20.multiply(mx01).add(x21.multiply(mx11)).add(x22.multiply(mx21)).subtract(m[2][1]).multiply(0.5));
  1111.             o[2][2] = x22.subtract(x20.multiply(mx02).add(x21.multiply(mx12)).add(x22.multiply(mx22)).subtract(m[2][2]).multiply(0.5));

  1113.             // correction on each elements
  1114.             final double corr00 = o[0][0].getReal() - m[0][0].getReal();
  1115.             final double corr01 = o[0][1].getReal() - m[0][1].getReal();
  1116.             final double corr02 = o[0][2].getReal() - m[0][2].getReal();
  1117.             final double corr10 = o[1][0].getReal() - m[1][0].getReal();
  1118.             final double corr11 = o[1][1].getReal() - m[1][1].getReal();
  1119.             final double corr12 = o[1][2].getReal() - m[1][2].getReal();
  1120.             final double corr20 = o[2][0].getReal() - m[2][0].getReal();
  1121.             final double corr21 = o[2][1].getReal() - m[2][1].getReal();
  1122.             final double corr22 = o[2][2].getReal() - m[2][2].getReal();

  1124.             // Frobenius norm of the correction
  1125.             fn1 = corr00 * corr00 + corr01 * corr01 + corr02 * corr02 +
  1126.                   corr10 * corr10 + corr11 * corr11 + corr12 * corr12 +
  1127.                   corr20 * corr20 + corr21 * corr21 + corr22 * corr22;

  1129.             // convergence test
  1130.             if (FastMath.abs(fn1 - fn) <= threshold) {
  1131.                 return o;
  1132.             }

  1134.             // prepare next iteration
  1135.             x00 = o[0][0];
  1136.             x01 = o[0][1];
  1137.             x02 = o[0][2];
  1138.             x10 = o[1][0];
  1139.             x11 = o[1][1];
  1140.             x12 = o[1][2];
  1141.             x20 = o[2][0];
  1142.             x21 = o[2][1];
  1143.             x22 = o[2][2];
  1144.             fn  = fn1;

  1146.         }

  1148.         // the algorithm did not converge after 10 iterations
  1149.         throw new NotARotationMatrixException(LocalizedFormats.UNABLE_TO_ORTHOGONOLIZE_MATRIX,
  1150.                                               i - 1);

  1152.     }

  1154.     /** Compute the <i>distance</i> between two rotations.
  1155.      * <p>The <i>distance</i> is intended here as a way to check if two
  1156.      * rotations are almost similar (i.e. they transform vectors the same way)
  1157.      * or very different. It is mathematically defined as the angle of
  1158.      * the rotation r that prepended to one of the rotations gives the other
  1159.      * one:</p>
  1160.      * <pre>
  1161.      *        r<sub>1</sub>(r) = r<sub>2</sub>
  1162.      * </pre>
  1163.      * <p>This distance is an angle between 0 and &pi;. Its value is the smallest
  1164.      * possible upper bound of the angle in radians between r<sub>1</sub>(v)
  1165.      * and r<sub>2</sub>(v) for all possible vectors v. This upper bound is
  1166.      * reached for some v. The distance is equal to 0 if and only if the two
  1167.      * rotations are identical.</p>
  1168.      * <p>Comparing two rotations should always be done using this value rather
  1169.      * than for example comparing the components of the quaternions. It is much
  1170.      * more stable, and has a geometric meaning. Also comparing quaternions
  1171.      * components is error prone since for example quaternions (0.36, 0.48, -0.48, -0.64)
  1172.      * and (-0.36, -0.48, 0.48, 0.64) represent exactly the same rotation despite
  1173.      * their components are different (they are exact opposites).</p>
  1174.      * @param r1 first rotation
  1175.      * @param r2 second rotation
  1176.      * @param <T> the type of the field elements
  1177.      * @return <i>distance</i> between r1 and r2
  1178.      */
  1179.     public static <T extends RealFieldElement<T>> T distance(final FieldRotation<T> r1, final FieldRotation<T> r2) {
  1180.         return r1.applyInverseTo(r2).getAngle();
  1181.     }

  1183. }
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