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+/*
+ * 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.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 implements rotations in a three-dimensional space.
+ *
+ * <p>Rotations can be represented by several different mathematical
+ * entities (matrices, axe and angle, Cardan or Euler angles,
+ * quaternions). This class presents an higher level abstraction, more
+ * user-oriented and hiding this implementation details. Well, for the
+ * curious, we use quaternions for the internal representation. The
+ * user can build a rotation from any of these representations, and
+ * any of these representations can be retrieved from a
+ * <code>Rotation</code> instance (see the various constructors and
+ * getters). In addition, a rotation can also be built implicitly
+ * from a set of vectors and their image.</p>
+ * <p>This implies that this class can be used to convert from one
+ * representation to another one. For example, converting a rotation
+ * matrix into a set of Cardan angles from can be done using the
+ * following single line of code:</p>
+ * <pre>
+ * double[] angles = new Rotation(matrix, 1.0e-10).getAngles(RotationOrder.XYZ);
+ * </pre>
+ * <p>Focus is oriented on what a rotation <em>do</em> rather than on its
+ * underlying representation. Once it has been built, and regardless of its
+ * internal representation, a rotation is an <em>operator</em> which basically
+ * transforms three dimensional {@link Vector3D vectors} into other three
+ * dimensional {@link Vector3D vectors}. Depending on the application, the
+ * meaning of these vectors may vary and the semantics of the rotation also.</p>
+ * <p>For example in an spacecraft attitude simulation tool, users will often
+ * consider the vectors are fixed (say the Earth direction for example) and the
+ * frames change. The rotation transforms the coordinates of the vector in inertial
+ * frame into the coordinates of the same vector in satellite frame. In this
+ * case, the rotation implicitly defines the relation between the two frames.</p>
+ * <p>Another example could be a telescope control application, where the rotation
+ * would transform the sighting direction at rest into the desired observing
+ * direction when the telescope is pointed towards an object of interest. In this
+ * case the rotation transforms the direction at rest in a topocentric frame
+ * into the sighting direction in the same topocentric frame. This implies in this
+ * case the frame is fixed and the vector moves.</p>
+ * <p>In many case, both approaches will be combined. In our telescope example,
+ * we will probably also need to transform the observing direction in the topocentric
+ * frame into the observing direction in inertial frame taking into account the observatory
+ * location and the Earth rotation, which would essentially be an application of the
+ * first approach.</p>
+ *
+ * <p>These examples show that a rotation is what the user wants it to be. This
+ * class does not push the user towards one specific definition and hence does not
+ * provide methods like <code>projectVectorIntoDestinationFrame</code> or
+ * <code>computeTransformedDirection</code>. It provides simpler and more generic
+ * methods: {@link #applyTo(Vector3D) applyTo(Vector3D)} and {@link
+ * #applyInverseTo(Vector3D) applyInverseTo(Vector3D)}.</p>
+ *
+ * <p>Since a rotation is basically a vectorial operator, several rotations can be
+ * composed together and the composite operation <code>r = r<sub>1</sub> o
+ * r<sub>2</sub></code> (which means that for each vector <code>u</code>,
+ * <code>r(u) = r<sub>1</sub>(r<sub>2</sub>(u))</code>) is also a rotation. Hence
+ * we can consider that in addition to vectors, a rotation can be applied to other
+ * rotations as well (or to itself). With our previous notations, we would say we
+ * can apply <code>r<sub>1</sub></code> to <code>r<sub>2</sub></code> and the result
+ * we get is <code>r = r<sub>1</sub> o r<sub>2</sub></code>. For this purpose, the
+ * class provides the methods: {@link #applyTo(Rotation) applyTo(Rotation)} and
+ * {@link #applyInverseTo(Rotation) applyInverseTo(Rotation)}.</p>
+ *
+ * <p>Rotations are guaranteed to be immutable objects.</p>
+ *
+ * @see Vector3D
+ * @see RotationOrder
+ * @since 1.2
+ */
+
+public class Rotation implements Serializable {
+
+ /** Identity rotation. */
+ public static final Rotation IDENTITY = new Rotation(1.0, 0.0, 0.0, 0.0, false);
+
+ /** Serializable version identifier */
+ private static final long serialVersionUID = -2153622329907944313L;
+
+ /** Scalar coordinate of the quaternion. */
+ private final double q0;
+
+ /** First coordinate of the vectorial part of the quaternion. */
+ private final double q1;
+
+ /** Second coordinate of the vectorial part of the quaternion. */
+ private final double q2;
+
+ /** Third coordinate of the vectorial part of the quaternion. */
+ private final double 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 Rotation(double q0, double q1, double q2, double q3,
+ boolean needsNormalization) {
+
+ if (needsNormalization) {
+ // normalization preprocessing
+ double inv = 1.0 / FastMath.sqrt(q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3);
+ q0 *= inv;
+ q1 *= inv;
+ q2 *= inv;
+ q3 *= inv;
+ }
+
+ this.q0 = q0;
+ this.q1 = q1;
+ this.q2 = q2;
+ this.q3 = q3;
+
+ }
+
+ /** Build a rotation from an axis and an angle.
+ * <p>
+ * Calling this constructor is equivalent to call
+ * {@link #Rotation(Vector3D, double, RotationConvention)
+ * new Rotation(axis, angle, RotationConvention.VECTOR_OPERATOR)}
+ * </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 #Rotation(Vector3D, double, RotationConvention)}
+ */
+ @Deprecated
+ public Rotation(Vector3D axis, double angle) throws MathIllegalArgumentException {
+ this(axis, angle, RotationConvention.VECTOR_OPERATOR);
+ }
+
+ /** Build a rotation from an axis and an angle.
+ * @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 Rotation(final Vector3D axis, final double angle, final RotationConvention convention)
+ throws MathIllegalArgumentException {
+
+ double norm = axis.getNorm();
+ if (norm == 0) {
+ throw new MathIllegalArgumentException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_AXIS);
+ }
+
+ double halfAngle = convention == RotationConvention.VECTOR_OPERATOR ? -0.5 * angle : +0.5 * angle;
+ double coeff = FastMath.sin(halfAngle) / norm;
+
+ q0 = FastMath.cos (halfAngle);
+ q1 = coeff * axis.getX();
+ q2 = coeff * axis.getY();
+ q3 = coeff * 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 Rotation(double[][] m, 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
+ double[][] ort = orthogonalizeMatrix(m, threshold);
+
+ // check the sign of the determinant
+ double det = ort[0][0] * (ort[1][1] * ort[2][2] - ort[2][1] * ort[1][2]) -
+ ort[1][0] * (ort[0][1] * ort[2][2] - ort[2][1] * ort[0][2]) +
+ ort[2][0] * (ort[0][1] * ort[1][2] - ort[1][1] * ort[0][2]);
+ if (det < 0.0) {
+ throw new NotARotationMatrixException(
+ LocalizedFormats.CLOSEST_ORTHOGONAL_MATRIX_HAS_NEGATIVE_DETERMINANT,
+ det);
+ }
+
+ double[] 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 (&pm;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 Rotation(Vector3D u1, Vector3D u2, Vector3D v1, Vector3D 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 Vector3D u3 = u1.crossProduct(u2).normalize();
+ u2 = u3.crossProduct(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 Vector3D v3 = v1.crossProduct(v2).normalize();
+ v2 = v3.crossProduct(v1).normalize();
+ v1 = v1.normalize();
+
+ // buid a matrix transforming the first base into the second one
+ final double[][] m = new double[][] {
+ {
+ MathArrays.linearCombination(u1.getX(), v1.getX(), u2.getX(), v2.getX(), u3.getX(), v3.getX()),
+ MathArrays.linearCombination(u1.getY(), v1.getX(), u2.getY(), v2.getX(), u3.getY(), v3.getX()),
+ MathArrays.linearCombination(u1.getZ(), v1.getX(), u2.getZ(), v2.getX(), u3.getZ(), v3.getX())
+ },
+ {
+ MathArrays.linearCombination(u1.getX(), v1.getY(), u2.getX(), v2.getY(), u3.getX(), v3.getY()),
+ MathArrays.linearCombination(u1.getY(), v1.getY(), u2.getY(), v2.getY(), u3.getY(), v3.getY()),
+ MathArrays.linearCombination(u1.getZ(), v1.getY(), u2.getZ(), v2.getY(), u3.getZ(), v3.getY())
+ },
+ {
+ MathArrays.linearCombination(u1.getX(), v1.getZ(), u2.getX(), v2.getZ(), u3.getX(), v3.getZ()),
+ MathArrays.linearCombination(u1.getY(), v1.getZ(), u2.getY(), v2.getZ(), u3.getY(), v3.getZ()),
+ MathArrays.linearCombination(u1.getZ(), v1.getZ(), u2.getZ(), v2.getZ(), u3.getZ(), v3.getZ())
+ }
+ };
+
+ double[] quat = mat2quat(m);
+ 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 Rotation(Vector3D u, Vector3D v) throws MathArithmeticException {
+
+ double normProduct = u.getNorm() * v.getNorm();
+ if (normProduct == 0) {
+ throw new MathArithmeticException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_DEFINING_VECTOR);
+ }
+
+ double dot = u.dotProduct(v);
+
+ if (dot < ((2.0e-15 - 1.0) * normProduct)) {
+ // special case u = -v: we select a PI angle rotation around
+ // an arbitrary vector orthogonal to u
+ Vector3D w = u.orthogonal();
+ q0 = 0.0;
+ q1 = -w.getX();
+ q2 = -w.getY();
+ q3 = -w.getZ();
+ } else {
+ // general case: (u, v) defines a plane, we select
+ // the shortest possible rotation: axis orthogonal to this plane
+ q0 = FastMath.sqrt(0.5 * (1.0 + dot / normProduct));
+ double coeff = 1.0 / (2.0 * q0 * normProduct);
+ Vector3D q = v.crossProduct(u);
+ q1 = coeff * q.getX();
+ q2 = coeff * q.getY();
+ q3 = coeff * q.getZ();
+ }
+
+ }
+
+ /** Build a rotation from three Cardan or Euler elementary rotations.
+
+ * <p>
+ * Calling this constructor is equivalent to call
+ * {@link #Rotation(RotationOrder, RotationConvention, double, double, double)
+ * new Rotation(order, RotationConvention.VECTOR_OPERATOR, alpha1, alpha2, alpha3)}
+ * </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
+ * #Rotation(RotationOrder, RotationConvention, double, double, double)}
+ */
+ @Deprecated
+ public Rotation(RotationOrder order,
+ double alpha1, double alpha2, double 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 Rotation(RotationOrder order, RotationConvention convention,
+ double alpha1, double alpha2, double alpha3) {
+ Rotation r1 = new Rotation(order.getA1(), alpha1, convention);
+ Rotation r2 = new Rotation(order.getA2(), alpha2, convention);
+ Rotation r3 = new Rotation(order.getA3(), alpha3, convention);
+ Rotation 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 static double[] mat2quat(final double[][] ort) {
+
+ final double[] quat = new double[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)
+ double s = ort[0][0] + ort[1][1] + ort[2][2];
+ if (s > -0.19) {
+ // compute q0 and deduce q1, q2 and q3
+ quat[0] = 0.5 * FastMath.sqrt(s + 1.0);
+ double inv = 0.25 / quat[0];
+ quat[1] = inv * (ort[1][2] - ort[2][1]);
+ quat[2] = inv * (ort[2][0] - ort[0][2]);
+ quat[3] = inv * (ort[0][1] - ort[1][0]);
+ } else {
+ s = ort[0][0] - ort[1][1] - ort[2][2];
+ if (s > -0.19) {
+ // compute q1 and deduce q0, q2 and q3
+ quat[1] = 0.5 * FastMath.sqrt(s + 1.0);
+ double inv = 0.25 / quat[1];
+ quat[0] = inv * (ort[1][2] - ort[2][1]);
+ quat[2] = inv * (ort[0][1] + ort[1][0]);
+ quat[3] = inv * (ort[0][2] + ort[2][0]);
+ } else {
+ s = ort[1][1] - ort[0][0] - ort[2][2];
+ if (s > -0.19) {
+ // compute q2 and deduce q0, q1 and q3
+ quat[2] = 0.5 * FastMath.sqrt(s + 1.0);
+ double inv = 0.25 / quat[2];
+ quat[0] = inv * (ort[2][0] - ort[0][2]);
+ quat[1] = inv * (ort[0][1] + ort[1][0]);
+ quat[3] = inv * (ort[2][1] + ort[1][2]);
+ } else {
+ // compute q3 and deduce q0, q1 and q2
+ s = ort[2][2] - ort[0][0] - ort[1][1];
+ quat[3] = 0.5 * FastMath.sqrt(s + 1.0);
+ double inv = 0.25 / quat[3];
+ quat[0] = inv * (ort[0][1] - ort[1][0]);
+ quat[1] = inv * (ort[0][2] + ort[2][0]);
+ quat[2] = inv * (ort[2][1] + 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 Rotation revert() {
+ return new Rotation(-q0, q1, q2, q3, false);
+ }
+
+ /** Get the scalar coordinate of the quaternion.
+ * @return scalar coordinate of the quaternion
+ */
+ public double 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 double 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 double 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 double getQ3() {
+ return q3;
+ }
+
+ /** Get the normalized axis of the rotation.
+ * <p>
+ * Calling this method is equivalent to call
+ * {@link #getAxis(RotationConvention) getAxis(RotationConvention.VECTOR_OPERATOR)}
+ * </p>
+ * @return normalized axis of the rotation
+ * @see #Rotation(Vector3D, double, RotationConvention)
+ * @deprecated as of 3.6, replaced with {@link #getAxis(RotationConvention)}
+ */
+ @Deprecated
+ public Vector3D 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 &pi;, 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 #Rotation(Vector3D, double, RotationConvention)
+ * @since 3.6
+ */
+ public Vector3D getAxis(final RotationConvention convention) {
+ final double squaredSine = q1 * q1 + q2 * q2 + q3 * q3;
+ if (squaredSine == 0) {
+ return convention == RotationConvention.VECTOR_OPERATOR ? Vector3D.PLUS_I : Vector3D.MINUS_I;
+ } else {
+ final double sgn = convention == RotationConvention.VECTOR_OPERATOR ? +1 : -1;
+ if (q0 < 0) {
+ final double inverse = sgn / FastMath.sqrt(squaredSine);
+ return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
+ }
+ final double inverse = -sgn / FastMath.sqrt(squaredSine);
+ return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
+ }
+ }
+
+ /** Get the angle of the rotation.
+ * @return angle of the rotation (between 0 and &pi;)
+ * @see #Rotation(Vector3D, double)
+ */
+ public double getAngle() {
+ if ((q0 < -0.1) || (q0 > 0.1)) {
+ return 2 * FastMath.asin(FastMath.sqrt(q1 * q1 + q2 * q2 + q3 * q3));
+ } else if (q0 < 0) {
+ return 2 * FastMath.acos(-q0);
+ }
+ return 2 * FastMath.acos(q0);
+ }
+
+ /** Get the Cardan or Euler angles corresponding to the instance.
+
+ * <p>
+ * Calling this method is equivalent to call
+ * {@link #getAngles(RotationOrder, RotationConvention)
+ * getAngles(order, RotationConvention.VECTOR_OPERATOR)}
+ * </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 double[] getAngles(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 &pi; + a<sub>1</sub>, &pi;
+ * - a<sub>2</sub> and &pi; + a<sub>3</sub>. This method implements
+ * the following arbitrary choices:</p>
+ * <ul>
+ * <li>for Cardan angles, the chosen set is the one for which the
+ * second angle is between -&pi;/2 and &pi;/2 (i.e its cosine is
+ * positive),</li>
+ * <li>for Euler angles, the chosen set is the one for which the
+ * second angle is between 0 and &pi; (i.e its sine is positive).</li>
+ * </ul>
+
+ * <p>Cardan and Euler angle have a very disappointing drawback: all
+ * of them have singularities. This means that if the instance is
+ * too close to the singularities corresponding to the given
+ * rotation order, it will be impossible to retrieve the angles. For
+ * Cardan angles, this is often called gimbal lock. There is
+ * <em>nothing</em> to do to prevent this, it is an intrinsic problem
+ * with Cardan and Euler representation (but not a problem with the
+ * rotation itself, which is perfectly well defined). For Cardan
+ * angles, singularities occur when the second angle is close to
+ * -&pi;/2 or +&pi;/2, for Euler angle singularities occur when the
+ * second angle is close to 0 or &pi;, this implies that the identity
+ * rotation is always singular for Euler angles!</p>
+
+ * @param order rotation order to use
+ * @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 double[] getAngles(RotationOrder order, RotationConvention convention)
+ throws CardanEulerSingularityException {
+
+ if (convention == RotationConvention.VECTOR_OPERATOR) {
+ if (order == RotationOrder.XYZ) {
+
+ // 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]
+ Vector3D v1 = applyTo(Vector3D.PLUS_K);
+ Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
+ if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
+ throw new CardanEulerSingularityException(true);
+ }
+ return new double[] {
+ FastMath.atan2(-(v1.getY()), v1.getZ()),
+ FastMath.asin(v2.getZ()),
+ FastMath.atan2(-(v2.getY()), v2.getX())
+ };
+
+ } else if (order == RotationOrder.XZY) {
+
+ // 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]
+ Vector3D v1 = applyTo(Vector3D.PLUS_J);
+ Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
+ if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
+ throw new CardanEulerSingularityException(true);
+ }
+ return new double[] {
+ FastMath.atan2(v1.getZ(), v1.getY()),
+ -FastMath.asin(v2.getY()),
+ FastMath.atan2(v2.getZ(), v2.getX())
+ };
+
+ } else if (order == RotationOrder.YXZ) {
+
+ // 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]
+ Vector3D v1 = applyTo(Vector3D.PLUS_K);
+ Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
+ if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
+ throw new CardanEulerSingularityException(true);
+ }
+ return new double[] {
+ FastMath.atan2(v1.getX(), v1.getZ()),
+ -FastMath.asin(v2.getZ()),
+ FastMath.atan2(v2.getX(), v2.getY())
+ };
+
+ } else if (order == RotationOrder.YZX) {
+
+ // 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]
+ Vector3D v1 = applyTo(Vector3D.PLUS_I);
+ Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
+ if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
+ throw new CardanEulerSingularityException(true);
+ }
+ return new double[] {
+ FastMath.atan2(-(v1.getZ()), v1.getX()),
+ FastMath.asin(v2.getX()),
+ FastMath.atan2(-(v2.getZ()), v2.getY())
+ };
+
+ } else if (order == RotationOrder.ZXY) {
+
+ // 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]
+ Vector3D v1 = applyTo(Vector3D.PLUS_J);
+ Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
+ if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
+ throw new CardanEulerSingularityException(true);
+ }
+ return new double[] {
+ FastMath.atan2(-(v1.getX()), v1.getY()),
+ FastMath.asin(v2.getY()),
+ FastMath.atan2(-(v2.getX()), v2.getZ())
+ };
+
+ } else if (order == RotationOrder.ZYX) {
+
+ // 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]
+ Vector3D v1 = applyTo(Vector3D.PLUS_I);
+ Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
+ if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
+ throw new CardanEulerSingularityException(true);
+ }
+ return new double[] {
+ FastMath.atan2(v1.getY(), v1.getX()),
+ -FastMath.asin(v2.getX()),
+ FastMath.atan2(v2.getY(), v2.getZ())
+ };
+
+ } else if (order == RotationOrder.XYX) {
+
+ // r (Vector3D.plusI) coordinates are :
+ // cos (theta), sin (phi1) sin (theta), -cos (phi1) sin (theta)
+ // (-r) (Vector3D.plusI) coordinates are :
+ // cos (theta), sin (theta) sin (phi2), sin (theta) cos (phi2)
+ // and we can choose to have theta in the interval [0 ; PI]
+ Vector3D v1 = applyTo(Vector3D.PLUS_I);
+ Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
+ if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
+ throw new CardanEulerSingularityException(false);
+ }
+ return new double[] {
+ FastMath.atan2(v1.getY(), -v1.getZ()),
+ FastMath.acos(v2.getX()),
+ FastMath.atan2(v2.getY(), v2.getZ())
+ };
+
+ } else if (order == RotationOrder.XZX) {
+
+ // r (Vector3D.plusI) coordinates are :
+ // cos (psi), cos (phi1) sin (psi), sin (phi1) sin (psi)
+ // (-r) (Vector3D.plusI) coordinates are :
+ // cos (psi), -sin (psi) cos (phi2), sin (psi) sin (phi2)
+ // and we can choose to have psi in the interval [0 ; PI]
+ Vector3D v1 = applyTo(Vector3D.PLUS_I);
+ Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
+ if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
+ throw new CardanEulerSingularityException(false);
+ }
+ return new double[] {
+ FastMath.atan2(v1.getZ(), v1.getY()),
+ FastMath.acos(v2.getX()),
+ FastMath.atan2(v2.getZ(), -v2.getY())
+ };
+
+ } else if (order == RotationOrder.YXY) {
+
+ // r (Vector3D.plusJ) coordinates are :
+ // sin (theta1) sin (phi), cos (phi), cos (theta1) sin (phi)
+ // (-r) (Vector3D.plusJ) coordinates are :
+ // sin (phi) sin (theta2), cos (phi), -sin (phi) cos (theta2)
+ // and we can choose to have phi in the interval [0 ; PI]
+ Vector3D v1 = applyTo(Vector3D.PLUS_J);
+ Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
+ if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
+ throw new CardanEulerSingularityException(false);
+ }
+ return new double[] {
+ FastMath.atan2(v1.getX(), v1.getZ()),
+ FastMath.acos(v2.getY()),
+ FastMath.atan2(v2.getX(), -v2.getZ())
+ };
+
+ } else if (order == RotationOrder.YZY) {
+
+ // r (Vector3D.plusJ) coordinates are :
+ // -cos (theta1) sin (psi), cos (psi), sin (theta1) sin (psi)
+ // (-r) (Vector3D.plusJ) coordinates are :
+ // sin (psi) cos (theta2), cos (psi), sin (psi) sin (theta2)
+ // and we can choose to have psi in the interval [0 ; PI]
+ Vector3D v1 = applyTo(Vector3D.PLUS_J);
+ Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
+ if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
+ throw new CardanEulerSingularityException(false);
+ }
+ return new double[] {
+ FastMath.atan2(v1.getZ(), -v1.getX()),
+ FastMath.acos(v2.getY()),
+ FastMath.atan2(v2.getZ(), v2.getX())
+ };
+
+ } else if (order == RotationOrder.ZXZ) {
+
+ // r (Vector3D.plusK) coordinates are :
+ // sin (psi1) sin (phi), -cos (psi1) sin (phi), cos (phi)
+ // (-r) (Vector3D.plusK) coordinates are :
+ // sin (phi) sin (psi2), sin (phi) cos (psi2), cos (phi)
+ // and we can choose to have phi in the interval [0 ; PI]
+ Vector3D v1 = applyTo(Vector3D.PLUS_K);
+ Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
+ if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
+ throw new CardanEulerSingularityException(false);
+ }
+ return new double[] {
+ FastMath.atan2(v1.getX(), -v1.getY()),
+ FastMath.acos(v2.getZ()),
+ FastMath.atan2(v2.getX(), v2.getY())
+ };
+
+ } else { // last possibility is ZYZ
+
+ // r (Vector3D.plusK) coordinates are :
+ // cos (psi1) sin (theta), sin (psi1) sin (theta), cos (theta)
+ // (-r) (Vector3D.plusK) coordinates are :
+ // -sin (theta) cos (psi2), sin (theta) sin (psi2), cos (theta)
+ // and we can choose to have theta in the interval [0 ; PI]
+ Vector3D v1 = applyTo(Vector3D.PLUS_K);
+ Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
+ if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
+ throw new CardanEulerSingularityException(false);
+ }
+ return new double[] {
+ FastMath.atan2(v1.getY(), v1.getX()),
+ FastMath.acos(v2.getZ()),
+ FastMath.atan2(v2.getY(), -v2.getX())
+ };
+
+ }
+ } 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]
+ Vector3D v1 = applyTo(Vector3D.PLUS_I);
+ Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
+ if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
+ throw new CardanEulerSingularityException(true);
+ }
+ return new double[] {
+ FastMath.atan2(-v2.getY(), v2.getZ()),
+ FastMath.asin(v2.getX()),
+ FastMath.atan2(-v1.getY(), 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]
+ Vector3D v1 = applyTo(Vector3D.PLUS_I);
+ Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
+ if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
+ throw new CardanEulerSingularityException(true);
+ }
+ return new double[] {
+ FastMath.atan2(v2.getZ(), v2.getY()),
+ -FastMath.asin(v2.getX()),
+ FastMath.atan2(v1.getZ(), 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]
+ Vector3D v1 = applyTo(Vector3D.PLUS_J);
+ Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
+ if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
+ throw new CardanEulerSingularityException(true);
+ }
+ return new double[] {
+ FastMath.atan2(v2.getX(), v2.getZ()),
+ -FastMath.asin(v2.getY()),
+ FastMath.atan2(v1.getX(), 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]
+ Vector3D v1 = applyTo(Vector3D.PLUS_J);
+ Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
+ if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
+ throw new CardanEulerSingularityException(true);
+ }
+ return new double[] {
+ FastMath.atan2(-v2.getZ(), v2.getX()),
+ FastMath.asin(v2.getY()),
+ FastMath.atan2(-v1.getZ(), 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]
+ Vector3D v1 = applyTo(Vector3D.PLUS_K);
+ Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
+ if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
+ throw new CardanEulerSingularityException(true);
+ }
+ return new double[] {
+ FastMath.atan2(-v2.getX(), v2.getY()),
+ FastMath.asin(v2.getZ()),
+ FastMath.atan2(-v1.getX(), 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]
+ Vector3D v1 = applyTo(Vector3D.PLUS_K);
+ Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
+ if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
+ throw new CardanEulerSingularityException(true);
+ }
+ return new double[] {
+ FastMath.atan2(v2.getY(), v2.getX()),
+ -FastMath.asin(v2.getZ()),
+ FastMath.atan2(v1.getY(), 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]
+ Vector3D v1 = applyTo(Vector3D.PLUS_I);
+ Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
+ if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
+ throw new CardanEulerSingularityException(false);
+ }
+ return new double[] {
+ FastMath.atan2(v2.getY(), -v2.getZ()),
+ FastMath.acos(v2.getX()),
+ FastMath.atan2(v1.getY(), 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]
+ Vector3D v1 = applyTo(Vector3D.PLUS_I);
+ Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
+ if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
+ throw new CardanEulerSingularityException(false);
+ }
+ return new double[] {
+ FastMath.atan2(v2.getZ(), v2.getY()),
+ FastMath.acos(v2.getX()),
+ FastMath.atan2(v1.getZ(), -v1.getY())
+ };
+
+ } 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]
+ Vector3D v1 = applyTo(Vector3D.PLUS_J);
+ Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
+ if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
+ throw new CardanEulerSingularityException(false);
+ }
+ return new double[] {
+ FastMath.atan2(v2.getX(), v2.getZ()),
+ FastMath.acos(v2.getY()),
+ FastMath.atan2(v1.getX(), -v1.getZ())
+ };
+
+ } 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]
+ Vector3D v1 = applyTo(Vector3D.PLUS_J);
+ Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
+ if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
+ throw new CardanEulerSingularityException(false);
+ }
+ return new double[] {
+ FastMath.atan2(v2.getZ(), -v2.getX()),
+ FastMath.acos(v2.getY()),
+ FastMath.atan2(v1.getZ(), 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]
+ Vector3D v1 = applyTo(Vector3D.PLUS_K);
+ Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
+ if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
+ throw new CardanEulerSingularityException(false);
+ }
+ return new double[] {
+ FastMath.atan2(v2.getX(), -v2.getY()),
+ FastMath.acos(v2.getZ()),
+ FastMath.atan2(v1.getX(), 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]
+ Vector3D v1 = applyTo(Vector3D.PLUS_K);
+ Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
+ if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
+ throw new CardanEulerSingularityException(false);
+ }
+ return new double[] {
+ FastMath.atan2(v2.getY(), v2.getX()),
+ FastMath.acos(v2.getZ()),
+ FastMath.atan2(v1.getY(), -v1.getX())
+ };
+
+ }
+ }
+
+ }
+
+ /** Get the 3X3 matrix corresponding to the instance
+ * @return the matrix corresponding to the instance
+ */
+ public double[][] getMatrix() {
+
+ // products
+ double q0q0 = q0 * q0;
+ double q0q1 = q0 * q1;
+ double q0q2 = q0 * q2;
+ double q0q3 = q0 * q3;
+ double q1q1 = q1 * q1;
+ double q1q2 = q1 * q2;
+ double q1q3 = q1 * q3;
+ double q2q2 = q2 * q2;
+ double q2q3 = q2 * q3;
+ double q3q3 = q3 * q3;
+
+ // create the matrix
+ double[][] m = new double[3][];
+ m[0] = new double[3];
+ m[1] = new double[3];
+ m[2] = new double[3];
+
+ m [0][0] = 2.0 * (q0q0 + q1q1) - 1.0;
+ m [1][0] = 2.0 * (q1q2 - q0q3);
+ m [2][0] = 2.0 * (q1q3 + q0q2);
+
+ m [0][1] = 2.0 * (q1q2 + q0q3);
+ m [1][1] = 2.0 * (q0q0 + q2q2) - 1.0;
+ m [2][1] = 2.0 * (q2q3 - q0q1);
+
+ m [0][2] = 2.0 * (q1q3 - q0q2);
+ m [1][2] = 2.0 * (q2q3 + q0q1);
+ m [2][2] = 2.0 * (q0q0 + q3q3) - 1.0;
+
+ return m;
+
+ }
+
+ /** 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 Vector3D applyTo(Vector3D u) {
+
+ double x = u.getX();
+ double y = u.getY();
+ double z = u.getZ();
+
+ double s = q1 * x + q2 * y + q3 * z;
+
+ return new Vector3D(2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x,
+ 2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y,
+ 2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - 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 double[] in, final double[] out) {
+
+ final double x = in[0];
+ final double y = in[1];
+ final double z = in[2];
+
+ final double s = q1 * x + q2 * y + q3 * z;
+
+ out[0] = 2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x;
+ out[1] = 2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y;
+ out[2] = 2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - 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 Vector3D applyInverseTo(Vector3D u) {
+
+ double x = u.getX();
+ double y = u.getY();
+ double z = u.getZ();
+
+ double s = q1 * x + q2 * y + q3 * z;
+ double m0 = -q0;
+
+ return new Vector3D(2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x,
+ 2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y,
+ 2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - 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 double[] in, final double[] out) {
+
+ final double x = in[0];
+ final double y = in[1];
+ final double z = in[2];
+
+ final double s = q1 * x + q2 * y + q3 * z;
+ final double m0 = -q0;
+
+ out[0] = 2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x;
+ out[1] = 2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y;
+ out[2] = 2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - z;
+
+ }
+
+ /** 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 Rotation applyTo(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 Rotation compose(final Rotation 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 Rotation composeInternal(final Rotation r) {
+ return new Rotation(r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),
+ r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),
+ r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),
+ r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),
+ 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 Rotation applyInverseTo(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 Rotation composeInverse(final Rotation 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 Rotation composeInverseInternal(Rotation r) {
+ return new Rotation(-r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),
+ -r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),
+ -r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),
+ -r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),
+ 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 double[][] orthogonalizeMatrix(double[][] m, double threshold)
+ throws NotARotationMatrixException {
+ double[] m0 = m[0];
+ double[] m1 = m[1];
+ double[] m2 = m[2];
+ double x00 = m0[0];
+ double x01 = m0[1];
+ double x02 = m0[2];
+ double x10 = m1[0];
+ double x11 = m1[1];
+ double x12 = m1[2];
+ double x20 = m2[0];
+ double x21 = m2[1];
+ double x22 = m2[2];
+ double fn = 0;
+ double fn1;
+
+ double[][] o = new double[3][3];
+ double[] o0 = o[0];
+ double[] o1 = o[1];
+ double[] o2 = o[2];
+
+ // iterative correction: Xn+1 = Xn - 0.5 * (Xn.Mt.Xn - M)
+ int i = 0;
+ while (++i < 11) {
+
+ // Mt.Xn
+ double mx00 = m0[0] * x00 + m1[0] * x10 + m2[0] * x20;
+ double mx10 = m0[1] * x00 + m1[1] * x10 + m2[1] * x20;
+ double mx20 = m0[2] * x00 + m1[2] * x10 + m2[2] * x20;
+ double mx01 = m0[0] * x01 + m1[0] * x11 + m2[0] * x21;
+ double mx11 = m0[1] * x01 + m1[1] * x11 + m2[1] * x21;
+ double mx21 = m0[2] * x01 + m1[2] * x11 + m2[2] * x21;
+ double mx02 = m0[0] * x02 + m1[0] * x12 + m2[0] * x22;
+ double mx12 = m0[1] * x02 + m1[1] * x12 + m2[1] * x22;
+ double mx22 = m0[2] * x02 + m1[2] * x12 + m2[2] * x22;
+
+ // Xn+1
+ o0[0] = x00 - 0.5 * (x00 * mx00 + x01 * mx10 + x02 * mx20 - m0[0]);
+ o0[1] = x01 - 0.5 * (x00 * mx01 + x01 * mx11 + x02 * mx21 - m0[1]);
+ o0[2] = x02 - 0.5 * (x00 * mx02 + x01 * mx12 + x02 * mx22 - m0[2]);
+ o1[0] = x10 - 0.5 * (x10 * mx00 + x11 * mx10 + x12 * mx20 - m1[0]);
+ o1[1] = x11 - 0.5 * (x10 * mx01 + x11 * mx11 + x12 * mx21 - m1[1]);
+ o1[2] = x12 - 0.5 * (x10 * mx02 + x11 * mx12 + x12 * mx22 - m1[2]);
+ o2[0] = x20 - 0.5 * (x20 * mx00 + x21 * mx10 + x22 * mx20 - m2[0]);
+ o2[1] = x21 - 0.5 * (x20 * mx01 + x21 * mx11 + x22 * mx21 - m2[1]);
+ o2[2] = x22 - 0.5 * (x20 * mx02 + x21 * mx12 + x22 * mx22 - m2[2]);
+
+ // correction on each elements
+ double corr00 = o0[0] - m0[0];
+ double corr01 = o0[1] - m0[1];
+ double corr02 = o0[2] - m0[2];
+ double corr10 = o1[0] - m1[0];
+ double corr11 = o1[1] - m1[1];
+ double corr12 = o1[2] - m1[2];
+ double corr20 = o2[0] - m2[0];
+ double corr21 = o2[1] - m2[1];
+ double corr22 = o2[2] - m2[2];
+
+ // 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 = o0[0];
+ x01 = o0[1];
+ x02 = o0[2];
+ x10 = o1[0];
+ x11 = o1[1];
+ x12 = o1[2];
+ x20 = o2[0];
+ x21 = o2[1];
+ x22 = o2[2];
+ fn = fn1;
+
+ }
+
+ // the algorithm did not converge after 10 iterations
+ throw new NotARotationMatrixException(
+ LocalizedFormats.UNABLE_TO_ORTHOGONOLIZE_MATRIX,
+ i - 1);
+ }
+
+ /** Compute the <i>distance</i> between two rotations.
+ * <p>The <i>distance</i> is intended here as a way to check if two
+ * rotations are almost similar (i.e. they transform vectors the same way)
+ * or very different. It is mathematically defined as the angle of
+ * the rotation r that prepended to one of the rotations gives the other
+ * one:</p>
+ * <pre>
+ * r<sub>1</sub>(r) = r<sub>2</sub>
+ * </pre>
+ * <p>This distance is an angle between 0 and &pi;. Its value is the smallest
+ * possible upper bound of the angle in radians between r<sub>1</sub>(v)
+ * and r<sub>2</sub>(v) for all possible vectors v. This upper bound is
+ * reached for some v. The distance is equal to 0 if and only if the two
+ * rotations are identical.</p>
+ * <p>Comparing two rotations should always be done using this value rather
+ * than for example comparing the components of the quaternions. It is much
+ * more stable, and has a geometric meaning. Also comparing quaternions
+ * components is error prone since for example quaternions (0.36, 0.48, -0.48, -0.64)
+ * and (-0.36, -0.48, 0.48, 0.64) represent exactly the same rotation despite
+ * their components are different (they are exact opposites).</p>
+ * @param r1 first rotation
+ * @param r2 second rotation
+ * @return <i>distance</i> between r1 and r2
+ */
+ public static double distance(Rotation r1, Rotation r2) {
+ return r1.composeInverseInternal(r2).getAngle();
+ }
+
+}
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