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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 new file mode 100644 index 0000000..4e2278b --- /dev/null +++ b/src/main/java/org/apache/commons/math3/geometry/euclidean/threed/FieldRotation.java @@ -0,0 +1,1663 @@ +/* + * 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 + * @deprecated as of 3.6, replaced with {@link + * #FieldRotation(FieldVector3D, RealFieldElement, RotationConvention)} + */ + @Deprecated + public FieldRotation(final FieldVector3D<T> axis, final T angle) + throws MathIllegalArgumentException { + this(axis, angle, RotationConvention.VECTOR_OPERATOR); + } + + /** 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. + * @param convention convention to use for the semantics of the angle + * @exception MathIllegalArgumentException if the axis norm is zero + * @since 3.6 + */ + public FieldRotation(final FieldVector3D<T> axis, final T angle, final RotationConvention convention) + 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(convention == RotationConvention.VECTOR_OPERATOR ? -0.5 : 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 vectors 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>) half-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 collinear) + */ + 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 collinear, 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 collinear, 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 collinear, 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 + * @deprecated as of 3.6, replaced with {@link + * #FieldRotation(RotationOrder, RotationConvention, + * RealFieldElement, RealFieldElement, RealFieldElement)} + */ + @Deprecated + public FieldRotation(final RotationOrder order, final T alpha1, final T alpha2, final T alpha3) { + this(order, RotationConvention.VECTOR_OPERATOR, alpha1, alpha2, alpha3); + } + + /** 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 compose, from left to right + * (i.e. we will use {@code r1.compose(r2.compose(r3, convention), convention)}) + * @param convention convention to use for the semantics of the angle + * @param alpha1 angle of the first elementary rotation + * @param alpha2 angle of the second elementary rotation + * @param alpha3 angle of the third elementary rotation + * @since 3.6 + */ + public FieldRotation(final RotationOrder order, final RotationConvention convention, + 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, convention); + final FieldRotation<T> r2 = new FieldRotation<T>(new FieldVector3D<T>(one, order.getA2()), alpha2, convention); + final FieldRotation<T> r3 = new FieldRotation<T>(new FieldVector3D<T>(one, order.getA3()), alpha3, convention); + final FieldRotation<T> composed = r1.compose(r2.compose(r3, convention), convention); + 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) + * @deprecated as of 3.6, replaced with {@link #getAxis(RotationConvention)} + */ + @Deprecated + public FieldVector3D<T> getAxis() { + return getAxis(RotationConvention.VECTOR_OPERATOR); + } + + /** Get the normalized axis of the rotation. + * <p> + * Note that as {@link #getAngle()} always returns an angle + * between 0 and π, changing the convention changes the + * direction of the axis, not the sign of the angle. + * </p> + * @param convention convention to use for the semantics of the angle + * @return normalized axis of the rotation + * @see #FieldRotation(FieldVector3D, RealFieldElement) + * @since 3.6 + */ + public FieldVector3D<T> getAxis(final RotationConvention convention) { + 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>(convention == RotationConvention.VECTOR_OPERATOR ? field.getOne(): field.getOne().negate(), + field.getZero(), + field.getZero()); + } else { + final double sgn = convention == RotationConvention.VECTOR_OPERATOR ? +1 : -1; + if (q0.getReal() < 0) { + T inverse = squaredSine.sqrt().reciprocal().multiply(sgn); + return new FieldVector3D<T>(q1.multiply(inverse), q2.multiply(inverse), q3.multiply(inverse)); + } + final T inverse = squaredSine.sqrt().reciprocal().negate().multiply(sgn); + 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 + * @deprecated as of 3.6, replaced with {@link #getAngles(RotationOrder, RotationConvention)} + */ + @Deprecated + public T[] getAngles(final RotationOrder order) + throws CardanEulerSingularityException { + return getAngles(order, RotationConvention.VECTOR_OPERATOR); + } + + /** 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 + * @param convention convention to use for the semantics of the angle + * @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 + * @since 3.6 + */ + public T[] getAngles(final RotationOrder order, RotationConvention convention) + throws CardanEulerSingularityException { + + if (convention == RotationConvention.VECTOR_OPERATOR) { + 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())); + + } + } else { + if (order == RotationOrder.XYZ) { + + // r (Vector3D.plusI) coordinates are : + // cos (theta) cos (psi), -cos (theta) sin (psi), sin (theta) + // (-r) (Vector3D.plusK) coordinates are : + // sin (theta), -sin (phi) cos (theta), cos (phi) cos (theta) + // and we can choose to have theta in the interval [-PI/2 ; +PI/2] + FieldVector3D<T> v1 = applyTo(Vector3D.PLUS_I); + FieldVector3D<T> v2 = applyInverseTo(Vector3D.PLUS_K); + if ((v2.getX().getReal() < -0.9999999999) || (v2.getX().getReal() > 0.9999999999)) { + throw new CardanEulerSingularityException(true); + } + return buildArray(v2.getY().negate().atan2(v2.getZ()), + v2.getX().asin(), + v1.getY().negate().atan2(v1.getX())); + + } else if (order == RotationOrder.XZY) { + + // r (Vector3D.plusI) coordinates are : + // cos (psi) cos (theta), -sin (psi), cos (psi) sin (theta) + // (-r) (Vector3D.plusJ) coordinates are : + // -sin (psi), cos (phi) cos (psi), sin (phi) cos (psi) + // and we can choose to have psi in the interval [-PI/2 ; +PI/2] + FieldVector3D<T> v1 = applyTo(Vector3D.PLUS_I); + FieldVector3D<T> v2 = applyInverseTo(Vector3D.PLUS_J); + if ((v2.getX().getReal() < -0.9999999999) || (v2.getX().getReal() > 0.9999999999)) { + throw new CardanEulerSingularityException(true); + } + return buildArray(v2.getZ().atan2(v2.getY()), + v2.getX().asin().negate(), + v1.getZ().atan2(v1.getX())); + + } else if (order == RotationOrder.YXZ) { + + // r (Vector3D.plusJ) coordinates are : + // cos (phi) sin (psi), cos (phi) cos (psi), -sin (phi) + // (-r) (Vector3D.plusK) coordinates are : + // sin (theta) cos (phi), -sin (phi), cos (theta) cos (phi) + // and we can choose to have phi in the interval [-PI/2 ; +PI/2] + FieldVector3D<T> v1 = applyTo(Vector3D.PLUS_J); + FieldVector3D<T> v2 = applyInverseTo(Vector3D.PLUS_K); + if ((v2.getY().getReal() < -0.9999999999) || (v2.getY().getReal() > 0.9999999999)) { + throw new CardanEulerSingularityException(true); + } + return buildArray(v2.getX().atan2(v2.getZ()), + v2.getY().asin().negate(), + v1.getX().atan2(v1.getY())); + + } else if (order == RotationOrder.YZX) { + + // r (Vector3D.plusJ) coordinates are : + // sin (psi), cos (psi) cos (phi), -cos (psi) sin (phi) + // (-r) (Vector3D.plusI) coordinates are : + // cos (theta) cos (psi), sin (psi), -sin (theta) cos (psi) + // and we can choose to have psi in the interval [-PI/2 ; +PI/2] + FieldVector3D<T> v1 = applyTo(Vector3D.PLUS_J); + FieldVector3D<T> v2 = applyInverseTo(Vector3D.PLUS_I); + if ((v2.getY().getReal() < -0.9999999999) || (v2.getY().getReal() > 0.9999999999)) { + throw new CardanEulerSingularityException(true); + } + return buildArray(v2.getZ().negate().atan2(v2.getX()), + v2.getY().asin(), + v1.getZ().negate().atan2(v1.getY())); + + } else if (order == RotationOrder.ZXY) { + + // r (Vector3D.plusK) coordinates are : + // -cos (phi) sin (theta), sin (phi), cos (phi) cos (theta) + // (-r) (Vector3D.plusJ) coordinates are : + // -sin (psi) cos (phi), cos (psi) cos (phi), sin (phi) + // and we can choose to have phi in the interval [-PI/2 ; +PI/2] + FieldVector3D<T> v1 = applyTo(Vector3D.PLUS_K); + FieldVector3D<T> v2 = applyInverseTo(Vector3D.PLUS_J); + if ((v2.getZ().getReal() < -0.9999999999) || (v2.getZ().getReal() > 0.9999999999)) { + throw new CardanEulerSingularityException(true); + } + return buildArray(v2.getX().negate().atan2(v2.getY()), + v2.getZ().asin(), + v1.getX().negate().atan2(v1.getZ())); + + } else if (order == RotationOrder.ZYX) { + + // r (Vector3D.plusK) coordinates are : + // -sin (theta), cos (theta) sin (phi), cos (theta) cos (phi) + // (-r) (Vector3D.plusI) coordinates are : + // cos (psi) cos (theta), sin (psi) cos (theta), -sin (theta) + // and we can choose to have theta in the interval [-PI/2 ; +PI/2] + FieldVector3D<T> v1 = applyTo(Vector3D.PLUS_K); + FieldVector3D<T> v2 = applyInverseTo(Vector3D.PLUS_I); + if ((v2.getZ().getReal() < -0.9999999999) || (v2.getZ().getReal() > 0.9999999999)) { + throw new CardanEulerSingularityException(true); + } + return buildArray(v2.getY().atan2(v2.getX()), + v2.getZ().asin().negate(), + v1.getY().atan2(v1.getZ())); + + } else if (order == RotationOrder.XYX) { + + // r (Vector3D.plusI) coordinates are : + // cos (theta), sin (phi2) sin (theta), cos (phi2) sin (theta) + // (-r) (Vector3D.plusI) coordinates are : + // cos (theta), sin (theta) sin (phi1), -sin (theta) cos (phi1) + // and we can choose to have theta in the interval [0 ; PI] + FieldVector3D<T> v1 = applyTo(Vector3D.PLUS_I); + FieldVector3D<T> v2 = applyInverseTo(Vector3D.PLUS_I); + if ((v2.getX().getReal() < -0.9999999999) || (v2.getX().getReal() > 0.9999999999)) { + throw new CardanEulerSingularityException(false); + } + return buildArray(v2.getY().atan2(v2.getZ().negate()), + v2.getX().acos(), + v1.getY().atan2(v1.getZ())); + + } else if (order == RotationOrder.XZX) { + + // r (Vector3D.plusI) coordinates are : + // cos (psi), -cos (phi2) sin (psi), sin (phi2) sin (psi) + // (-r) (Vector3D.plusI) coordinates are : + // cos (psi), sin (psi) cos (phi1), sin (psi) sin (phi1) + // and we can choose to have psi in the interval [0 ; PI] + FieldVector3D<T> v1 = applyTo(Vector3D.PLUS_I); + FieldVector3D<T> v2 = applyInverseTo(Vector3D.PLUS_I); + if ((v2.getX().getReal() < -0.9999999999) || (v2.getX().getReal() > 0.9999999999)) { + throw new CardanEulerSingularityException(false); + } + return buildArray(v2.getZ().atan2(v2.getY()), + v2.getX().acos(), + v1.getZ().atan2(v1.getY().negate())); + + } else if (order == RotationOrder.YXY) { + + // r (Vector3D.plusJ) coordinates are : + // sin (phi) sin (theta2), cos (phi), -sin (phi) cos (theta2) + // (-r) (Vector3D.plusJ) coordinates are : + // sin (theta1) sin (phi), cos (phi), cos (theta1) sin (phi) + // and we can choose to have phi in the interval [0 ; PI] + FieldVector3D<T> v1 = applyTo(Vector3D.PLUS_J); + FieldVector3D<T> v2 = applyInverseTo(Vector3D.PLUS_J); + if ((v2.getY().getReal() < -0.9999999999) || (v2.getY().getReal() > 0.9999999999)) { + throw new CardanEulerSingularityException(false); + } + return buildArray(v2.getX().atan2(v2.getZ()), + v2.getY().acos(), + v1.getX().atan2(v1.getZ().negate())); + + } else if (order == RotationOrder.YZY) { + + // r (Vector3D.plusJ) coordinates are : + // sin (psi) cos (theta2), cos (psi), sin (psi) sin (theta2) + // (-r) (Vector3D.plusJ) coordinates are : + // -cos (theta1) sin (psi), cos (psi), sin (theta1) sin (psi) + // and we can choose to have psi in the interval [0 ; PI] + FieldVector3D<T> v1 = applyTo(Vector3D.PLUS_J); + FieldVector3D<T> v2 = applyInverseTo(Vector3D.PLUS_J); + if ((v2.getY().getReal() < -0.9999999999) || (v2.getY().getReal() > 0.9999999999)) { + throw new CardanEulerSingularityException(false); + } + return buildArray(v2.getZ().atan2(v2.getX().negate()), + v2.getY().acos(), + v1.getZ().atan2(v1.getX())); + + } else if (order == RotationOrder.ZXZ) { + + // r (Vector3D.plusK) coordinates are : + // sin (phi) sin (psi2), sin (phi) cos (psi2), cos (phi) + // (-r) (Vector3D.plusK) coordinates are : + // sin (psi1) sin (phi), -cos (psi1) sin (phi), cos (phi) + // and we can choose to have phi in the interval [0 ; PI] + FieldVector3D<T> v1 = applyTo(Vector3D.PLUS_K); + FieldVector3D<T> v2 = applyInverseTo(Vector3D.PLUS_K); + if ((v2.getZ().getReal() < -0.9999999999) || (v2.getZ().getReal() > 0.9999999999)) { + throw new CardanEulerSingularityException(false); + } + return buildArray(v2.getX().atan2(v2.getY().negate()), + v2.getZ().acos(), + v1.getX().atan2(v1.getY())); + + } else { // last possibility is ZYZ + + // r (Vector3D.plusK) coordinates are : + // -sin (theta) cos (psi2), sin (theta) sin (psi2), cos (theta) + // (-r) (Vector3D.plusK) coordinates are : + // cos (psi1) sin (theta), sin (psi1) sin (theta), cos (theta) + // and we can choose to have theta in the interval [0 ; PI] + FieldVector3D<T> v1 = applyTo(Vector3D.PLUS_K); + FieldVector3D<T> v2 = applyInverseTo(Vector3D.PLUS_K); + if ((v2.getZ().getReal() < -0.9999999999) || (v2.getZ().getReal() > 0.9999999999)) { + throw new CardanEulerSingularityException(false); + } + return buildArray(v2.getY().atan2(v2.getX()), + v2.getZ().acos(), + v1.getY().atan2(v1.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. + * <p> + * Calling this method is equivalent to call + * {@link #compose(FieldRotation, RotationConvention) + * compose(r, RotationConvention.VECTOR_OPERATOR)}. + * </p> + * @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 compose(r, RotationConvention.VECTOR_OPERATOR); + } + + /** Compose the instance with another rotation. + * <p> + * If the semantics of the rotations composition corresponds to a + * {@link RotationConvention#VECTOR_OPERATOR vector operator} convention, + * applying the instance to a rotation is computing the composition + * in an order compliant with the following rule : let {@code u} be any + * vector and {@code v} its image by {@code r1} (i.e. + * {@code r1.applyTo(u) = v}). Let {@code w} be the image of {@code v} by + * rotation {@code r2} (i.e. {@code r2.applyTo(v) = w}). Then + * {@code w = comp.applyTo(u)}, where + * {@code comp = r2.compose(r1, RotationConvention.VECTOR_OPERATOR)}. + * </p> + * <p> + * If the semantics of the rotations composition corresponds to a + * {@link RotationConvention#FRAME_TRANSFORM frame transform} convention, + * the application order will be reversed. So keeping the exact same + * meaning of all {@code r1}, {@code r2}, {@code u}, {@code v}, {@code w} + * and {@code comp} as above, {@code comp} could also be computed as + * {@code comp = r1.compose(r2, RotationConvention.FRAME_TRANSFORM)}. + * </p> + * @param r rotation to apply the rotation to + * @param convention convention to use for the semantics of the angle + * @return a new rotation which is the composition of r by the instance + */ + public FieldRotation<T> compose(final FieldRotation<T> r, final RotationConvention convention) { + return convention == RotationConvention.VECTOR_OPERATOR ? + composeInternal(r) : r.composeInternal(this); + } + + /** Compose the instance with another rotation using vector operator convention. + * @param r rotation to apply the rotation to + * @return a new rotation which is the composition of r by the instance + * using vector operator convention + */ + private FieldRotation<T> composeInternal(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. + * <p> + * Calling this method is equivalent to call + * {@link #compose(Rotation, RotationConvention) + * compose(r, RotationConvention.VECTOR_OPERATOR)}. + * </p> + * @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 compose(r, RotationConvention.VECTOR_OPERATOR); + } + + /** Compose the instance with another rotation. + * <p> + * If the semantics of the rotations composition corresponds to a + * {@link RotationConvention#VECTOR_OPERATOR vector operator} convention, + * applying the instance to a rotation is computing the composition + * in an order compliant with the following rule : let {@code u} be any + * vector and {@code v} its image by {@code r1} (i.e. + * {@code r1.applyTo(u) = v}). Let {@code w} be the image of {@code v} by + * rotation {@code r2} (i.e. {@code r2.applyTo(v) = w}). Then + * {@code w = comp.applyTo(u)}, where + * {@code comp = r2.compose(r1, RotationConvention.VECTOR_OPERATOR)}. + * </p> + * <p> + * If the semantics of the rotations composition corresponds to a + * {@link RotationConvention#FRAME_TRANSFORM frame transform} convention, + * the application order will be reversed. So keeping the exact same + * meaning of all {@code r1}, {@code r2}, {@code u}, {@code v}, {@code w} + * and {@code comp} as above, {@code comp} could also be computed as + * {@code comp = r1.compose(r2, RotationConvention.FRAME_TRANSFORM)}. + * </p> + * @param r rotation to apply the rotation to + * @param convention convention to use for the semantics of the angle + * @return a new rotation which is the composition of r by the instance + */ + public FieldRotation<T> compose(final Rotation r, final RotationConvention convention) { + return convention == RotationConvention.VECTOR_OPERATOR ? + composeInternal(r) : applyTo(r, this); + } + + /** Compose the instance with another rotation using vector operator convention. + * @param r rotation to apply the rotation to + * @return a new rotation which is the composition of r by the instance + * using vector operator convention + */ + private FieldRotation<T> composeInternal(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. + * <p> + * Calling this method is equivalent to call + * {@link #composeInverse(FieldRotation, RotationConvention) + * composeInverse(r, RotationConvention.VECTOR_OPERATOR)}. + * </p> + * @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 composeInverse(r, RotationConvention.VECTOR_OPERATOR); + } + + /** Compose the inverse of the instance with another rotation. + * <p> + * If the semantics of the rotations composition corresponds to a + * {@link RotationConvention#VECTOR_OPERATOR vector operator} convention, + * applying the inverse of the instance to a rotation is computing + * the composition in an order compliant with the following rule : + * let {@code u} be any vector and {@code v} its image by {@code r1} + * (i.e. {@code r1.applyTo(u) = v}). Let {@code w} be the inverse image + * of {@code v} by {@code r2} (i.e. {@code r2.applyInverseTo(v) = w}). + * Then {@code w = comp.applyTo(u)}, where + * {@code comp = r2.composeInverse(r1)}. + * </p> + * <p> + * If the semantics of the rotations composition corresponds to a + * {@link RotationConvention#FRAME_TRANSFORM frame transform} convention, + * the application order will be reversed, which means it is the + * <em>innermost</em> rotation that will be reversed. So keeping the exact same + * meaning of all {@code r1}, {@code r2}, {@code u}, {@code v}, {@code w} + * and {@code comp} as above, {@code comp} could also be computed as + * {@code comp = r1.revert().composeInverse(r2.revert(), RotationConvention.FRAME_TRANSFORM)}. + * </p> + * @param r rotation to apply the rotation to + * @param convention convention to use for the semantics of the angle + * @return a new rotation which is the composition of r by the inverse + * of the instance + */ + public FieldRotation<T> composeInverse(final FieldRotation<T> r, final RotationConvention convention) { + return convention == RotationConvention.VECTOR_OPERATOR ? + composeInverseInternal(r) : r.composeInternal(revert()); + } + + /** Compose the inverse of the instance with another rotation + * using vector operator convention. + * @param r rotation to apply the rotation to + * @return a new rotation which is the composition of r by the inverse + * of the instance using vector operator convention + */ + private FieldRotation<T> composeInverseInternal(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. + * <p> + * Calling this method is equivalent to call + * {@link #composeInverse(Rotation, RotationConvention) + * composeInverse(r, RotationConvention.VECTOR_OPERATOR)}. + * </p> + * @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 composeInverse(r, RotationConvention.VECTOR_OPERATOR); + } + + /** Compose the inverse of the instance with another rotation. + * <p> + * If the semantics of the rotations composition corresponds to a + * {@link RotationConvention#VECTOR_OPERATOR vector operator} convention, + * applying the inverse of the instance to a rotation is computing + * the composition in an order compliant with the following rule : + * let {@code u} be any vector and {@code v} its image by {@code r1} + * (i.e. {@code r1.applyTo(u) = v}). Let {@code w} be the inverse image + * of {@code v} by {@code r2} (i.e. {@code r2.applyInverseTo(v) = w}). + * Then {@code w = comp.applyTo(u)}, where + * {@code comp = r2.composeInverse(r1)}. + * </p> + * <p> + * If the semantics of the rotations composition corresponds to a + * {@link RotationConvention#FRAME_TRANSFORM frame transform} convention, + * the application order will be reversed, which means it is the + * <em>innermost</em> rotation that will be reversed. So keeping the exact same + * meaning of all {@code r1}, {@code r2}, {@code u}, {@code v}, {@code w} + * and {@code comp} as above, {@code comp} could also be computed as + * {@code comp = r1.revert().composeInverse(r2.revert(), RotationConvention.FRAME_TRANSFORM)}. + * </p> + * @param r rotation to apply the rotation to + * @param convention convention to use for the semantics of the angle + * @return a new rotation which is the composition of r by the inverse + * of the instance + */ + public FieldRotation<T> composeInverse(final Rotation r, final RotationConvention convention) { + return convention == RotationConvention.VECTOR_OPERATOR ? + composeInverseInternal(r) : applyTo(r, revert()); + } + + /** Compose the inverse of the instance with another rotation + * using vector operator convention. + * @param r rotation to apply the rotation to + * @return a new rotation which is the composition of r by the inverse + * of the instance using vector operator convention + */ + private FieldRotation<T> composeInverseInternal(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.composeInverseInternal(r2).getAngle(); + } + +} |