Java tutorial
/* * 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 π/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 θ about the unit vector (x, y, z) is the same as the * rotation build from quaternion components { cos(-θ/2), * x * sin(-θ/2), y * sin(-θ/2), z * sin(-θ/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 π) * @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 π + a<sub>1</sub>, π * - a<sub>2</sub> and π + 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 -π/2 and π/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 π (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 * -π/2 or +π/2, for Euler angle singularities occur when the * second angle is close to 0 or π, 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 π. 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(); } }