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.  *      https://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. /*
  19.  * This is not the original file distributed by the Apache Software Foundation
  20.  * It has been modified by the Hipparchus project
  21.  */

  23. package org.hipparchus.geometry.euclidean.threed;

  25. import java.io.Serializable;

  27. import org.hipparchus.CalculusFieldElement;
  28. import org.hipparchus.Field;
  29. import org.hipparchus.exception.MathIllegalArgumentException;
  30. import org.hipparchus.exception.MathRuntimeException;
  31. import org.hipparchus.geometry.LocalizedGeometryFormats;
  32. import org.hipparchus.util.FastMath;
  33. import org.hipparchus.util.FieldSinCos;
  34. import org.hipparchus.util.MathArrays;

  36. /**
  37.  * This class is a re-implementation of {@link Rotation} using {@link CalculusFieldElement}.
  38.  * <p>Instance of this class are guaranteed to be immutable.</p>
  39.  *
  40.  * @param <T> the type of the field elements
  41.  * @see FieldVector3D
  42.  * @see RotationOrder
  43.  */

  45. public class FieldRotation<T extends CalculusFieldElement<T>> implements Serializable {

  47.     /** Serializable version identifier */
  48.     private static final long serialVersionUID = 20130224L;

  50.     /** Scalar coordinate of the quaternion. */
  51.     private final T q0;

  53.     /** First coordinate of the vectorial part of the quaternion. */
  54.     private final T q1;

  56.     /** Second coordinate of the vectorial part of the quaternion. */
  57.     private final T q2;

  59.     /** Third coordinate of the vectorial part of the quaternion. */
  60.     private final T q3;

  62.     /** Build a rotation from the quaternion coordinates.
  63.      * <p>A rotation can be built from a <em>normalized</em> quaternion,
  64.      * i.e. a quaternion for which q<sub>0</sub><sup>2</sup> +
  65.      * q<sub>1</sub><sup>2</sup> + q<sub>2</sub><sup>2</sup> +
  66.      * q<sub>3</sub><sup>2</sup> = 1. If the quaternion is not normalized,
  67.      * the constructor can normalize it in a preprocessing step.</p>
  68.      * <p>Note that some conventions put the scalar part of the quaternion
  69.      * as the 4<sup>th</sup> component and the vector part as the first three
  70.      * components. This is <em>not</em> our convention. We put the scalar part
  71.      * as the first component.</p>
  72.      * @param q0 scalar part of the quaternion
  73.      * @param q1 first coordinate of the vectorial part of the quaternion
  74.      * @param q2 second coordinate of the vectorial part of the quaternion
  75.      * @param q3 third coordinate of the vectorial part of the quaternion
  76.      * @param needsNormalization if true, the coordinates are considered
  77.      * not to be normalized, a normalization preprocessing step is performed
  78.      * before using them
  79.      */
  80.     public FieldRotation(final T q0, final T q1, final T q2, final T q3, final boolean needsNormalization) {

  82.         if (needsNormalization) {
  83.             // normalization preprocessing
  84.             final T inv =
  85.                     q0.square().add(q1.square()).add(q2.square()).add(q3.square()).sqrt().reciprocal();
  86.             this.q0 = inv.multiply(q0);
  87.             this.q1 = inv.multiply(q1);
  88.             this.q2 = inv.multiply(q2);
  89.             this.q3 = inv.multiply(q3);
  90.         } else {
  91.             this.q0 = q0;
  92.             this.q1 = q1;
  93.             this.q2 = q2;
  94.             this.q3 = q3;
  95.         }

  97.     }

  99.     /** Build a rotation from an axis and an angle.
  100.      * <p>We use the convention that angles are oriented according to
  101.      * the effect of the rotation on vectors around the axis. That means
  102.      * that if (i, j, k) is a direct frame and if we first provide +k as
  103.      * the axis and &pi;/2 as the angle to this constructor, and then
  104.      * {@link #applyTo(FieldVector3D) apply} the instance to +i, we will get
  105.      * +j.</p>
  106.      * <p>Another way to represent our convention is to say that a rotation
  107.      * of angle &theta; about the unit vector (x, y, z) is the same as the
  108.      * rotation build from quaternion components { cos(-&theta;/2),
  109.      * x * sin(-&theta;/2), y * sin(-&theta;/2), z * sin(-&theta;/2) }.
  110.      * Note the minus sign on the angle!</p>
  111.      * <p>On the one hand this convention is consistent with a vectorial
  112.      * perspective (moving vectors in fixed frames), on the other hand it
  113.      * is different from conventions with a frame perspective (fixed vectors
  114.      * viewed from different frames) like the ones used for example in spacecraft
  115.      * attitude community or in the graphics community.</p>
  116.      * @param axis axis around which to rotate
  117.      * @param angle rotation angle.
  118.      * @param convention convention to use for the semantics of the angle
  119.      * @exception MathIllegalArgumentException if the axis norm is zero
  120.      */
  121.     public FieldRotation(final FieldVector3D<T> axis, final T angle, final RotationConvention convention)
  122.         throws MathIllegalArgumentException {

  124.         final T norm = axis.getNorm();
  125.         if (norm.getReal() == 0) {
  126.             throw new MathIllegalArgumentException(LocalizedGeometryFormats.ZERO_NORM_FOR_ROTATION_AXIS);
  127.         }

  129.         final T halfAngle = angle.multiply(convention == RotationConvention.VECTOR_OPERATOR ? -0.5 : 0.5);
  130.         final FieldSinCos<T> sinCos = FastMath.sinCos(halfAngle);
  131.         final T coeff = sinCos.sin().divide(norm);

  133.         q0 = sinCos.cos();
  134.         q1 = coeff.multiply(axis.getX());
  135.         q2 = coeff.multiply(axis.getY());
  136.         q3 = coeff.multiply(axis.getZ());

  138.     }

  140.     /** Build a {@link FieldRotation} from a {@link Rotation}.
  141.      * @param field field for the components
  142.      * @param r rotation to convert
  143.      */
  144.     public FieldRotation(final Field<T> field, final Rotation r) {
  145.         this.q0 = field.getZero().add(r.getQ0());
  146.         this.q1 = field.getZero().add(r.getQ1());
  147.         this.q2 = field.getZero().add(r.getQ2());
  148.         this.q3 = field.getZero().add(r.getQ3());
  149.     }

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

  153.      * <p>Rotation matrices are orthogonal matrices, i.e. unit matrices
  154.      * (which are matrices for which m.m<sup>T</sup> = I) with real
  155.      * coefficients. The module of the determinant of unit matrices is
  156.      * 1, among the orthogonal 3X3 matrices, only the ones having a
  157.      * positive determinant (+1) are rotation matrices.</p>

  159.      * <p>When a rotation is defined by a matrix with truncated values
  160.      * (typically when it is extracted from a technical sheet where only
  161.      * four to five significant digits are available), the matrix is not
  162.      * orthogonal anymore. This constructor handles this case
  163.      * transparently by using a copy of the given matrix and applying a
  164.      * correction to the copy in order to perfect its orthogonality. If
  165.      * the Frobenius norm of the correction needed is above the given
  166.      * threshold, then the matrix is considered to be too far from a
  167.      * true rotation matrix and an exception is thrown.</p>

  169.      * @param m rotation matrix
  170.      * @param threshold convergence threshold for the iterative
  171.      * orthogonality correction (convergence is reached when the
  172.      * difference between two steps of the Frobenius norm of the
  173.      * correction is below this threshold)

  175.      * @exception MathIllegalArgumentException if the matrix is not a 3X3
  176.      * matrix, or if it cannot be transformed into an orthogonal matrix
  177.      * with the given threshold, or if the determinant of the resulting
  178.      * orthogonal matrix is negative

  180.      */
  181.     public FieldRotation(final T[][] m, final double threshold)
  182.         throws MathIllegalArgumentException {

  184.         // dimension check
  185.         if ((m.length != 3) || (m[0].length != 3) ||
  186.                 (m[1].length != 3) || (m[2].length != 3)) {
  187.             throw new MathIllegalArgumentException(LocalizedGeometryFormats.ROTATION_MATRIX_DIMENSIONS,
  188.                                                    m.length, m[0].length);
  189.         }

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

  194.         // check the sign of the determinant
  195.         final T d0 = ort[1][1].multiply(ort[2][2]).subtract(ort[2][1].multiply(ort[1][2]));
  196.         final T d1 = ort[0][1].multiply(ort[2][2]).subtract(ort[2][1].multiply(ort[0][2]));
  197.         final T d2 = ort[0][1].multiply(ort[1][2]).subtract(ort[1][1].multiply(ort[0][2]));
  198.         final T det =
  199.                 ort[0][0].multiply(d0).subtract(ort[1][0].multiply(d1)).add(ort[2][0].multiply(d2));
  200.         if (det.getReal() < 0.0) {
  201.             throw new MathIllegalArgumentException(LocalizedGeometryFormats.CLOSEST_ORTHOGONAL_MATRIX_HAS_NEGATIVE_DETERMINANT,
  202.                                                    det);
  203.         }

  205.         final T[] quat = mat2quat(ort);
  206.         q0 = quat[0];
  207.         q1 = quat[1];
  208.         q2 = quat[2];
  209.         q3 = quat[3];

  211.     }

  213.     /** Build the rotation that transforms a pair of vectors into another pair.

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

  219.      * <p>If the angular separation between u<sub>1</sub> and u<sub>2</sub> is
  220.      * not the same as the angular separation between v<sub>1</sub> and
  221.      * v<sub>2</sub>, then a corrected v'<sub>2</sub> will be used rather than
  222.      * v<sub>2</sub>, the corrected vector will be in the (±v<sub>1</sub>,
  223.      * +v<sub>2</sub>) half-plane.</p>
  224.      * @param u1 first vector of the origin pair
  225.      * @param u2 second vector of the origin pair
  226.      * @param v1 desired image of u1 by the rotation
  227.      * @param v2 desired image of u2 by the rotation
  228.      * @exception MathRuntimeException if the norm of one of the vectors is zero,
  229.      * or if one of the pair is degenerated (i.e. the vectors of the pair are collinear)
  230.      */
  231.     public FieldRotation(FieldVector3D<T> u1, FieldVector3D<T> u2, FieldVector3D<T> v1, FieldVector3D<T> v2)
  232.         throws MathRuntimeException {

  234.         // build orthonormalized base from u1, u2
  235.         // this fails when vectors are null or collinear, which is forbidden to define a rotation
  236.         final FieldVector3D<T> u3 = FieldVector3D.crossProduct(u1, u2).normalize();
  237.         u2 = FieldVector3D.crossProduct(u3, u1).normalize();
  238.         u1 = u1.normalize();

  240.         // build an orthonormalized base from v1, v2
  241.         // this fails when vectors are null or collinear, which is forbidden to define a rotation
  242.         final FieldVector3D<T> v3 = FieldVector3D.crossProduct(v1, v2).normalize();
  243.         v2 = FieldVector3D.crossProduct(v3, v1).normalize();
  244.         v1 = v1.normalize();

  246.         // buid a matrix transforming the first base into the second one
  247.         final T[][] array = MathArrays.buildArray(u1.getX().getField(), 3, 3);
  248.         array[0][0] = u1.getX().multiply(v1.getX()).add(u2.getX().multiply(v2.getX())).add(u3.getX().multiply(v3.getX()));
  249.         array[0][1] = u1.getY().multiply(v1.getX()).add(u2.getY().multiply(v2.getX())).add(u3.getY().multiply(v3.getX()));
  250.         array[0][2] = u1.getZ().multiply(v1.getX()).add(u2.getZ().multiply(v2.getX())).add(u3.getZ().multiply(v3.getX()));
  251.         array[1][0] = u1.getX().multiply(v1.getY()).add(u2.getX().multiply(v2.getY())).add(u3.getX().multiply(v3.getY()));
  252.         array[1][1] = u1.getY().multiply(v1.getY()).add(u2.getY().multiply(v2.getY())).add(u3.getY().multiply(v3.getY()));
  253.         array[1][2] = u1.getZ().multiply(v1.getY()).add(u2.getZ().multiply(v2.getY())).add(u3.getZ().multiply(v3.getY()));
  254.         array[2][0] = u1.getX().multiply(v1.getZ()).add(u2.getX().multiply(v2.getZ())).add(u3.getX().multiply(v3.getZ()));
  255.         array[2][1] = u1.getY().multiply(v1.getZ()).add(u2.getY().multiply(v2.getZ())).add(u3.getY().multiply(v3.getZ()));
  256.         array[2][2] = u1.getZ().multiply(v1.getZ()).add(u2.getZ().multiply(v2.getZ())).add(u3.getZ().multiply(v3.getZ()));

  258.         T[] quat = mat2quat(array);
  259.         q0 = quat[0];
  260.         q1 = quat[1];
  261.         q2 = quat[2];
  262.         q3 = quat[3];

  264.     }

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

  268.      * <p>Except for a possible scale factor, if the instance were
  269.      * applied to the vector u it will produce the vector v. There is an
  270.      * infinite number of such rotations, this constructor choose the
  271.      * one with the smallest associated angle (i.e. the one whose axis
  272.      * is orthogonal to the (u, v) plane). If u and v are collinear, an
  273.      * arbitrary rotation axis is chosen.</p>

  275.      * @param u origin vector
  276.      * @param v desired image of u by the rotation
  277.      * @exception MathRuntimeException if the norm of one of the vectors is zero
  278.      */
  279.     public FieldRotation(final FieldVector3D<T> u, final FieldVector3D<T> v) throws MathRuntimeException {

  281.         final T normProduct = u.getNorm().multiply(v.getNorm());
  282.         if (normProduct.getReal() == 0) {
  283.             throw new MathRuntimeException(LocalizedGeometryFormats.ZERO_NORM_FOR_ROTATION_DEFINING_VECTOR);
  284.         }

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

  288.         if (dot.getReal() < ((2.0e-15 - 1.0) * normProduct.getReal())) {
  289.             // special case u = -v: we select a PI angle rotation around
  290.             // an arbitrary vector orthogonal to u
  291.             final FieldVector3D<T> w = u.orthogonal();
  292.             q0 = normProduct.getField().getZero();
  293.             q1 = w.getX().negate();
  294.             q2 = w.getY().negate();
  295.             q3 = w.getZ().negate();
  296.         } else {
  297.             // general case: (u, v) defines a plane, we select
  298.             // the shortest possible rotation: axis orthogonal to this plane
  299.             q0 = dot.divide(normProduct).add(1.0).multiply(0.5).sqrt();
  300.             final T coeff = q0.multiply(normProduct).multiply(2.0).reciprocal();
  301.             final FieldVector3D<T> q = FieldVector3D.crossProduct(v, u);
  302.             q1 = coeff.multiply(q.getX());
  303.             q2 = coeff.multiply(q.getY());
  304.             q3 = coeff.multiply(q.getZ());
  305.         }

  307.     }

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

  311.      * <p>Cardan rotations are three successive rotations around the
  312.      * canonical axes X, Y and Z, each axis being used once. There are
  313.      * 6 such sets of rotations (XYZ, XZY, YXZ, YZX, ZXY and ZYX). Euler
  314.      * rotations are three successive rotations around the canonical
  315.      * axes X, Y and Z, the first and last rotations being around the
  316.      * same axis. There are 6 such sets of rotations (XYX, XZX, YXY,
  317.      * YZY, ZXZ and ZYZ), the most popular one being ZXZ.</p>
  318.      * <p>Beware that many people routinely use the term Euler angles even
  319.      * for what really are Cardan angles (this confusion is especially
  320.      * widespread in the aerospace business where Roll, Pitch and Yaw angles
  321.      * are often wrongly tagged as Euler angles).</p>

  323.      * @param order order of rotations to compose, from left to right
  324.      * (i.e. we will use {@code r1.compose(r2.compose(r3, convention), convention)})
  325.      * @param convention convention to use for the semantics of the angle
  326.      * @param alpha1 angle of the first elementary rotation
  327.      * @param alpha2 angle of the second elementary rotation
  328.      * @param alpha3 angle of the third elementary rotation
  329.      */
  330.     public FieldRotation(final RotationOrder order, final RotationConvention convention,
  331.                          final T alpha1, final T alpha2, final T alpha3) {
  332.         final Field<T> field = alpha1.getField();
  333.         final FieldRotation<T> r1 = new FieldRotation<>(new FieldVector3D<>(field, order.getA1()), alpha1, convention);
  334.         final FieldRotation<T> r2 = new FieldRotation<>(new FieldVector3D<>(field, order.getA2()), alpha2, convention);
  335.         final FieldRotation<T> r3 = new FieldRotation<>(new FieldVector3D<>(field, order.getA3()), alpha3, convention);
  336.         final FieldRotation<T> composed = r1.compose(r2.compose(r3, convention), convention);
  337.         q0 = composed.q0;
  338.         q1 = composed.q1;
  339.         q2 = composed.q2;
  340.         q3 = composed.q3;
  341.     }

  343.     /** Get identity rotation.
  344.      * @param field field for the components
  345.      * @return a new rotation
  346.      * @param <T> the type of the field elements
  347.      */
  348.     public static <T extends CalculusFieldElement<T>> FieldRotation<T> getIdentity(final Field<T> field) {
  349.         return new FieldRotation<>(field, Rotation.IDENTITY);
  350.     }

  352.     /** Convert an orthogonal rotation matrix to a quaternion.
  353.      * @param ort orthogonal rotation matrix
  354.      * @return quaternion corresponding to the matrix
  355.      */
  356.     private T[] mat2quat(final T[][] ort) {

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

  360.         // There are different ways to compute the quaternions elements
  361.         // from the matrix. They all involve computing one element from
  362.         // the diagonal of the matrix, and computing the three other ones
  363.         // using a formula involving a division by the first element,
  364.         // which unfortunately can be zero. Since the norm of the
  365.         // quaternion is 1, we know at least one element has an absolute
  366.         // value greater or equal to 0.5, so it is always possible to
  367.         // select the right formula and avoid division by zero and even
  368.         // numerical inaccuracy. Checking the elements in turn and using
  369.         // the first one greater than 0.45 is safe (this leads to a simple
  370.         // test since qi = 0.45 implies 4 qi^2 - 1 = -0.19)
  371.         T s = ort[0][0].add(ort[1][1]).add(ort[2][2]);
  372.         if (s.getReal() > -0.19) {
  373.             // compute q0 and deduce q1, q2 and q3
  374.             quat[0] = s.add(1.0).sqrt().multiply(0.5);
  375.             T inv = quat[0].reciprocal().multiply(0.25);
  376.             quat[1] = inv.multiply(ort[1][2].subtract(ort[2][1]));
  377.             quat[2] = inv.multiply(ort[2][0].subtract(ort[0][2]));
  378.             quat[3] = inv.multiply(ort[0][1].subtract(ort[1][0]));
  379.         } else {
  380.             s = ort[0][0].subtract(ort[1][1]).subtract(ort[2][2]);
  381.             if (s.getReal() > -0.19) {
  382.                 // compute q1 and deduce q0, q2 and q3
  383.                 quat[1] = s.add(1.0).sqrt().multiply(0.5);
  384.                 T inv = quat[1].reciprocal().multiply(0.25);
  385.                 quat[0] = inv.multiply(ort[1][2].subtract(ort[2][1]));
  386.                 quat[2] = inv.multiply(ort[0][1].add(ort[1][0]));
  387.                 quat[3] = inv.multiply(ort[0][2].add(ort[2][0]));
  388.             } else {
  389.                 s = ort[1][1].subtract(ort[0][0]).subtract(ort[2][2]);
  390.                 if (s.getReal() > -0.19) {
  391.                     // compute q2 and deduce q0, q1 and q3
  392.                     quat[2] = s.add(1.0).sqrt().multiply(0.5);
  393.                     T inv = quat[2].reciprocal().multiply(0.25);
  394.                     quat[0] = inv.multiply(ort[2][0].subtract(ort[0][2]));
  395.                     quat[1] = inv.multiply(ort[0][1].add(ort[1][0]));
  396.                     quat[3] = inv.multiply(ort[2][1].add(ort[1][2]));
  397.                 } else {
  398.                     // compute q3 and deduce q0, q1 and q2
  399.                     s = ort[2][2].subtract(ort[0][0]).subtract(ort[1][1]);
  400.                     quat[3] = s.add(1.0).sqrt().multiply(0.5);
  401.                     T inv = quat[3].reciprocal().multiply(0.25);
  402.                     quat[0] = inv.multiply(ort[0][1].subtract(ort[1][0]));
  403.                     quat[1] = inv.multiply(ort[0][2].add(ort[2][0]));
  404.                     quat[2] = inv.multiply(ort[2][1].add(ort[1][2]));
  405.                 }
  406.             }
  407.         }

  409.         return quat;

  411.     }

  413.     /** Revert a rotation.
  414.      * Build a rotation which reverse the effect of another
  415.      * rotation. This means that if r(u) = v, then r.revert(v) = u. The
  416.      * instance is not changed.
  417.      * @return a new rotation whose effect is the reverse of the effect
  418.      * of the instance
  419.      */
  420.     public FieldRotation<T> revert() {
  421.         return new FieldRotation<>(q0.negate(), q1, q2, q3, false);
  422.     }

  424.     /** Get the scalar coordinate of the quaternion.
  425.      * @return scalar coordinate of the quaternion
  426.      */
  427.     public T getQ0() {
  428.         return q0;
  429.     }

  431.     /** Get the first coordinate of the vectorial part of the quaternion.
  432.      * @return first coordinate of the vectorial part of the quaternion
  433.      */
  434.     public T getQ1() {
  435.         return q1;
  436.     }

  438.     /** Get the second coordinate of the vectorial part of the quaternion.
  439.      * @return second coordinate of the vectorial part of the quaternion
  440.      */
  441.     public T getQ2() {
  442.         return q2;
  443.     }

  445.     /** Get the third coordinate of the vectorial part of the quaternion.
  446.      * @return third coordinate of the vectorial part of the quaternion
  447.      */
  448.     public T getQ3() {
  449.         return q3;
  450.     }

  452.     /** Get the normalized axis of the rotation.
  453.      * <p>
  454.      * Note that as {@link #getAngle()} always returns an angle
  455.      * between 0 and &pi;, changing the convention changes the
  456.      * direction of the axis, not the sign of the angle.
  457.      * </p>
  458.      * @param convention convention to use for the semantics of the angle
  459.      * @return normalized axis of the rotation
  460.      * @see #FieldRotation(FieldVector3D, CalculusFieldElement, RotationConvention)
  461.      */
  462.     public FieldVector3D<T> getAxis(final RotationConvention convention) {
  463.         final T squaredSine = q1.square().add(q2.square()).add(q3.square());
  464.         if (squaredSine.getReal() == 0) {
  465.             final Field<T> field = squaredSine.getField();
  466.             return new FieldVector3D<>(convention == RotationConvention.VECTOR_OPERATOR ? field.getOne(): field.getOne().negate(),
  467.                                        field.getZero(),
  468.                                        field.getZero());
  469.         } else {
  470.             final double sgn = convention == RotationConvention.VECTOR_OPERATOR ? +1 : -1;
  471.             if (q0.getReal() < 0) {
  472.                 T inverse = squaredSine.sqrt().reciprocal().multiply(sgn);
  473.                 return new FieldVector3D<>(q1.multiply(inverse), q2.multiply(inverse), q3.multiply(inverse));
  474.             }
  475.             final T inverse = squaredSine.sqrt().reciprocal().negate().multiply(sgn);
  476.             return new FieldVector3D<>(q1.multiply(inverse), q2.multiply(inverse), q3.multiply(inverse));
  477.         }
  478.     }

  480.     /** Get the angle of the rotation.
  481.      * @return angle of the rotation (between 0 and &pi;)
  482.      * @see #FieldRotation(FieldVector3D, CalculusFieldElement, RotationConvention)
  483.      */
  484.     public T getAngle() {
  485.         if ((q0.getReal() < -0.1) || (q0.getReal() > 0.1)) {
  486.             return q1.square().add(q2.square()).add(q3.square()).sqrt().asin().multiply(2);
  487.         } else if (q0.getReal() < 0) {
  488.             return q0.negate().acos().multiply(2);
  489.         }
  490.         return q0.acos().multiply(2);
  491.     }

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

  495.      * <p>The equations show that each rotation can be defined by two
  496.      * different values of the Cardan or Euler angles set. For example
  497.      * if Cardan angles are used, the rotation defined by the angles
  498.      * a<sub>1</sub>, a<sub>2</sub> and a<sub>3</sub> is the same as
  499.      * the rotation defined by the angles &pi; + a<sub>1</sub>, &pi;
  500.      * - a<sub>2</sub> and &pi; + a<sub>3</sub>. This method implements
  501.      * the following arbitrary choices:</p>
  502.      * <ul>
  503.      *   <li>for Cardan angles, the chosen set is the one for which the
  504.      *   second angle is between -&pi;/2 and &pi;/2 (i.e its cosine is
  505.      *   positive),</li>
  506.      *   <li>for Euler angles, the chosen set is the one for which the
  507.      *   second angle is between 0 and &pi; (i.e its sine is positive).</li>
  508.      * </ul>

  510.      * <p>
  511.      * The algorithm used here works even when the rotation is exactly at the
  512.      * the singularity of the rotation order and convention. In this case, one of
  513.      * the angles in the singular pair is arbitrarily set to exactly 0 and the
  514.      * second angle is computed. The angle set to 0 in the singular case is the
  515.      * angle of the first rotation in the case of Cardan orders, and it is the angle
  516.      * of the last rotation in the case of Euler orders. This implies that extracting
  517.      * the angles of a rotation never fails (it used to trigger an exception in singular
  518.      * cases up to Hipparchus 3.0).
  519.      * </p>

  521.      * @param order rotation order to use
  522.      * @param convention convention to use for the semantics of the angle
  523.      * @return an array of three angles, in the order specified by the set
  524.      */
  525.     public T[] getAngles(final RotationOrder order, RotationConvention convention) {
  526.         return order.getAngles(this, convention);
  527.     }

  529.     /** Get the 3X3 matrix corresponding to the instance
  530.      * @return the matrix corresponding to the instance
  531.      */
  532.     public T[][] getMatrix() {

  534.         // products
  535.         final T q0q0  = q0.square();
  536.         final T q0q1  = q0.multiply(q1);
  537.         final T q0q2  = q0.multiply(q2);
  538.         final T q0q3  = q0.multiply(q3);
  539.         final T q1q1  = q1.square();
  540.         final T q1q2  = q1.multiply(q2);
  541.         final T q1q3  = q1.multiply(q3);
  542.         final T q2q2  = q2.square();
  543.         final T q2q3  = q2.multiply(q3);
  544.         final T q3q3  = q3.square();

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

  549.         m [0][0] = q0q0.add(q1q1).multiply(2).subtract(1);
  550.         m [1][0] = q1q2.subtract(q0q3).multiply(2);
  551.         m [2][0] = q1q3.add(q0q2).multiply(2);

  553.         m [0][1] = q1q2.add(q0q3).multiply(2);
  554.         m [1][1] = q0q0.add(q2q2).multiply(2).subtract(1);
  555.         m [2][1] = q2q3.subtract(q0q1).multiply(2);

  557.         m [0][2] = q1q3.subtract(q0q2).multiply(2);
  558.         m [1][2] = q2q3.add(q0q1).multiply(2);
  559.         m [2][2] = q0q0.add(q3q3).multiply(2).subtract(1);

  561.         return m;

  563.     }

  565.     /** Convert to a constant vector without derivatives.
  566.      * @return a constant vector
  567.      */
  568.     public Rotation toRotation() {
  569.         return new Rotation(q0.getReal(), q1.getReal(), q2.getReal(), q3.getReal(), false);
  570.     }

  572.     /** Apply the rotation to a vector.
  573.      * @param u vector to apply the rotation to
  574.      * @return a new vector which is the image of u by the rotation
  575.      */
  576.     public FieldVector3D<T> applyTo(final FieldVector3D<T> u) {

  578.         final T x = u.getX();
  579.         final T y = u.getY();
  580.         final T z = u.getZ();

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

  584.         return new FieldVector3D<>(q0.multiply(x.multiply(q0).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x),
  585.                                    q0.multiply(y.multiply(q0).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y),
  586.                                    q0.multiply(z.multiply(q0).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z));

  588.     }

  590.     /** Apply the rotation to a vector.
  591.      * @param u vector to apply the rotation to
  592.      * @return a new vector which is the image of u by the rotation
  593.      */
  594.     public FieldVector3D<T> applyTo(final Vector3D u) {

  596.         final double x = u.getX();
  597.         final double y = u.getY();
  598.         final double z = u.getZ();

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

  602.         return new FieldVector3D<>(q0.multiply(q0.multiply(x).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x),
  603.                                    q0.multiply(q0.multiply(y).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y),
  604.                                    q0.multiply(q0.multiply(z).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z));

  606.     }

  608.     /** Apply the rotation to a vector stored in an array.
  609.      * @param in an array with three items which stores vector to rotate
  610.      * @param out an array with three items to put result to (it can be the same
  611.      * array as in)
  612.      */
  613.     public void applyTo(final T[] in, final T[] out) {

  615.         final T x = in[0];
  616.         final T y = in[1];
  617.         final T z = in[2];

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

  621.         out[0] = q0.multiply(x.multiply(q0).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x);
  622.         out[1] = q0.multiply(y.multiply(q0).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y);
  623.         out[2] = q0.multiply(z.multiply(q0).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z);

  625.     }

  627.     /** Apply the rotation to a vector stored in an array.
  628.      * @param in an array with three items which stores vector to rotate
  629.      * @param out an array with three items to put result to
  630.      */
  631.     public void applyTo(final double[] in, final T[] out) {

  633.         final double x = in[0];
  634.         final double y = in[1];
  635.         final double z = in[2];

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

  639.         out[0] = q0.multiply(q0.multiply(x).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x);
  640.         out[1] = q0.multiply(q0.multiply(y).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y);
  641.         out[2] = q0.multiply(q0.multiply(z).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z);

  643.     }

  645.     /** Apply a rotation to a vector.
  646.      * @param r rotation to apply
  647.      * @param u vector to apply the rotation to
  648.      * @param <T> the type of the field elements
  649.      * @return a new vector which is the image of u by the rotation
  650.      */
  651.     public static <T extends CalculusFieldElement<T>> FieldVector3D<T> applyTo(final Rotation r, final FieldVector3D<T> u) {

  653.         final T x = u.getX();
  654.         final T y = u.getY();
  655.         final T z = u.getZ();

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

  659.         return new FieldVector3D<>(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),
  660.                                    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),
  661.                                    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));

  663.     }

  665.     /** Apply the inverse of the rotation to a vector.
  666.      * @param u vector to apply the inverse of the rotation to
  667.      * @return a new vector which such that u is its image by the rotation
  668.      */
  669.     public FieldVector3D<T> applyInverseTo(final FieldVector3D<T> u) {

  671.         final T x = u.getX();
  672.         final T y = u.getY();
  673.         final T z = u.getZ();

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

  678.         return new FieldVector3D<>(m0.multiply(x.multiply(m0).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x),
  679.                                    m0.multiply(y.multiply(m0).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y),
  680.                                    m0.multiply(z.multiply(m0).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z));

  682.     }

  684.     /** Apply the inverse of the rotation to a vector.
  685.      * @param u vector to apply the inverse of the rotation to
  686.      * @return a new vector which such that u is its image by the rotation
  687.      */
  688.     public FieldVector3D<T> applyInverseTo(final Vector3D u) {

  690.         final double x = u.getX();
  691.         final double y = u.getY();
  692.         final double z = u.getZ();

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

  697.         return new FieldVector3D<>(m0.multiply(m0.multiply(x).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x),
  698.                                    m0.multiply(m0.multiply(y).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y),
  699.                                    m0.multiply(m0.multiply(z).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z));

  701.     }

  703.     /** Apply the inverse of the rotation to a vector stored in an array.
  704.      * @param in an array with three items which stores vector to rotate
  705.      * @param out an array with three items to put result to (it can be the same
  706.      * array as in)
  707.      */
  708.     public void applyInverseTo(final T[] in, final T[] out) {

  710.         final T x = in[0];
  711.         final T y = in[1];
  712.         final T z = in[2];

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

  717.         out[0] = m0.multiply(x.multiply(m0).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x);
  718.         out[1] = m0.multiply(y.multiply(m0).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y);
  719.         out[2] = m0.multiply(z.multiply(m0).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z);

  721.     }

  723.     /** Apply the inverse of the rotation to a vector stored in an array.
  724.      * @param in an array with three items which stores vector to rotate
  725.      * @param out an array with three items to put result to
  726.      */
  727.     public void applyInverseTo(final double[] in, final T[] out) {

  729.         final double x = in[0];
  730.         final double y = in[1];
  731.         final double z = in[2];

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

  736.         out[0] = m0.multiply(m0.multiply(x).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x);
  737.         out[1] = m0.multiply(m0.multiply(y).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y);
  738.         out[2] = m0.multiply(m0.multiply(z).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z);

  740.     }

  742.     /** Apply the inverse of a rotation to a vector.
  743.      * @param r rotation to apply
  744.      * @param u vector to apply the inverse of the rotation to
  745.      * @param <T> the type of the field elements
  746.      * @return a new vector which such that u is its image by the rotation
  747.      */
  748.     public static <T extends CalculusFieldElement<T>> FieldVector3D<T> applyInverseTo(final Rotation r, final FieldVector3D<T> u) {

  750.         final T x = u.getX();
  751.         final T y = u.getY();
  752.         final T z = u.getZ();

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

  757.         return new FieldVector3D<>(x.multiply(m0).subtract(z.multiply(r.getQ2()).subtract(y.multiply(r.getQ3()))).multiply(m0).add(s.multiply(r.getQ1())).multiply(2).subtract(x),
  758.                                    y.multiply(m0).subtract(x.multiply(r.getQ3()).subtract(z.multiply(r.getQ1()))).multiply(m0).add(s.multiply(r.getQ2())).multiply(2).subtract(y),
  759.                                    z.multiply(m0).subtract(y.multiply(r.getQ1()).subtract(x.multiply(r.getQ2()))).multiply(m0).add(s.multiply(r.getQ3())).multiply(2).subtract(z));

  761.     }

  763.     /** Apply the instance to another rotation.
  764.      * <p>
  765.      * Calling this method is equivalent to call
  766.      * {@link #compose(FieldRotation, RotationConvention)
  767.      * compose(r, RotationConvention.VECTOR_OPERATOR)}.
  768.      * </p>
  769.      * @param r rotation to apply the rotation to
  770.      * @return a new rotation which is the composition of r by the instance
  771.      */
  772.     public FieldRotation<T> applyTo(final FieldRotation<T> r) {
  773.         return compose(r, RotationConvention.VECTOR_OPERATOR);
  774.     }

  776.     /** Compose the instance with another rotation.
  777.      * <p>
  778.      * If the semantics of the rotations composition corresponds to a
  779.      * {@link RotationConvention#VECTOR_OPERATOR vector operator} convention,
  780.      * applying the instance to a rotation is computing the composition
  781.      * in an order compliant with the following rule : let {@code u} be any
  782.      * vector and {@code v} its image by {@code r1} (i.e.
  783.      * {@code r1.applyTo(u) = v}). Let {@code w} be the image of {@code v} by
  784.      * rotation {@code r2} (i.e. {@code r2.applyTo(v) = w}). Then
  785.      * {@code w = comp.applyTo(u)}, where
  786.      * {@code comp = r2.compose(r1, RotationConvention.VECTOR_OPERATOR)}.
  787.      * </p>
  788.      * <p>
  789.      * If the semantics of the rotations composition corresponds to a
  790.      * {@link RotationConvention#FRAME_TRANSFORM frame transform} convention,
  791.      * the application order will be reversed. So keeping the exact same
  792.      * meaning of all {@code r1}, {@code r2}, {@code u}, {@code v}, {@code w}
  793.      * and  {@code comp} as above, {@code comp} could also be computed as
  794.      * {@code comp = r1.compose(r2, RotationConvention.FRAME_TRANSFORM)}.
  795.      * </p>
  796.      * @param r rotation to apply the rotation to
  797.      * @param convention convention to use for the semantics of the angle
  798.      * @return a new rotation which is the composition of r by the instance
  799.      */
  800.     public FieldRotation<T> compose(final FieldRotation<T> r, final RotationConvention convention) {
  801.         return convention == RotationConvention.VECTOR_OPERATOR ?
  802.                              composeInternal(r) : r.composeInternal(this);
  803.     }

  805.     /** Compose the instance with another rotation using vector operator convention.
  806.      * @param r rotation to apply the rotation to
  807.      * @return a new rotation which is the composition of r by the instance
  808.      * using vector operator convention
  809.      */
  810.     private FieldRotation<T> composeInternal(final FieldRotation<T> r) {
  811.         return new FieldRotation<>(r.q0.multiply(q0).subtract(r.q1.multiply(q1).add(r.q2.multiply(q2)).add(r.q3.multiply(q3))),
  812.                                    r.q1.multiply(q0).add(r.q0.multiply(q1)).add(r.q2.multiply(q3).subtract(r.q3.multiply(q2))),
  813.                                    r.q2.multiply(q0).add(r.q0.multiply(q2)).add(r.q3.multiply(q1).subtract(r.q1.multiply(q3))),
  814.                                    r.q3.multiply(q0).add(r.q0.multiply(q3)).add(r.q1.multiply(q2).subtract(r.q2.multiply(q1))),
  815.                                    false);
  816.     }

  818.     /** Apply the instance to another rotation.
  819.      * <p>
  820.      * Calling this method is equivalent to call
  821.      * {@link #compose(Rotation, RotationConvention)
  822.      * compose(r, RotationConvention.VECTOR_OPERATOR)}.
  823.      * </p>
  824.      * @param r rotation to apply the rotation to
  825.      * @return a new rotation which is the composition of r by the instance
  826.      */
  827.     public FieldRotation<T> applyTo(final Rotation r) {
  828.         return compose(r, RotationConvention.VECTOR_OPERATOR);
  829.     }

  831.     /** Compose the instance with another rotation.
  832.      * <p>
  833.      * If the semantics of the rotations composition corresponds to a
  834.      * {@link RotationConvention#VECTOR_OPERATOR vector operator} convention,
  835.      * applying the instance to a rotation is computing the composition
  836.      * in an order compliant with the following rule : let {@code u} be any
  837.      * vector and {@code v} its image by {@code r1} (i.e.
  838.      * {@code r1.applyTo(u) = v}). Let {@code w} be the image of {@code v} by
  839.      * rotation {@code r2} (i.e. {@code r2.applyTo(v) = w}). Then
  840.      * {@code w = comp.applyTo(u)}, where
  841.      * {@code comp = r2.compose(r1, RotationConvention.VECTOR_OPERATOR)}.
  842.      * </p>
  843.      * <p>
  844.      * If the semantics of the rotations composition corresponds to a
  845.      * {@link RotationConvention#FRAME_TRANSFORM frame transform} convention,
  846.      * the application order will be reversed. So keeping the exact same
  847.      * meaning of all {@code r1}, {@code r2}, {@code u}, {@code v}, {@code w}
  848.      * and  {@code comp} as above, {@code comp} could also be computed as
  849.      * {@code comp = r1.compose(r2, RotationConvention.FRAME_TRANSFORM)}.
  850.      * </p>
  851.      * @param r rotation to apply the rotation to
  852.      * @param convention convention to use for the semantics of the angle
  853.      * @return a new rotation which is the composition of r by the instance
  854.      */
  855.     public FieldRotation<T> compose(final Rotation r, final RotationConvention convention) {
  856.         return convention == RotationConvention.VECTOR_OPERATOR ?
  857.                              composeInternal(r) : applyTo(r, this);
  858.     }

  860.     /** Compose the instance with another rotation using vector operator convention.
  861.      * @param r rotation to apply the rotation to
  862.      * @return a new rotation which is the composition of r by the instance
  863.      * using vector operator convention
  864.      */
  865.     private FieldRotation<T> composeInternal(final Rotation r) {
  866.         return new FieldRotation<>(q0.multiply(r.getQ0()).subtract(q1.multiply(r.getQ1()).add(q2.multiply(r.getQ2())).add(q3.multiply(r.getQ3()))),
  867.                                    q0.multiply(r.getQ1()).add(q1.multiply(r.getQ0())).add(q3.multiply(r.getQ2()).subtract(q2.multiply(r.getQ3()))),
  868.                                    q0.multiply(r.getQ2()).add(q2.multiply(r.getQ0())).add(q1.multiply(r.getQ3()).subtract(q3.multiply(r.getQ1()))),
  869.                                    q0.multiply(r.getQ3()).add(q3.multiply(r.getQ0())).add(q2.multiply(r.getQ1()).subtract(q1.multiply(r.getQ2()))),
  870.                                    false);
  871.     }

  873.     /** Apply a rotation to another rotation.
  874.      * Applying a rotation to another rotation is computing the composition
  875.      * in an order compliant with the following rule : let u be any
  876.      * vector and v its image by rInner (i.e. rInner.applyTo(u) = v), let w be the image
  877.      * of v by rOuter (i.e. rOuter.applyTo(v) = w), then w = comp.applyTo(u),
  878.      * where comp = applyTo(rOuter, rInner).
  879.      * @param r1 rotation to apply
  880.      * @param rInner rotation to apply the rotation to
  881.      * @param <T> the type of the field elements
  882.      * @return a new rotation which is the composition of r by the instance
  883.      */
  884.     public static <T extends CalculusFieldElement<T>> FieldRotation<T> applyTo(final Rotation r1, final FieldRotation<T> rInner) {
  885.         return new FieldRotation<>(rInner.q0.multiply(r1.getQ0()).subtract(rInner.q1.multiply(r1.getQ1()).add(rInner.q2.multiply(r1.getQ2())).add(rInner.q3.multiply(r1.getQ3()))),
  886.                                    rInner.q1.multiply(r1.getQ0()).add(rInner.q0.multiply(r1.getQ1())).add(rInner.q2.multiply(r1.getQ3()).subtract(rInner.q3.multiply(r1.getQ2()))),
  887.                                    rInner.q2.multiply(r1.getQ0()).add(rInner.q0.multiply(r1.getQ2())).add(rInner.q3.multiply(r1.getQ1()).subtract(rInner.q1.multiply(r1.getQ3()))),
  888.                                    rInner.q3.multiply(r1.getQ0()).add(rInner.q0.multiply(r1.getQ3())).add(rInner.q1.multiply(r1.getQ2()).subtract(rInner.q2.multiply(r1.getQ1()))),
  889.                                    false);
  890.     }

  892.     /** Apply the inverse of the instance to another rotation.
  893.      * <p>
  894.      * Calling this method is equivalent to call
  895.      * {@link #composeInverse(FieldRotation, RotationConvention)
  896.      * composeInverse(r, RotationConvention.VECTOR_OPERATOR)}.
  897.      * </p>
  898.      * @param r rotation to apply the rotation to
  899.      * @return a new rotation which is the composition of r by the inverse
  900.      * of the instance
  901.      */
  902.     public FieldRotation<T> applyInverseTo(final FieldRotation<T> r) {
  903.         return composeInverse(r, RotationConvention.VECTOR_OPERATOR);
  904.     }

  906.     /** Compose the inverse of the instance with another rotation.
  907.      * <p>
  908.      * If the semantics of the rotations composition corresponds to a
  909.      * {@link RotationConvention#VECTOR_OPERATOR vector operator} convention,
  910.      * applying the inverse of the instance to a rotation is computing
  911.      * the composition in an order compliant with the following rule :
  912.      * let {@code u} be any vector and {@code v} its image by {@code r1}
  913.      * (i.e. {@code r1.applyTo(u) = v}). Let {@code w} be the inverse image
  914.      * of {@code v} by {@code r2} (i.e. {@code r2.applyInverseTo(v) = w}).
  915.      * Then {@code w = comp.applyTo(u)}, where
  916.      * {@code comp = r2.composeInverse(r1)}.
  917.      * </p>
  918.      * <p>
  919.      * If the semantics of the rotations composition corresponds to a
  920.      * {@link RotationConvention#FRAME_TRANSFORM frame transform} convention,
  921.      * the application order will be reversed, which means it is the
  922.      * <em>innermost</em> rotation that will be reversed. So keeping the exact same
  923.      * meaning of all {@code r1}, {@code r2}, {@code u}, {@code v}, {@code w}
  924.      * and  {@code comp} as above, {@code comp} could also be computed as
  925.      * {@code comp = r1.revert().composeInverse(r2.revert(), RotationConvention.FRAME_TRANSFORM)}.
  926.      * </p>
  927.      * @param r rotation to apply the rotation to
  928.      * @param convention convention to use for the semantics of the angle
  929.      * @return a new rotation which is the composition of r by the inverse
  930.      * of the instance
  931.      */
  932.     public FieldRotation<T> composeInverse(final FieldRotation<T> r, final RotationConvention convention) {
  933.         return convention == RotationConvention.VECTOR_OPERATOR ?
  934.                              composeInverseInternal(r) : r.composeInternal(revert());
  935.     }

  937.     /** Compose the inverse of the instance with another rotation
  938.      * using vector operator convention.
  939.      * @param r rotation to apply the rotation to
  940.      * @return a new rotation which is the composition of r by the inverse
  941.      * of the instance using vector operator convention
  942.      */
  943.     private FieldRotation<T> composeInverseInternal(FieldRotation<T> r) {
  944.         return new FieldRotation<>(r.q0.multiply(q0).add(r.q1.multiply(q1)).add(r.q2.multiply(q2)).add(r.q3.multiply(q3)).negate(),
  945.                                    r.q0.multiply(q1).add(r.q2.multiply(q3).subtract(r.q3.multiply(q2))).subtract(r.q1.multiply(q0)),
  946.                                    r.q0.multiply(q2).add(r.q3.multiply(q1).subtract(r.q1.multiply(q3))).subtract(r.q2.multiply(q0)),
  947.                                    r.q0.multiply(q3).add(r.q1.multiply(q2).subtract(r.q2.multiply(q1))).subtract(r.q3.multiply(q0)),
  948.                                    false);
  949.     }

  951.     /** Apply the inverse of the instance to another rotation.
  952.      * <p>
  953.      * Calling this method is equivalent to call
  954.      * {@link #composeInverse(Rotation, RotationConvention)
  955.      * composeInverse(r, RotationConvention.VECTOR_OPERATOR)}.
  956.      * </p>
  957.      * @param r rotation to apply the rotation to
  958.      * @return a new rotation which is the composition of r by the inverse
  959.      * of the instance
  960.      */
  961.     public FieldRotation<T> applyInverseTo(final Rotation r) {
  962.         return composeInverse(r, RotationConvention.VECTOR_OPERATOR);
  963.     }

  965.     /** Compose the inverse of the instance with another rotation.
  966.      * <p>
  967.      * If the semantics of the rotations composition corresponds to a
  968.      * {@link RotationConvention#VECTOR_OPERATOR vector operator} convention,
  969.      * applying the inverse of the instance to a rotation is computing
  970.      * the composition in an order compliant with the following rule :
  971.      * let {@code u} be any vector and {@code v} its image by {@code r1}
  972.      * (i.e. {@code r1.applyTo(u) = v}). Let {@code w} be the inverse image
  973.      * of {@code v} by {@code r2} (i.e. {@code r2.applyInverseTo(v) = w}).
  974.      * Then {@code w = comp.applyTo(u)}, where
  975.      * {@code comp = r2.composeInverse(r1)}.
  976.      * </p>
  977.      * <p>
  978.      * If the semantics of the rotations composition corresponds to a
  979.      * {@link RotationConvention#FRAME_TRANSFORM frame transform} convention,
  980.      * the application order will be reversed, which means it is the
  981.      * <em>innermost</em> rotation that will be reversed. So keeping the exact same
  982.      * meaning of all {@code r1}, {@code r2}, {@code u}, {@code v}, {@code w}
  983.      * and  {@code comp} as above, {@code comp} could also be computed as
  984.      * {@code comp = r1.revert().composeInverse(r2.revert(), RotationConvention.FRAME_TRANSFORM)}.
  985.      * </p>
  986.      * @param r rotation to apply the rotation to
  987.      * @param convention convention to use for the semantics of the angle
  988.      * @return a new rotation which is the composition of r by the inverse
  989.      * of the instance
  990.      */
  991.     public FieldRotation<T> composeInverse(final Rotation r, final RotationConvention convention) {
  992.         return convention == RotationConvention.VECTOR_OPERATOR ?
  993.                              composeInverseInternal(r) : applyTo(r, revert());
  994.     }

  996.     /** Compose the inverse of the instance with another rotation
  997.      * using vector operator convention.
  998.      * @param r rotation to apply the rotation to
  999.      * @return a new rotation which is the composition of r by the inverse
  1000.      * of the instance using vector operator convention
  1001.      */
  1002.     private FieldRotation<T> composeInverseInternal(Rotation r) {
  1003.         return new FieldRotation<>(q0.multiply(r.getQ0()).add(q1.multiply(r.getQ1()).add(q2.multiply(r.getQ2())).add(q3.multiply(r.getQ3()))).negate(),
  1004.                                    q1.multiply(r.getQ0()).add(q3.multiply(r.getQ2()).subtract(q2.multiply(r.getQ3()))).subtract(q0.multiply(r.getQ1())),
  1005.                                    q2.multiply(r.getQ0()).add(q1.multiply(r.getQ3()).subtract(q3.multiply(r.getQ1()))).subtract(q0.multiply(r.getQ2())),
  1006.                                    q3.multiply(r.getQ0()).add(q2.multiply(r.getQ1()).subtract(q1.multiply(r.getQ2()))).subtract(q0.multiply(r.getQ3())),
  1007.                                    false);
  1008.     }

  1010.     /** Apply the inverse of a rotation to another rotation.
  1011.      * Applying the inverse of a rotation to another rotation is computing
  1012.      * the composition in an order compliant with the following rule :
  1013.      * let u be any vector and v its image by rInner (i.e. rInner.applyTo(u) = v),
  1014.      * let w be the inverse image of v by rOuter
  1015.      * (i.e. rOuter.applyInverseTo(v) = w), then w = comp.applyTo(u), where
  1016.      * comp = applyInverseTo(rOuter, rInner).
  1017.      * @param rOuter rotation to apply the rotation to
  1018.      * @param rInner rotation to apply the rotation to
  1019.      * @param <T> the type of the field elements
  1020.      * @return a new rotation which is the composition of r by the inverse
  1021.      * of the instance
  1022.      */
  1023.     public static <T extends CalculusFieldElement<T>> FieldRotation<T> applyInverseTo(final Rotation rOuter, final FieldRotation<T> rInner) {
  1024.         return new FieldRotation<>(rInner.q0.multiply(rOuter.getQ0()).add(rInner.q1.multiply(rOuter.getQ1()).add(rInner.q2.multiply(rOuter.getQ2())).add(rInner.q3.multiply(rOuter.getQ3()))).negate(),
  1025.                                    rInner.q0.multiply(rOuter.getQ1()).add(rInner.q2.multiply(rOuter.getQ3()).subtract(rInner.q3.multiply(rOuter.getQ2()))).subtract(rInner.q1.multiply(rOuter.getQ0())),
  1026.                                    rInner.q0.multiply(rOuter.getQ2()).add(rInner.q3.multiply(rOuter.getQ1()).subtract(rInner.q1.multiply(rOuter.getQ3()))).subtract(rInner.q2.multiply(rOuter.getQ0())),
  1027.                                    rInner.q0.multiply(rOuter.getQ3()).add(rInner.q1.multiply(rOuter.getQ2()).subtract(rInner.q2.multiply(rOuter.getQ1()))).subtract(rInner.q3.multiply(rOuter.getQ0())),
  1028.                                    false);
  1029.     }

  1031.     /** Perfect orthogonality on a 3X3 matrix.
  1032.      * @param m initial matrix (not exactly orthogonal)
  1033.      * @param threshold convergence threshold for the iterative
  1034.      * orthogonality correction (convergence is reached when the
  1035.      * difference between two steps of the Frobenius norm of the
  1036.      * correction is below this threshold)
  1037.      * @return an orthogonal matrix close to m
  1038.      * @exception MathIllegalArgumentException if the matrix cannot be
  1039.      * orthogonalized with the given threshold after 10 iterations
  1040.      */
  1041.     private T[][] orthogonalizeMatrix(final T[][] m, final double threshold)
  1042.         throws MathIllegalArgumentException {

  1044.         T x00 = m[0][0];
  1045.         T x01 = m[0][1];
  1046.         T x02 = m[0][2];
  1047.         T x10 = m[1][0];
  1048.         T x11 = m[1][1];
  1049.         T x12 = m[1][2];
  1050.         T x20 = m[2][0];
  1051.         T x21 = m[2][1];
  1052.         T x22 = m[2][2];
  1053.         double fn = 0;
  1054.         double fn1;

  1056.         final T[][] o = MathArrays.buildArray(m[0][0].getField(), 3, 3);

  1058.         // iterative correction: Xn+1 = Xn - 0.5 * (Xn.Mt.Xn - M)
  1059.         int i;
  1060.         for (i = 0; i < 11; ++i) {

  1062.             // Mt.Xn
  1063.             final T mx00 = m[0][0].multiply(x00).add(m[1][0].multiply(x10)).add(m[2][0].multiply(x20));
  1064.             final T mx10 = m[0][1].multiply(x00).add(m[1][1].multiply(x10)).add(m[2][1].multiply(x20));
  1065.             final T mx20 = m[0][2].multiply(x00).add(m[1][2].multiply(x10)).add(m[2][2].multiply(x20));
  1066.             final T mx01 = m[0][0].multiply(x01).add(m[1][0].multiply(x11)).add(m[2][0].multiply(x21));
  1067.             final T mx11 = m[0][1].multiply(x01).add(m[1][1].multiply(x11)).add(m[2][1].multiply(x21));
  1068.             final T mx21 = m[0][2].multiply(x01).add(m[1][2].multiply(x11)).add(m[2][2].multiply(x21));
  1069.             final T mx02 = m[0][0].multiply(x02).add(m[1][0].multiply(x12)).add(m[2][0].multiply(x22));
  1070.             final T mx12 = m[0][1].multiply(x02).add(m[1][1].multiply(x12)).add(m[2][1].multiply(x22));
  1071.             final T mx22 = m[0][2].multiply(x02).add(m[1][2].multiply(x12)).add(m[2][2].multiply(x22));

  1073.             // Xn+1
  1074.             o[0][0] = x00.subtract(x00.multiply(mx00).add(x01.multiply(mx10)).add(x02.multiply(mx20)).subtract(m[0][0]).multiply(0.5));
  1075.             o[0][1] = x01.subtract(x00.multiply(mx01).add(x01.multiply(mx11)).add(x02.multiply(mx21)).subtract(m[0][1]).multiply(0.5));
  1076.             o[0][2] = x02.subtract(x00.multiply(mx02).add(x01.multiply(mx12)).add(x02.multiply(mx22)).subtract(m[0][2]).multiply(0.5));
  1077.             o[1][0] = x10.subtract(x10.multiply(mx00).add(x11.multiply(mx10)).add(x12.multiply(mx20)).subtract(m[1][0]).multiply(0.5));
  1078.             o[1][1] = x11.subtract(x10.multiply(mx01).add(x11.multiply(mx11)).add(x12.multiply(mx21)).subtract(m[1][1]).multiply(0.5));
  1079.             o[1][2] = x12.subtract(x10.multiply(mx02).add(x11.multiply(mx12)).add(x12.multiply(mx22)).subtract(m[1][2]).multiply(0.5));
  1080.             o[2][0] = x20.subtract(x20.multiply(mx00).add(x21.multiply(mx10)).add(x22.multiply(mx20)).subtract(m[2][0]).multiply(0.5));
  1081.             o[2][1] = x21.subtract(x20.multiply(mx01).add(x21.multiply(mx11)).add(x22.multiply(mx21)).subtract(m[2][1]).multiply(0.5));
  1082.             o[2][2] = x22.subtract(x20.multiply(mx02).add(x21.multiply(mx12)).add(x22.multiply(mx22)).subtract(m[2][2]).multiply(0.5));

  1084.             // correction on each elements
  1085.             final double corr00 = o[0][0].getReal() - m[0][0].getReal();
  1086.             final double corr01 = o[0][1].getReal() - m[0][1].getReal();
  1087.             final double corr02 = o[0][2].getReal() - m[0][2].getReal();
  1088.             final double corr10 = o[1][0].getReal() - m[1][0].getReal();
  1089.             final double corr11 = o[1][1].getReal() - m[1][1].getReal();
  1090.             final double corr12 = o[1][2].getReal() - m[1][2].getReal();
  1091.             final double corr20 = o[2][0].getReal() - m[2][0].getReal();
  1092.             final double corr21 = o[2][1].getReal() - m[2][1].getReal();
  1093.             final double corr22 = o[2][2].getReal() - m[2][2].getReal();

  1095.             // Frobenius norm of the correction
  1096.             fn1 = corr00 * corr00 + corr01 * corr01 + corr02 * corr02 +
  1097.                   corr10 * corr10 + corr11 * corr11 + corr12 * corr12 +
  1098.                   corr20 * corr20 + corr21 * corr21 + corr22 * corr22;

  1100.             // convergence test
  1101.             if (FastMath.abs(fn1 - fn) <= threshold) {
  1102.                 return o;
  1103.             }

  1105.             // prepare next iteration
  1106.             x00 = o[0][0];
  1107.             x01 = o[0][1];
  1108.             x02 = o[0][2];
  1109.             x10 = o[1][0];
  1110.             x11 = o[1][1];
  1111.             x12 = o[1][2];
  1112.             x20 = o[2][0];
  1113.             x21 = o[2][1];
  1114.             x22 = o[2][2];
  1115.             fn  = fn1;

  1117.         }

  1119.         // the algorithm did not converge after 10 iterations
  1120.         throw new MathIllegalArgumentException(LocalizedGeometryFormats.UNABLE_TO_ORTHOGONOLIZE_MATRIX,
  1121.                                                i - 1);

  1123.     }

  1125.     /** Compute the <i>distance</i> between two rotations.
  1126.      * <p>The <i>distance</i> is intended here as a way to check if two
  1127.      * rotations are almost similar (i.e. they transform vectors the same way)
  1128.      * or very different. It is mathematically defined as the angle of
  1129.      * the rotation r that prepended to one of the rotations gives the other
  1130.      * one: \(r_1(r) = r_2\)
  1131.      * </p>
  1132.      * <p>This distance is an angle between 0 and &pi;. Its value is the smallest
  1133.      * possible upper bound of the angle in radians between r<sub>1</sub>(v)
  1134.      * and r<sub>2</sub>(v) for all possible vectors v. This upper bound is
  1135.      * reached for some v. The distance is equal to 0 if and only if the two
  1136.      * rotations are identical.</p>
  1137.      * <p>Comparing two rotations should always be done using this value rather
  1138.      * than for example comparing the components of the quaternions. It is much
  1139.      * more stable, and has a geometric meaning. Also comparing quaternions
  1140.      * components is error prone since for example quaternions (0.36, 0.48, -0.48, -0.64)
  1141.      * and (-0.36, -0.48, 0.48, 0.64) represent exactly the same rotation despite
  1142.      * their components are different (they are exact opposites).</p>
  1143.      * @param r1 first rotation
  1144.      * @param r2 second rotation
  1145.      * @param <T> the type of the field elements
  1146.      * @return <i>distance</i> between r1 and r2
  1147.      */
  1148.     public static <T extends CalculusFieldElement<T>> T distance(final FieldRotation<T> r1, final FieldRotation<T> r2) {
  1149.         return r1.composeInverseInternal(r2).getAngle();
  1150.     }

  1152. }
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