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From t.@apache.org
Subject [20/82] [partial] [math] Update for next development iteration: commons-math4
Date Mon, 16 Feb 2015 22:39:50 GMT
http://git-wip-us.apache.org/repos/asf/commons-math/blob/a7b4803f/src/main/java/org/apache/commons/math3/geometry/euclidean/threed/FieldRotation.java
----------------------------------------------------------------------
diff --git a/src/main/java/org/apache/commons/math3/geometry/euclidean/threed/FieldRotation.java b/src/main/java/org/apache/commons/math3/geometry/euclidean/threed/FieldRotation.java
deleted file mode 100644
index dd6f3e4..0000000
--- a/src/main/java/org/apache/commons/math3/geometry/euclidean/threed/FieldRotation.java
+++ /dev/null
@@ -1,1183 +0,0 @@
-/*
- * Licensed to the Apache Software Foundation (ASF) under one or more
- * contributor license agreements.  See the NOTICE file distributed with
- * this work for additional information regarding copyright ownership.
- * The ASF licenses this file to You under the Apache License, Version 2.0
- * (the "License"); you may not use this file except in compliance with
- * the License.  You may obtain a copy of the License at
- *
- *      http://www.apache.org/licenses/LICENSE-2.0
- *
- * Unless required by applicable law or agreed to in writing, software
- * distributed under the License is distributed on an "AS IS" BASIS,
- * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
- * See the License for the specific language governing permissions and
- * limitations under the License.
- */
-
-package org.apache.commons.math3.geometry.euclidean.threed;
-
-import java.io.Serializable;
-
-import org.apache.commons.math3.RealFieldElement;
-import org.apache.commons.math3.Field;
-import org.apache.commons.math3.exception.MathArithmeticException;
-import org.apache.commons.math3.exception.MathIllegalArgumentException;
-import org.apache.commons.math3.exception.util.LocalizedFormats;
-import org.apache.commons.math3.util.FastMath;
-import org.apache.commons.math3.util.MathArrays;
-
-/**
- * This class is a re-implementation of {@link Rotation} using {@link RealFieldElement}.
- * <p>Instance of this class are guaranteed to be immutable.</p>
- *
- * @param <T> the type of the field elements
- * @see FieldVector3D
- * @see RotationOrder
- * @since 3.2
- */
-
-public class FieldRotation<T extends RealFieldElement<T>> implements Serializable {
-
-    /** Serializable version identifier */
-    private static final long serialVersionUID = 20130224l;
-
-    /** Scalar coordinate of the quaternion. */
-    private final T q0;
-
-    /** First coordinate of the vectorial part of the quaternion. */
-    private final T q1;
-
-    /** Second coordinate of the vectorial part of the quaternion. */
-    private final T q2;
-
-    /** Third coordinate of the vectorial part of the quaternion. */
-    private final T q3;
-
-    /** Build a rotation from the quaternion coordinates.
-     * <p>A rotation can be built from a <em>normalized</em> quaternion,
-     * i.e. a quaternion for which q<sub>0</sub><sup>2</sup> +
-     * q<sub>1</sub><sup>2</sup> + q<sub>2</sub><sup>2</sup> +
-     * q<sub>3</sub><sup>2</sup> = 1. If the quaternion is not normalized,
-     * the constructor can normalize it in a preprocessing step.</p>
-     * <p>Note that some conventions put the scalar part of the quaternion
-     * as the 4<sup>th</sup> component and the vector part as the first three
-     * components. This is <em>not</em> our convention. We put the scalar part
-     * as the first component.</p>
-     * @param q0 scalar part of the quaternion
-     * @param q1 first coordinate of the vectorial part of the quaternion
-     * @param q2 second coordinate of the vectorial part of the quaternion
-     * @param q3 third coordinate of the vectorial part of the quaternion
-     * @param needsNormalization if true, the coordinates are considered
-     * not to be normalized, a normalization preprocessing step is performed
-     * before using them
-     */
-    public FieldRotation(final T q0, final T q1, final T q2, final T q3, final boolean needsNormalization) {
-
-        if (needsNormalization) {
-            // normalization preprocessing
-            final T inv =
-                    q0.multiply(q0).add(q1.multiply(q1)).add(q2.multiply(q2)).add(q3.multiply(q3)).sqrt().reciprocal();
-            this.q0 = inv.multiply(q0);
-            this.q1 = inv.multiply(q1);
-            this.q2 = inv.multiply(q2);
-            this.q3 = inv.multiply(q3);
-        } else {
-            this.q0 = q0;
-            this.q1 = q1;
-            this.q2 = q2;
-            this.q3 = q3;
-        }
-
-    }
-
-    /** Build a rotation from an axis and an angle.
-     * <p>We use the convention that angles are oriented according to
-     * the effect of the rotation on vectors around the axis. That means
-     * that if (i, j, k) is a direct frame and if we first provide +k as
-     * the axis and &pi;/2 as the angle to this constructor, and then
-     * {@link #applyTo(FieldVector3D) apply} the instance to +i, we will get
-     * +j.</p>
-     * <p>Another way to represent our convention is to say that a rotation
-     * of angle &theta; about the unit vector (x, y, z) is the same as the
-     * rotation build from quaternion components { cos(-&theta;/2),
-     * x * sin(-&theta;/2), y * sin(-&theta;/2), z * sin(-&theta;/2) }.
-     * Note the minus sign on the angle!</p>
-     * <p>On the one hand this convention is consistent with a vectorial
-     * perspective (moving vectors in fixed frames), on the other hand it
-     * is different from conventions with a frame perspective (fixed vectors
-     * viewed from different frames) like the ones used for example in spacecraft
-     * attitude community or in the graphics community.</p>
-     * @param axis axis around which to rotate
-     * @param angle rotation angle.
-     * @exception MathIllegalArgumentException if the axis norm is zero
-     */
-    public FieldRotation(final FieldVector3D<T> axis, final T angle)
-        throws MathIllegalArgumentException {
-
-        final T norm = axis.getNorm();
-        if (norm.getReal() == 0) {
-            throw new MathIllegalArgumentException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_AXIS);
-        }
-
-        final T halfAngle = angle.multiply(-0.5);
-        final T coeff = halfAngle.sin().divide(norm);
-
-        q0 = halfAngle.cos();
-        q1 = coeff.multiply(axis.getX());
-        q2 = coeff.multiply(axis.getY());
-        q3 = coeff.multiply(axis.getZ());
-
-    }
-
-    /** Build a rotation from a 3X3 matrix.
-
-     * <p>Rotation matrices are orthogonal matrices, i.e. unit matrices
-     * (which are matrices for which m.m<sup>T</sup> = I) with real
-     * coefficients. The module of the determinant of unit matrices is
-     * 1, among the orthogonal 3X3 matrices, only the ones having a
-     * positive determinant (+1) are rotation matrices.</p>
-
-     * <p>When a rotation is defined by a matrix with truncated values
-     * (typically when it is extracted from a technical sheet where only
-     * four to five significant digits are available), the matrix is not
-     * orthogonal anymore. This constructor handles this case
-     * transparently by using a copy of the given matrix and applying a
-     * correction to the copy in order to perfect its orthogonality. If
-     * the Frobenius norm of the correction needed is above the given
-     * threshold, then the matrix is considered to be too far from a
-     * true rotation matrix and an exception is thrown.<p>
-
-     * @param m rotation matrix
-     * @param threshold convergence threshold for the iterative
-     * orthogonality correction (convergence is reached when the
-     * difference between two steps of the Frobenius norm of the
-     * correction is below this threshold)
-
-     * @exception NotARotationMatrixException if the matrix is not a 3X3
-     * matrix, or if it cannot be transformed into an orthogonal matrix
-     * with the given threshold, or if the determinant of the resulting
-     * orthogonal matrix is negative
-
-     */
-    public FieldRotation(final T[][] m, final double threshold)
-        throws NotARotationMatrixException {
-
-        // dimension check
-        if ((m.length != 3) || (m[0].length != 3) ||
-                (m[1].length != 3) || (m[2].length != 3)) {
-            throw new NotARotationMatrixException(
-                                                  LocalizedFormats.ROTATION_MATRIX_DIMENSIONS,
-                                                  m.length, m[0].length);
-        }
-
-        // compute a "close" orthogonal matrix
-        final T[][] ort = orthogonalizeMatrix(m, threshold);
-
-        // check the sign of the determinant
-        final T d0 = ort[1][1].multiply(ort[2][2]).subtract(ort[2][1].multiply(ort[1][2]));
-        final T d1 = ort[0][1].multiply(ort[2][2]).subtract(ort[2][1].multiply(ort[0][2]));
-        final T d2 = ort[0][1].multiply(ort[1][2]).subtract(ort[1][1].multiply(ort[0][2]));
-        final T det =
-                ort[0][0].multiply(d0).subtract(ort[1][0].multiply(d1)).add(ort[2][0].multiply(d2));
-        if (det.getReal() < 0.0) {
-            throw new NotARotationMatrixException(
-                                                  LocalizedFormats.CLOSEST_ORTHOGONAL_MATRIX_HAS_NEGATIVE_DETERMINANT,
-                                                  det);
-        }
-
-        final T[] quat = mat2quat(ort);
-        q0 = quat[0];
-        q1 = quat[1];
-        q2 = quat[2];
-        q3 = quat[3];
-
-    }
-
-    /** Build the rotation that transforms a pair of vector into another pair.
-
-     * <p>Except for possible scale factors, if the instance were applied to
-     * the pair (u<sub>1</sub>, u<sub>2</sub>) it will produce the pair
-     * (v<sub>1</sub>, v<sub>2</sub>).</p>
-
-     * <p>If the angular separation between u<sub>1</sub> and u<sub>2</sub> is
-     * not the same as the angular separation between v<sub>1</sub> and
-     * v<sub>2</sub>, then a corrected v'<sub>2</sub> will be used rather than
-     * v<sub>2</sub>, the corrected vector will be in the (v<sub>1</sub>,
-     * v<sub>2</sub>) plane.</p>
-
-     * @param u1 first vector of the origin pair
-     * @param u2 second vector of the origin pair
-     * @param v1 desired image of u1 by the rotation
-     * @param v2 desired image of u2 by the rotation
-     * @exception MathArithmeticException if the norm of one of the vectors is zero,
-     * or if one of the pair is degenerated (i.e. the vectors of the pair are colinear)
-     */
-    public FieldRotation(FieldVector3D<T> u1, FieldVector3D<T> u2, FieldVector3D<T> v1, FieldVector3D<T> v2)
-        throws MathArithmeticException {
-
-        // build orthonormalized base from u1, u2
-        // this fails when vectors are null or colinear, which is forbidden to define a rotation
-        final FieldVector3D<T> u3 = FieldVector3D.crossProduct(u1, u2).normalize();
-        u2 = FieldVector3D.crossProduct(u3, u1).normalize();
-        u1 = u1.normalize();
-
-        // build an orthonormalized base from v1, v2
-        // this fails when vectors are null or colinear, which is forbidden to define a rotation
-        final FieldVector3D<T> v3 = FieldVector3D.crossProduct(v1, v2).normalize();
-        v2 = FieldVector3D.crossProduct(v3, v1).normalize();
-        v1 = v1.normalize();
-
-        // buid a matrix transforming the first base into the second one
-        final T[][] array = MathArrays.buildArray(u1.getX().getField(), 3, 3);
-        array[0][0] = u1.getX().multiply(v1.getX()).add(u2.getX().multiply(v2.getX())).add(u3.getX().multiply(v3.getX()));
-        array[0][1] = u1.getY().multiply(v1.getX()).add(u2.getY().multiply(v2.getX())).add(u3.getY().multiply(v3.getX()));
-        array[0][2] = u1.getZ().multiply(v1.getX()).add(u2.getZ().multiply(v2.getX())).add(u3.getZ().multiply(v3.getX()));
-        array[1][0] = u1.getX().multiply(v1.getY()).add(u2.getX().multiply(v2.getY())).add(u3.getX().multiply(v3.getY()));
-        array[1][1] = u1.getY().multiply(v1.getY()).add(u2.getY().multiply(v2.getY())).add(u3.getY().multiply(v3.getY()));
-        array[1][2] = u1.getZ().multiply(v1.getY()).add(u2.getZ().multiply(v2.getY())).add(u3.getZ().multiply(v3.getY()));
-        array[2][0] = u1.getX().multiply(v1.getZ()).add(u2.getX().multiply(v2.getZ())).add(u3.getX().multiply(v3.getZ()));
-        array[2][1] = u1.getY().multiply(v1.getZ()).add(u2.getY().multiply(v2.getZ())).add(u3.getY().multiply(v3.getZ()));
-        array[2][2] = u1.getZ().multiply(v1.getZ()).add(u2.getZ().multiply(v2.getZ())).add(u3.getZ().multiply(v3.getZ()));
-
-        T[] quat = mat2quat(array);
-        q0 = quat[0];
-        q1 = quat[1];
-        q2 = quat[2];
-        q3 = quat[3];
-
-    }
-
-    /** Build one of the rotations that transform one vector into another one.
-
-     * <p>Except for a possible scale factor, if the instance were
-     * applied to the vector u it will produce the vector v. There is an
-     * infinite number of such rotations, this constructor choose the
-     * one with the smallest associated angle (i.e. the one whose axis
-     * is orthogonal to the (u, v) plane). If u and v are colinear, an
-     * arbitrary rotation axis is chosen.</p>
-
-     * @param u origin vector
-     * @param v desired image of u by the rotation
-     * @exception MathArithmeticException if the norm of one of the vectors is zero
-     */
-    public FieldRotation(final FieldVector3D<T> u, final FieldVector3D<T> v) throws MathArithmeticException {
-
-        final T normProduct = u.getNorm().multiply(v.getNorm());
-        if (normProduct.getReal() == 0) {
-            throw new MathArithmeticException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_DEFINING_VECTOR);
-        }
-
-        final T dot = FieldVector3D.dotProduct(u, v);
-
-        if (dot.getReal() < ((2.0e-15 - 1.0) * normProduct.getReal())) {
-            // special case u = -v: we select a PI angle rotation around
-            // an arbitrary vector orthogonal to u
-            final FieldVector3D<T> w = u.orthogonal();
-            q0 = normProduct.getField().getZero();
-            q1 = w.getX().negate();
-            q2 = w.getY().negate();
-            q3 = w.getZ().negate();
-        } else {
-            // general case: (u, v) defines a plane, we select
-            // the shortest possible rotation: axis orthogonal to this plane
-            q0 = dot.divide(normProduct).add(1.0).multiply(0.5).sqrt();
-            final T coeff = q0.multiply(normProduct).multiply(2.0).reciprocal();
-            final FieldVector3D<T> q = FieldVector3D.crossProduct(v, u);
-            q1 = coeff.multiply(q.getX());
-            q2 = coeff.multiply(q.getY());
-            q3 = coeff.multiply(q.getZ());
-        }
-
-    }
-
-    /** Build a rotation from three Cardan or Euler elementary rotations.
-
-     * <p>Cardan rotations are three successive rotations around the
-     * canonical axes X, Y and Z, each axis being used once. There are
-     * 6 such sets of rotations (XYZ, XZY, YXZ, YZX, ZXY and ZYX). Euler
-     * rotations are three successive rotations around the canonical
-     * axes X, Y and Z, the first and last rotations being around the
-     * same axis. There are 6 such sets of rotations (XYX, XZX, YXY,
-     * YZY, ZXZ and ZYZ), the most popular one being ZXZ.</p>
-     * <p>Beware that many people routinely use the term Euler angles even
-     * for what really are Cardan angles (this confusion is especially
-     * widespread in the aerospace business where Roll, Pitch and Yaw angles
-     * are often wrongly tagged as Euler angles).</p>
-
-     * @param order order of rotations to use
-     * @param alpha1 angle of the first elementary rotation
-     * @param alpha2 angle of the second elementary rotation
-     * @param alpha3 angle of the third elementary rotation
-     */
-    public FieldRotation(final RotationOrder order, final T alpha1, final T alpha2, final T alpha3) {
-        final T one = alpha1.getField().getOne();
-        final FieldRotation<T> r1 = new FieldRotation<T>(new FieldVector3D<T>(one, order.getA1()), alpha1);
-        final FieldRotation<T> r2 = new FieldRotation<T>(new FieldVector3D<T>(one, order.getA2()), alpha2);
-        final FieldRotation<T> r3 = new FieldRotation<T>(new FieldVector3D<T>(one, order.getA3()), alpha3);
-        final FieldRotation<T> composed = r1.applyTo(r2.applyTo(r3));
-        q0 = composed.q0;
-        q1 = composed.q1;
-        q2 = composed.q2;
-        q3 = composed.q3;
-    }
-
-    /** Convert an orthogonal rotation matrix to a quaternion.
-     * @param ort orthogonal rotation matrix
-     * @return quaternion corresponding to the matrix
-     */
-    private T[] mat2quat(final T[][] ort) {
-
-        final T[] quat = MathArrays.buildArray(ort[0][0].getField(), 4);
-
-        // There are different ways to compute the quaternions elements
-        // from the matrix. They all involve computing one element from
-        // the diagonal of the matrix, and computing the three other ones
-        // using a formula involving a division by the first element,
-        // which unfortunately can be zero. Since the norm of the
-        // quaternion is 1, we know at least one element has an absolute
-        // value greater or equal to 0.5, so it is always possible to
-        // select the right formula and avoid division by zero and even
-        // numerical inaccuracy. Checking the elements in turn and using
-        // the first one greater than 0.45 is safe (this leads to a simple
-        // test since qi = 0.45 implies 4 qi^2 - 1 = -0.19)
-        T s = ort[0][0].add(ort[1][1]).add(ort[2][2]);
-        if (s.getReal() > -0.19) {
-            // compute q0 and deduce q1, q2 and q3
-            quat[0] = s.add(1.0).sqrt().multiply(0.5);
-            T inv = quat[0].reciprocal().multiply(0.25);
-            quat[1] = inv.multiply(ort[1][2].subtract(ort[2][1]));
-            quat[2] = inv.multiply(ort[2][0].subtract(ort[0][2]));
-            quat[3] = inv.multiply(ort[0][1].subtract(ort[1][0]));
-        } else {
-            s = ort[0][0].subtract(ort[1][1]).subtract(ort[2][2]);
-            if (s.getReal() > -0.19) {
-                // compute q1 and deduce q0, q2 and q3
-                quat[1] = s.add(1.0).sqrt().multiply(0.5);
-                T inv = quat[1].reciprocal().multiply(0.25);
-                quat[0] = inv.multiply(ort[1][2].subtract(ort[2][1]));
-                quat[2] = inv.multiply(ort[0][1].add(ort[1][0]));
-                quat[3] = inv.multiply(ort[0][2].add(ort[2][0]));
-            } else {
-                s = ort[1][1].subtract(ort[0][0]).subtract(ort[2][2]);
-                if (s.getReal() > -0.19) {
-                    // compute q2 and deduce q0, q1 and q3
-                    quat[2] = s.add(1.0).sqrt().multiply(0.5);
-                    T inv = quat[2].reciprocal().multiply(0.25);
-                    quat[0] = inv.multiply(ort[2][0].subtract(ort[0][2]));
-                    quat[1] = inv.multiply(ort[0][1].add(ort[1][0]));
-                    quat[3] = inv.multiply(ort[2][1].add(ort[1][2]));
-                } else {
-                    // compute q3 and deduce q0, q1 and q2
-                    s = ort[2][2].subtract(ort[0][0]).subtract(ort[1][1]);
-                    quat[3] = s.add(1.0).sqrt().multiply(0.5);
-                    T inv = quat[3].reciprocal().multiply(0.25);
-                    quat[0] = inv.multiply(ort[0][1].subtract(ort[1][0]));
-                    quat[1] = inv.multiply(ort[0][2].add(ort[2][0]));
-                    quat[2] = inv.multiply(ort[2][1].add(ort[1][2]));
-                }
-            }
-        }
-
-        return quat;
-
-    }
-
-    /** Revert a rotation.
-     * Build a rotation which reverse the effect of another
-     * rotation. This means that if r(u) = v, then r.revert(v) = u. The
-     * instance is not changed.
-     * @return a new rotation whose effect is the reverse of the effect
-     * of the instance
-     */
-    public FieldRotation<T> revert() {
-        return new FieldRotation<T>(q0.negate(), q1, q2, q3, false);
-    }
-
-    /** Get the scalar coordinate of the quaternion.
-     * @return scalar coordinate of the quaternion
-     */
-    public T getQ0() {
-        return q0;
-    }
-
-    /** Get the first coordinate of the vectorial part of the quaternion.
-     * @return first coordinate of the vectorial part of the quaternion
-     */
-    public T getQ1() {
-        return q1;
-    }
-
-    /** Get the second coordinate of the vectorial part of the quaternion.
-     * @return second coordinate of the vectorial part of the quaternion
-     */
-    public T getQ2() {
-        return q2;
-    }
-
-    /** Get the third coordinate of the vectorial part of the quaternion.
-     * @return third coordinate of the vectorial part of the quaternion
-     */
-    public T getQ3() {
-        return q3;
-    }
-
-    /** Get the normalized axis of the rotation.
-     * @return normalized axis of the rotation
-     * @see #FieldRotation(FieldVector3D, RealFieldElement)
-     */
-    public FieldVector3D<T> getAxis() {
-        final T squaredSine = q1.multiply(q1).add(q2.multiply(q2)).add(q3.multiply(q3));
-        if (squaredSine.getReal() == 0) {
-            final Field<T> field = squaredSine.getField();
-            return new FieldVector3D<T>(field.getOne(), field.getZero(), field.getZero());
-        } else if (q0.getReal() < 0) {
-            T inverse = squaredSine.sqrt().reciprocal();
-            return new FieldVector3D<T>(q1.multiply(inverse), q2.multiply(inverse), q3.multiply(inverse));
-        }
-        final T inverse = squaredSine.sqrt().reciprocal().negate();
-        return new FieldVector3D<T>(q1.multiply(inverse), q2.multiply(inverse), q3.multiply(inverse));
-    }
-
-    /** Get the angle of the rotation.
-     * @return angle of the rotation (between 0 and &pi;)
-     * @see #FieldRotation(FieldVector3D, RealFieldElement)
-     */
-    public T getAngle() {
-        if ((q0.getReal() < -0.1) || (q0.getReal() > 0.1)) {
-            return q1.multiply(q1).add(q2.multiply(q2)).add(q3.multiply(q3)).sqrt().asin().multiply(2);
-        } else if (q0.getReal() < 0) {
-            return q0.negate().acos().multiply(2);
-        }
-        return q0.acos().multiply(2);
-    }
-
-    /** Get the Cardan or Euler angles corresponding to the instance.
-
-     * <p>The equations show that each rotation can be defined by two
-     * different values of the Cardan or Euler angles set. For example
-     * if Cardan angles are used, the rotation defined by the angles
-     * a<sub>1</sub>, a<sub>2</sub> and a<sub>3</sub> is the same as
-     * the rotation defined by the angles &pi; + a<sub>1</sub>, &pi;
-     * - a<sub>2</sub> and &pi; + a<sub>3</sub>. This method implements
-     * the following arbitrary choices:</p>
-     * <ul>
-     *   <li>for Cardan angles, the chosen set is the one for which the
-     *   second angle is between -&pi;/2 and &pi;/2 (i.e its cosine is
-     *   positive),</li>
-     *   <li>for Euler angles, the chosen set is the one for which the
-     *   second angle is between 0 and &pi; (i.e its sine is positive).</li>
-     * </ul>
-
-     * <p>Cardan and Euler angle have a very disappointing drawback: all
-     * of them have singularities. This means that if the instance is
-     * too close to the singularities corresponding to the given
-     * rotation order, it will be impossible to retrieve the angles. For
-     * Cardan angles, this is often called gimbal lock. There is
-     * <em>nothing</em> to do to prevent this, it is an intrinsic problem
-     * with Cardan and Euler representation (but not a problem with the
-     * rotation itself, which is perfectly well defined). For Cardan
-     * angles, singularities occur when the second angle is close to
-     * -&pi;/2 or +&pi;/2, for Euler angle singularities occur when the
-     * second angle is close to 0 or &pi;, this implies that the identity
-     * rotation is always singular for Euler angles!</p>
-
-     * @param order rotation order to use
-     * @return an array of three angles, in the order specified by the set
-     * @exception CardanEulerSingularityException if the rotation is
-     * singular with respect to the angles set specified
-     */
-    public T[] getAngles(final RotationOrder order)
-        throws CardanEulerSingularityException {
-
-        if (order == RotationOrder.XYZ) {
-
-            // r (+K) coordinates are :
-            //  sin (theta), -cos (theta) sin (phi), cos (theta) cos (phi)
-            // (-r) (+I) coordinates are :
-            // cos (psi) cos (theta), -sin (psi) cos (theta), sin (theta)
-            final // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
-            FieldVector3D<T> v1 = applyTo(vector(0, 0, 1));
-            final FieldVector3D<T> v2 = applyInverseTo(vector(1, 0, 0));
-            if  ((v2.getZ().getReal() < -0.9999999999) || (v2.getZ().getReal() > 0.9999999999)) {
-                throw new CardanEulerSingularityException(true);
-            }
-            return buildArray(v1.getY().negate().atan2(v1.getZ()),
-                              v2.getZ().asin(),
-                              v2.getY().negate().atan2(v2.getX()));
-
-        } else if (order == RotationOrder.XZY) {
-
-            // r (+J) coordinates are :
-            // -sin (psi), cos (psi) cos (phi), cos (psi) sin (phi)
-            // (-r) (+I) coordinates are :
-            // cos (theta) cos (psi), -sin (psi), sin (theta) cos (psi)
-            // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
-            final FieldVector3D<T> v1 = applyTo(vector(0, 1, 0));
-            final FieldVector3D<T> v2 = applyInverseTo(vector(1, 0, 0));
-            if ((v2.getY().getReal() < -0.9999999999) || (v2.getY().getReal() > 0.9999999999)) {
-                throw new CardanEulerSingularityException(true);
-            }
-            return buildArray(v1.getZ().atan2(v1.getY()),
-                              v2.getY().asin().negate(),
-                              v2.getZ().atan2(v2.getX()));
-
-        } else if (order == RotationOrder.YXZ) {
-
-            // r (+K) coordinates are :
-            //  cos (phi) sin (theta), -sin (phi), cos (phi) cos (theta)
-            // (-r) (+J) coordinates are :
-            // sin (psi) cos (phi), cos (psi) cos (phi), -sin (phi)
-            // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
-            final FieldVector3D<T> v1 = applyTo(vector(0, 0, 1));
-            final FieldVector3D<T> v2 = applyInverseTo(vector(0, 1, 0));
-            if ((v2.getZ().getReal() < -0.9999999999) || (v2.getZ().getReal() > 0.9999999999)) {
-                throw new CardanEulerSingularityException(true);
-            }
-            return buildArray(v1.getX().atan2(v1.getZ()),
-                              v2.getZ().asin().negate(),
-                              v2.getX().atan2(v2.getY()));
-
-        } else if (order == RotationOrder.YZX) {
-
-            // r (+I) coordinates are :
-            // cos (psi) cos (theta), sin (psi), -cos (psi) sin (theta)
-            // (-r) (+J) coordinates are :
-            // sin (psi), cos (phi) cos (psi), -sin (phi) cos (psi)
-            // and we can choose to have psi in the interval [-PI/2 ; +PI/2]
-            final FieldVector3D<T> v1 = applyTo(vector(1, 0, 0));
-            final FieldVector3D<T> v2 = applyInverseTo(vector(0, 1, 0));
-            if ((v2.getX().getReal() < -0.9999999999) || (v2.getX().getReal() > 0.9999999999)) {
-                throw new CardanEulerSingularityException(true);
-            }
-            return buildArray(v1.getZ().negate().atan2(v1.getX()),
-                              v2.getX().asin(),
-                              v2.getZ().negate().atan2(v2.getY()));
-
-        } else if (order == RotationOrder.ZXY) {
-
-            // r (+J) coordinates are :
-            // -cos (phi) sin (psi), cos (phi) cos (psi), sin (phi)
-            // (-r) (+K) coordinates are :
-            // -sin (theta) cos (phi), sin (phi), cos (theta) cos (phi)
-            // and we can choose to have phi in the interval [-PI/2 ; +PI/2]
-            final FieldVector3D<T> v1 = applyTo(vector(0, 1, 0));
-            final FieldVector3D<T> v2 = applyInverseTo(vector(0, 0, 1));
-            if ((v2.getY().getReal() < -0.9999999999) || (v2.getY().getReal() > 0.9999999999)) {
-                throw new CardanEulerSingularityException(true);
-            }
-            return buildArray(v1.getX().negate().atan2(v1.getY()),
-                              v2.getY().asin(),
-                              v2.getX().negate().atan2(v2.getZ()));
-
-        } else if (order == RotationOrder.ZYX) {
-
-            // r (+I) coordinates are :
-            //  cos (theta) cos (psi), cos (theta) sin (psi), -sin (theta)
-            // (-r) (+K) coordinates are :
-            // -sin (theta), sin (phi) cos (theta), cos (phi) cos (theta)
-            // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
-            final FieldVector3D<T> v1 = applyTo(vector(1, 0, 0));
-            final FieldVector3D<T> v2 = applyInverseTo(vector(0, 0, 1));
-            if ((v2.getX().getReal() < -0.9999999999) || (v2.getX().getReal() > 0.9999999999)) {
-                throw new CardanEulerSingularityException(true);
-            }
-            return buildArray(v1.getY().atan2(v1.getX()),
-                              v2.getX().asin().negate(),
-                              v2.getY().atan2(v2.getZ()));
-
-        } else if (order == RotationOrder.XYX) {
-
-            // r (+I) coordinates are :
-            //  cos (theta), sin (phi1) sin (theta), -cos (phi1) sin (theta)
-            // (-r) (+I) coordinates are :
-            // cos (theta), sin (theta) sin (phi2), sin (theta) cos (phi2)
-            // and we can choose to have theta in the interval [0 ; PI]
-            final FieldVector3D<T> v1 = applyTo(vector(1, 0, 0));
-            final FieldVector3D<T> v2 = applyInverseTo(vector(1, 0, 0));
-            if ((v2.getX().getReal() < -0.9999999999) || (v2.getX().getReal() > 0.9999999999)) {
-                throw new CardanEulerSingularityException(false);
-            }
-            return buildArray(v1.getY().atan2(v1.getZ().negate()),
-                              v2.getX().acos(),
-                              v2.getY().atan2(v2.getZ()));
-
-        } else if (order == RotationOrder.XZX) {
-
-            // r (+I) coordinates are :
-            //  cos (psi), cos (phi1) sin (psi), sin (phi1) sin (psi)
-            // (-r) (+I) coordinates are :
-            // cos (psi), -sin (psi) cos (phi2), sin (psi) sin (phi2)
-            // and we can choose to have psi in the interval [0 ; PI]
-            final FieldVector3D<T> v1 = applyTo(vector(1, 0, 0));
-            final FieldVector3D<T> v2 = applyInverseTo(vector(1, 0, 0));
-            if ((v2.getX().getReal() < -0.9999999999) || (v2.getX().getReal() > 0.9999999999)) {
-                throw new CardanEulerSingularityException(false);
-            }
-            return buildArray(v1.getZ().atan2(v1.getY()),
-                              v2.getX().acos(),
-                              v2.getZ().atan2(v2.getY().negate()));
-
-        } else if (order == RotationOrder.YXY) {
-
-            // r (+J) coordinates are :
-            //  sin (theta1) sin (phi), cos (phi), cos (theta1) sin (phi)
-            // (-r) (+J) coordinates are :
-            // sin (phi) sin (theta2), cos (phi), -sin (phi) cos (theta2)
-            // and we can choose to have phi in the interval [0 ; PI]
-            final FieldVector3D<T> v1 = applyTo(vector(0, 1, 0));
-            final FieldVector3D<T> v2 = applyInverseTo(vector(0, 1, 0));
-            if ((v2.getY().getReal() < -0.9999999999) || (v2.getY().getReal() > 0.9999999999)) {
-                throw new CardanEulerSingularityException(false);
-            }
-            return buildArray(v1.getX().atan2(v1.getZ()),
-                              v2.getY().acos(),
-                              v2.getX().atan2(v2.getZ().negate()));
-
-        } else if (order == RotationOrder.YZY) {
-
-            // r (+J) coordinates are :
-            //  -cos (theta1) sin (psi), cos (psi), sin (theta1) sin (psi)
-            // (-r) (+J) coordinates are :
-            // sin (psi) cos (theta2), cos (psi), sin (psi) sin (theta2)
-            // and we can choose to have psi in the interval [0 ; PI]
-            final FieldVector3D<T> v1 = applyTo(vector(0, 1, 0));
-            final FieldVector3D<T> v2 = applyInverseTo(vector(0, 1, 0));
-            if ((v2.getY().getReal() < -0.9999999999) || (v2.getY().getReal() > 0.9999999999)) {
-                throw new CardanEulerSingularityException(false);
-            }
-            return buildArray(v1.getZ().atan2(v1.getX().negate()),
-                              v2.getY().acos(),
-                              v2.getZ().atan2(v2.getX()));
-
-        } else if (order == RotationOrder.ZXZ) {
-
-            // r (+K) coordinates are :
-            //  sin (psi1) sin (phi), -cos (psi1) sin (phi), cos (phi)
-            // (-r) (+K) coordinates are :
-            // sin (phi) sin (psi2), sin (phi) cos (psi2), cos (phi)
-            // and we can choose to have phi in the interval [0 ; PI]
-            final FieldVector3D<T> v1 = applyTo(vector(0, 0, 1));
-            final FieldVector3D<T> v2 = applyInverseTo(vector(0, 0, 1));
-            if ((v2.getZ().getReal() < -0.9999999999) || (v2.getZ().getReal() > 0.9999999999)) {
-                throw new CardanEulerSingularityException(false);
-            }
-            return buildArray(v1.getX().atan2(v1.getY().negate()),
-                              v2.getZ().acos(),
-                              v2.getX().atan2(v2.getY()));
-
-        } else { // last possibility is ZYZ
-
-            // r (+K) coordinates are :
-            //  cos (psi1) sin (theta), sin (psi1) sin (theta), cos (theta)
-            // (-r) (+K) coordinates are :
-            // -sin (theta) cos (psi2), sin (theta) sin (psi2), cos (theta)
-            // and we can choose to have theta in the interval [0 ; PI]
-            final FieldVector3D<T> v1 = applyTo(vector(0, 0, 1));
-            final FieldVector3D<T> v2 = applyInverseTo(vector(0, 0, 1));
-            if ((v2.getZ().getReal() < -0.9999999999) || (v2.getZ().getReal() > 0.9999999999)) {
-                throw new CardanEulerSingularityException(false);
-            }
-            return buildArray(v1.getY().atan2(v1.getX()),
-                              v2.getZ().acos(),
-                              v2.getY().atan2(v2.getX().negate()));
-
-        }
-
-    }
-
-    /** Create a dimension 3 array.
-     * @param a0 first array element
-     * @param a1 second array element
-     * @param a2 third array element
-     * @return new array
-     */
-    private T[] buildArray(final T a0, final T a1, final T a2) {
-        final T[] array = MathArrays.buildArray(a0.getField(), 3);
-        array[0] = a0;
-        array[1] = a1;
-        array[2] = a2;
-        return array;
-    }
-
-    /** Create a constant vector.
-     * @param x abscissa
-     * @param y ordinate
-     * @param z height
-     * @return a constant vector
-     */
-    private FieldVector3D<T> vector(final double x, final double y, final double z) {
-        final T zero = q0.getField().getZero();
-        return new FieldVector3D<T>(zero.add(x), zero.add(y), zero.add(z));
-    }
-
-    /** Get the 3X3 matrix corresponding to the instance
-     * @return the matrix corresponding to the instance
-     */
-    public T[][] getMatrix() {
-
-        // products
-        final T q0q0  = q0.multiply(q0);
-        final T q0q1  = q0.multiply(q1);
-        final T q0q2  = q0.multiply(q2);
-        final T q0q3  = q0.multiply(q3);
-        final T q1q1  = q1.multiply(q1);
-        final T q1q2  = q1.multiply(q2);
-        final T q1q3  = q1.multiply(q3);
-        final T q2q2  = q2.multiply(q2);
-        final T q2q3  = q2.multiply(q3);
-        final T q3q3  = q3.multiply(q3);
-
-        // create the matrix
-        final T[][] m = MathArrays.buildArray(q0.getField(), 3, 3);
-
-        m [0][0] = q0q0.add(q1q1).multiply(2).subtract(1);
-        m [1][0] = q1q2.subtract(q0q3).multiply(2);
-        m [2][0] = q1q3.add(q0q2).multiply(2);
-
-        m [0][1] = q1q2.add(q0q3).multiply(2);
-        m [1][1] = q0q0.add(q2q2).multiply(2).subtract(1);
-        m [2][1] = q2q3.subtract(q0q1).multiply(2);
-
-        m [0][2] = q1q3.subtract(q0q2).multiply(2);
-        m [1][2] = q2q3.add(q0q1).multiply(2);
-        m [2][2] = q0q0.add(q3q3).multiply(2).subtract(1);
-
-        return m;
-
-    }
-
-    /** Convert to a constant vector without derivatives.
-     * @return a constant vector
-     */
-    public Rotation toRotation() {
-        return new Rotation(q0.getReal(), q1.getReal(), q2.getReal(), q3.getReal(), false);
-    }
-
-    /** Apply the rotation to a vector.
-     * @param u vector to apply the rotation to
-     * @return a new vector which is the image of u by the rotation
-     */
-    public FieldVector3D<T> applyTo(final FieldVector3D<T> u) {
-
-        final T x = u.getX();
-        final T y = u.getY();
-        final T z = u.getZ();
-
-        final T s = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z));
-
-        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),
-                                    q0.multiply(y.multiply(q0).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y),
-                                    q0.multiply(z.multiply(q0).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z));
-
-    }
-
-    /** Apply the rotation to a vector.
-     * @param u vector to apply the rotation to
-     * @return a new vector which is the image of u by the rotation
-     */
-    public FieldVector3D<T> applyTo(final Vector3D u) {
-
-        final double x = u.getX();
-        final double y = u.getY();
-        final double z = u.getZ();
-
-        final T s = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z));
-
-        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),
-                                    q0.multiply(q0.multiply(y).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y),
-                                    q0.multiply(q0.multiply(z).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z));
-
-    }
-
-    /** Apply the rotation to a vector stored in an array.
-     * @param in an array with three items which stores vector to rotate
-     * @param out an array with three items to put result to (it can be the same
-     * array as in)
-     */
-    public void applyTo(final T[] in, final T[] out) {
-
-        final T x = in[0];
-        final T y = in[1];
-        final T z = in[2];
-
-        final T s = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z));
-
-        out[0] = q0.multiply(x.multiply(q0).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x);
-        out[1] = q0.multiply(y.multiply(q0).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y);
-        out[2] = q0.multiply(z.multiply(q0).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z);
-
-    }
-
-    /** Apply the rotation to a vector stored in an array.
-     * @param in an array with three items which stores vector to rotate
-     * @param out an array with three items to put result to
-     */
-    public void applyTo(final double[] in, final T[] out) {
-
-        final double x = in[0];
-        final double y = in[1];
-        final double z = in[2];
-
-        final T s = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z));
-
-        out[0] = q0.multiply(q0.multiply(x).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x);
-        out[1] = q0.multiply(q0.multiply(y).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y);
-        out[2] = q0.multiply(q0.multiply(z).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z);
-
-    }
-
-    /** Apply a rotation to a vector.
-     * @param r rotation to apply
-     * @param u vector to apply the rotation to
-     * @param <T> the type of the field elements
-     * @return a new vector which is the image of u by the rotation
-     */
-    public static <T extends RealFieldElement<T>> FieldVector3D<T> applyTo(final Rotation r, final FieldVector3D<T> u) {
-
-        final T x = u.getX();
-        final T y = u.getY();
-        final T z = u.getZ();
-
-        final T s = x.multiply(r.getQ1()).add(y.multiply(r.getQ2())).add(z.multiply(r.getQ3()));
-
-        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),
-                                    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),
-                                    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));
-
-    }
-
-    /** Apply the inverse of the rotation to a vector.
-     * @param u vector to apply the inverse of the rotation to
-     * @return a new vector which such that u is its image by the rotation
-     */
-    public FieldVector3D<T> applyInverseTo(final FieldVector3D<T> u) {
-
-        final T x = u.getX();
-        final T y = u.getY();
-        final T z = u.getZ();
-
-        final T s  = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z));
-        final T m0 = q0.negate();
-
-        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),
-                                    m0.multiply(y.multiply(m0).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y),
-                                    m0.multiply(z.multiply(m0).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z));
-
-    }
-
-    /** Apply the inverse of the rotation to a vector.
-     * @param u vector to apply the inverse of the rotation to
-     * @return a new vector which such that u is its image by the rotation
-     */
-    public FieldVector3D<T> applyInverseTo(final Vector3D u) {
-
-        final double x = u.getX();
-        final double y = u.getY();
-        final double z = u.getZ();
-
-        final T s  = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z));
-        final T m0 = q0.negate();
-
-        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),
-                                    m0.multiply(m0.multiply(y).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y),
-                                    m0.multiply(m0.multiply(z).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z));
-
-    }
-
-    /** Apply the inverse of the rotation to a vector stored in an array.
-     * @param in an array with three items which stores vector to rotate
-     * @param out an array with three items to put result to (it can be the same
-     * array as in)
-     */
-    public void applyInverseTo(final T[] in, final T[] out) {
-
-        final T x = in[0];
-        final T y = in[1];
-        final T z = in[2];
-
-        final T s = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z));
-        final T m0 = q0.negate();
-
-        out[0] = m0.multiply(x.multiply(m0).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x);
-        out[1] = m0.multiply(y.multiply(m0).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y);
-        out[2] = m0.multiply(z.multiply(m0).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z);
-
-    }
-
-    /** Apply the inverse of the rotation to a vector stored in an array.
-     * @param in an array with three items which stores vector to rotate
-     * @param out an array with three items to put result to
-     */
-    public void applyInverseTo(final double[] in, final T[] out) {
-
-        final double x = in[0];
-        final double y = in[1];
-        final double z = in[2];
-
-        final T s = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z));
-        final T m0 = q0.negate();
-
-        out[0] = m0.multiply(m0.multiply(x).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x);
-        out[1] = m0.multiply(m0.multiply(y).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y);
-        out[2] = m0.multiply(m0.multiply(z).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z);
-
-    }
-
-    /** Apply the inverse of a rotation to a vector.
-     * @param r rotation to apply
-     * @param u vector to apply the inverse of the rotation to
-     * @param <T> the type of the field elements
-     * @return a new vector which such that u is its image by the rotation
-     */
-    public static <T extends RealFieldElement<T>> FieldVector3D<T> applyInverseTo(final Rotation r, final FieldVector3D<T> u) {
-
-        final T x = u.getX();
-        final T y = u.getY();
-        final T z = u.getZ();
-
-        final T s  = x.multiply(r.getQ1()).add(y.multiply(r.getQ2())).add(z.multiply(r.getQ3()));
-        final double m0 = -r.getQ0();
-
-        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),
-                                    y.multiply(m0).subtract(x.multiply(r.getQ3()).subtract(z.multiply(r.getQ1()))).multiply(m0).add(s.multiply(r.getQ2())).multiply(2).subtract(y),
-                                    z.multiply(m0).subtract(y.multiply(r.getQ1()).subtract(x.multiply(r.getQ2()))).multiply(m0).add(s.multiply(r.getQ3())).multiply(2).subtract(z));
-
-    }
-
-    /** Apply the instance to another rotation.
-     * Applying the instance to a rotation is computing the composition
-     * in an order compliant with the following rule : let u be any
-     * vector and v its image by r (i.e. r.applyTo(u) = v), let w be the image
-     * of v by the instance (i.e. applyTo(v) = w), then w = comp.applyTo(u),
-     * where comp = applyTo(r).
-     * @param r rotation to apply the rotation to
-     * @return a new rotation which is the composition of r by the instance
-     */
-    public FieldRotation<T> applyTo(final FieldRotation<T> r) {
-        return new FieldRotation<T>(r.q0.multiply(q0).subtract(r.q1.multiply(q1).add(r.q2.multiply(q2)).add(r.q3.multiply(q3))),
-                                    r.q1.multiply(q0).add(r.q0.multiply(q1)).add(r.q2.multiply(q3).subtract(r.q3.multiply(q2))),
-                                    r.q2.multiply(q0).add(r.q0.multiply(q2)).add(r.q3.multiply(q1).subtract(r.q1.multiply(q3))),
-                                    r.q3.multiply(q0).add(r.q0.multiply(q3)).add(r.q1.multiply(q2).subtract(r.q2.multiply(q1))),
-                                    false);
-    }
-
-    /** Apply the instance to another rotation.
-     * Applying the instance to a rotation is computing the composition
-     * in an order compliant with the following rule : let u be any
-     * vector and v its image by r (i.e. r.applyTo(u) = v), let w be the image
-     * of v by the instance (i.e. applyTo(v) = w), then w = comp.applyTo(u),
-     * where comp = applyTo(r).
-     * @param r rotation to apply the rotation to
-     * @return a new rotation which is the composition of r by the instance
-     */
-    public FieldRotation<T> applyTo(final Rotation r) {
-        return new FieldRotation<T>(q0.multiply(r.getQ0()).subtract(q1.multiply(r.getQ1()).add(q2.multiply(r.getQ2())).add(q3.multiply(r.getQ3()))),
-                                    q0.multiply(r.getQ1()).add(q1.multiply(r.getQ0())).add(q3.multiply(r.getQ2()).subtract(q2.multiply(r.getQ3()))),
-                                    q0.multiply(r.getQ2()).add(q2.multiply(r.getQ0())).add(q1.multiply(r.getQ3()).subtract(q3.multiply(r.getQ1()))),
-                                    q0.multiply(r.getQ3()).add(q3.multiply(r.getQ0())).add(q2.multiply(r.getQ1()).subtract(q1.multiply(r.getQ2()))),
-                                    false);
-    }
-
-    /** Apply a rotation to another rotation.
-     * Applying a rotation to another rotation is computing the composition
-     * in an order compliant with the following rule : let u be any
-     * vector and v its image by rInner (i.e. rInner.applyTo(u) = v), let w be the image
-     * of v by rOuter (i.e. rOuter.applyTo(v) = w), then w = comp.applyTo(u),
-     * where comp = applyTo(rOuter, rInner).
-     * @param r1 rotation to apply
-     * @param rInner rotation to apply the rotation to
-     * @param <T> the type of the field elements
-     * @return a new rotation which is the composition of r by the instance
-     */
-    public static <T extends RealFieldElement<T>> FieldRotation<T> applyTo(final Rotation r1, final FieldRotation<T> rInner) {
-        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()))),
-                                    rInner.q1.multiply(r1.getQ0()).add(rInner.q0.multiply(r1.getQ1())).add(rInner.q2.multiply(r1.getQ3()).subtract(rInner.q3.multiply(r1.getQ2()))),
-                                    rInner.q2.multiply(r1.getQ0()).add(rInner.q0.multiply(r1.getQ2())).add(rInner.q3.multiply(r1.getQ1()).subtract(rInner.q1.multiply(r1.getQ3()))),
-                                    rInner.q3.multiply(r1.getQ0()).add(rInner.q0.multiply(r1.getQ3())).add(rInner.q1.multiply(r1.getQ2()).subtract(rInner.q2.multiply(r1.getQ1()))),
-                                    false);
-    }
-
-    /** Apply the inverse of the instance to another rotation.
-     * Applying the inverse of the instance to a rotation is computing
-     * the composition in an order compliant with the following rule :
-     * let u be any vector and v its image by r (i.e. r.applyTo(u) = v),
-     * let w be the inverse image of v by the instance
-     * (i.e. applyInverseTo(v) = w), then w = comp.applyTo(u), where
-     * comp = applyInverseTo(r).
-     * @param r rotation to apply the rotation to
-     * @return a new rotation which is the composition of r by the inverse
-     * of the instance
-     */
-    public FieldRotation<T> applyInverseTo(final FieldRotation<T> r) {
-        return new FieldRotation<T>(r.q0.multiply(q0).add(r.q1.multiply(q1).add(r.q2.multiply(q2)).add(r.q3.multiply(q3))).negate(),
-                                    r.q0.multiply(q1).add(r.q2.multiply(q3).subtract(r.q3.multiply(q2))).subtract(r.q1.multiply(q0)),
-                                    r.q0.multiply(q2).add(r.q3.multiply(q1).subtract(r.q1.multiply(q3))).subtract(r.q2.multiply(q0)),
-                                    r.q0.multiply(q3).add(r.q1.multiply(q2).subtract(r.q2.multiply(q1))).subtract(r.q3.multiply(q0)),
-                                    false);
-    }
-
-    /** Apply the inverse of the instance to another rotation.
-     * Applying the inverse of the instance to a rotation is computing
-     * the composition in an order compliant with the following rule :
-     * let u be any vector and v its image by r (i.e. r.applyTo(u) = v),
-     * let w be the inverse image of v by the instance
-     * (i.e. applyInverseTo(v) = w), then w = comp.applyTo(u), where
-     * comp = applyInverseTo(r).
-     * @param r rotation to apply the rotation to
-     * @return a new rotation which is the composition of r by the inverse
-     * of the instance
-     */
-    public FieldRotation<T> applyInverseTo(final Rotation r) {
-        return new FieldRotation<T>(q0.multiply(r.getQ0()).add(q1.multiply(r.getQ1()).add(q2.multiply(r.getQ2())).add(q3.multiply(r.getQ3()))).negate(),
-                                    q1.multiply(r.getQ0()).add(q3.multiply(r.getQ2()).subtract(q2.multiply(r.getQ3()))).subtract(q0.multiply(r.getQ1())),
-                                    q2.multiply(r.getQ0()).add(q1.multiply(r.getQ3()).subtract(q3.multiply(r.getQ1()))).subtract(q0.multiply(r.getQ2())),
-                                    q3.multiply(r.getQ0()).add(q2.multiply(r.getQ1()).subtract(q1.multiply(r.getQ2()))).subtract(q0.multiply(r.getQ3())),
-                                    false);
-    }
-
-    /** Apply the inverse of a rotation to another rotation.
-     * Applying the inverse of a rotation to another rotation is computing
-     * the composition in an order compliant with the following rule :
-     * let u be any vector and v its image by rInner (i.e. rInner.applyTo(u) = v),
-     * let w be the inverse image of v by rOuter
-     * (i.e. rOuter.applyInverseTo(v) = w), then w = comp.applyTo(u), where
-     * comp = applyInverseTo(rOuter, rInner).
-     * @param rOuter rotation to apply the rotation to
-     * @param rInner rotation to apply the rotation to
-     * @param <T> the type of the field elements
-     * @return a new rotation which is the composition of r by the inverse
-     * of the instance
-     */
-    public static <T extends RealFieldElement<T>> FieldRotation<T> applyInverseTo(final Rotation rOuter, final FieldRotation<T> rInner) {
-        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(),
-                                    rInner.q0.multiply(rOuter.getQ1()).add(rInner.q2.multiply(rOuter.getQ3()).subtract(rInner.q3.multiply(rOuter.getQ2()))).subtract(rInner.q1.multiply(rOuter.getQ0())),
-                                    rInner.q0.multiply(rOuter.getQ2()).add(rInner.q3.multiply(rOuter.getQ1()).subtract(rInner.q1.multiply(rOuter.getQ3()))).subtract(rInner.q2.multiply(rOuter.getQ0())),
-                                    rInner.q0.multiply(rOuter.getQ3()).add(rInner.q1.multiply(rOuter.getQ2()).subtract(rInner.q2.multiply(rOuter.getQ1()))).subtract(rInner.q3.multiply(rOuter.getQ0())),
-                                    false);
-    }
-
-    /** Perfect orthogonality on a 3X3 matrix.
-     * @param m initial matrix (not exactly orthogonal)
-     * @param threshold convergence threshold for the iterative
-     * orthogonality correction (convergence is reached when the
-     * difference between two steps of the Frobenius norm of the
-     * correction is below this threshold)
-     * @return an orthogonal matrix close to m
-     * @exception NotARotationMatrixException if the matrix cannot be
-     * orthogonalized with the given threshold after 10 iterations
-     */
-    private T[][] orthogonalizeMatrix(final T[][] m, final double threshold)
-        throws NotARotationMatrixException {
-
-        T x00 = m[0][0];
-        T x01 = m[0][1];
-        T x02 = m[0][2];
-        T x10 = m[1][0];
-        T x11 = m[1][1];
-        T x12 = m[1][2];
-        T x20 = m[2][0];
-        T x21 = m[2][1];
-        T x22 = m[2][2];
-        double fn = 0;
-        double fn1;
-
-        final T[][] o = MathArrays.buildArray(m[0][0].getField(), 3, 3);
-
-        // iterative correction: Xn+1 = Xn - 0.5 * (Xn.Mt.Xn - M)
-        int i = 0;
-        while (++i < 11) {
-
-            // Mt.Xn
-            final T mx00 = m[0][0].multiply(x00).add(m[1][0].multiply(x10)).add(m[2][0].multiply(x20));
-            final T mx10 = m[0][1].multiply(x00).add(m[1][1].multiply(x10)).add(m[2][1].multiply(x20));
-            final T mx20 = m[0][2].multiply(x00).add(m[1][2].multiply(x10)).add(m[2][2].multiply(x20));
-            final T mx01 = m[0][0].multiply(x01).add(m[1][0].multiply(x11)).add(m[2][0].multiply(x21));
-            final T mx11 = m[0][1].multiply(x01).add(m[1][1].multiply(x11)).add(m[2][1].multiply(x21));
-            final T mx21 = m[0][2].multiply(x01).add(m[1][2].multiply(x11)).add(m[2][2].multiply(x21));
-            final T mx02 = m[0][0].multiply(x02).add(m[1][0].multiply(x12)).add(m[2][0].multiply(x22));
-            final T mx12 = m[0][1].multiply(x02).add(m[1][1].multiply(x12)).add(m[2][1].multiply(x22));
-            final T mx22 = m[0][2].multiply(x02).add(m[1][2].multiply(x12)).add(m[2][2].multiply(x22));
-
-            // Xn+1
-            o[0][0] = x00.subtract(x00.multiply(mx00).add(x01.multiply(mx10)).add(x02.multiply(mx20)).subtract(m[0][0]).multiply(0.5));
-            o[0][1] = x01.subtract(x00.multiply(mx01).add(x01.multiply(mx11)).add(x02.multiply(mx21)).subtract(m[0][1]).multiply(0.5));
-            o[0][2] = x02.subtract(x00.multiply(mx02).add(x01.multiply(mx12)).add(x02.multiply(mx22)).subtract(m[0][2]).multiply(0.5));
-            o[1][0] = x10.subtract(x10.multiply(mx00).add(x11.multiply(mx10)).add(x12.multiply(mx20)).subtract(m[1][0]).multiply(0.5));
-            o[1][1] = x11.subtract(x10.multiply(mx01).add(x11.multiply(mx11)).add(x12.multiply(mx21)).subtract(m[1][1]).multiply(0.5));
-            o[1][2] = x12.subtract(x10.multiply(mx02).add(x11.multiply(mx12)).add(x12.multiply(mx22)).subtract(m[1][2]).multiply(0.5));
-            o[2][0] = x20.subtract(x20.multiply(mx00).add(x21.multiply(mx10)).add(x22.multiply(mx20)).subtract(m[2][0]).multiply(0.5));
-            o[2][1] = x21.subtract(x20.multiply(mx01).add(x21.multiply(mx11)).add(x22.multiply(mx21)).subtract(m[2][1]).multiply(0.5));
-            o[2][2] = x22.subtract(x20.multiply(mx02).add(x21.multiply(mx12)).add(x22.multiply(mx22)).subtract(m[2][2]).multiply(0.5));
-
-            // correction on each elements
-            final double corr00 = o[0][0].getReal() - m[0][0].getReal();
-            final double corr01 = o[0][1].getReal() - m[0][1].getReal();
-            final double corr02 = o[0][2].getReal() - m[0][2].getReal();
-            final double corr10 = o[1][0].getReal() - m[1][0].getReal();
-            final double corr11 = o[1][1].getReal() - m[1][1].getReal();
-            final double corr12 = o[1][2].getReal() - m[1][2].getReal();
-            final double corr20 = o[2][0].getReal() - m[2][0].getReal();
-            final double corr21 = o[2][1].getReal() - m[2][1].getReal();
-            final double corr22 = o[2][2].getReal() - m[2][2].getReal();
-
-            // Frobenius norm of the correction
-            fn1 = corr00 * corr00 + corr01 * corr01 + corr02 * corr02 +
-                  corr10 * corr10 + corr11 * corr11 + corr12 * corr12 +
-                  corr20 * corr20 + corr21 * corr21 + corr22 * corr22;
-
-            // convergence test
-            if (FastMath.abs(fn1 - fn) <= threshold) {
-                return o;
-            }
-
-            // prepare next iteration
-            x00 = o[0][0];
-            x01 = o[0][1];
-            x02 = o[0][2];
-            x10 = o[1][0];
-            x11 = o[1][1];
-            x12 = o[1][2];
-            x20 = o[2][0];
-            x21 = o[2][1];
-            x22 = o[2][2];
-            fn  = fn1;
-
-        }
-
-        // the algorithm did not converge after 10 iterations
-        throw new NotARotationMatrixException(LocalizedFormats.UNABLE_TO_ORTHOGONOLIZE_MATRIX,
-                                              i - 1);
-
-    }
-
-    /** Compute the <i>distance</i> between two rotations.
-     * <p>The <i>distance</i> is intended here as a way to check if two
-     * rotations are almost similar (i.e. they transform vectors the same way)
-     * or very different. It is mathematically defined as the angle of
-     * the rotation r that prepended to one of the rotations gives the other
-     * one:</p>
-     * <pre>
-     *        r<sub>1</sub>(r) = r<sub>2</sub>
-     * </pre>
-     * <p>This distance is an angle between 0 and &pi;. Its value is the smallest
-     * possible upper bound of the angle in radians between r<sub>1</sub>(v)
-     * and r<sub>2</sub>(v) for all possible vectors v. This upper bound is
-     * reached for some v. The distance is equal to 0 if and only if the two
-     * rotations are identical.</p>
-     * <p>Comparing two rotations should always be done using this value rather
-     * than for example comparing the components of the quaternions. It is much
-     * more stable, and has a geometric meaning. Also comparing quaternions
-     * components is error prone since for example quaternions (0.36, 0.48, -0.48, -0.64)
-     * and (-0.36, -0.48, 0.48, 0.64) represent exactly the same rotation despite
-     * their components are different (they are exact opposites).</p>
-     * @param r1 first rotation
-     * @param r2 second rotation
-     * @param <T> the type of the field elements
-     * @return <i>distance</i> between r1 and r2
-     */
-    public static <T extends RealFieldElement<T>> T distance(final FieldRotation<T> r1, final FieldRotation<T> r2) {
-        return r1.applyInverseTo(r2).getAngle();
-    }
-
-}


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