| /* |
| * 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.math.geometry; |
| |
| import java.io.Serializable; |
| |
| import org.apache.commons.math.MathRuntimeException; |
| import org.apache.commons.math.exception.util.LocalizedFormats; |
| import org.apache.commons.math.util.FastMath; |
| |
| /** |
| * 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> |
| * |
| * @version $Revision: 1067500 $ $Date: 2011-02-05 21:11:30 +0100 (sam. 05 févr. 2011) $ |
| * @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>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(Vector3D) 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 ArithmeticException if the axis norm is zero |
| */ |
| public Rotation(Vector3D axis, double angle) { |
| |
| double norm = axis.getNorm(); |
| if (norm == 0) { |
| throw MathRuntimeException.createArithmeticException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_AXIS); |
| } |
| |
| double halfAngle = -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); |
| } |
| |
| // 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 |
| q0 = 0.5 * FastMath.sqrt(s + 1.0); |
| double inv = 0.25 / q0; |
| q1 = inv * (ort[1][2] - ort[2][1]); |
| q2 = inv * (ort[2][0] - ort[0][2]); |
| q3 = 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 |
| q1 = 0.5 * FastMath.sqrt(s + 1.0); |
| double inv = 0.25 / q1; |
| q0 = inv * (ort[1][2] - ort[2][1]); |
| q2 = inv * (ort[0][1] + ort[1][0]); |
| q3 = 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 |
| q2 = 0.5 * FastMath.sqrt(s + 1.0); |
| double inv = 0.25 / q2; |
| q0 = inv * (ort[2][0] - ort[0][2]); |
| q1 = inv * (ort[0][1] + ort[1][0]); |
| q3 = 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]; |
| q3 = 0.5 * FastMath.sqrt(s + 1.0); |
| double inv = 0.25 / q3; |
| q0 = inv * (ort[0][1] - ort[1][0]); |
| q1 = inv * (ort[0][2] + ort[2][0]); |
| q2 = inv * (ort[2][1] + ort[1][2]); |
| } |
| } |
| } |
| |
| } |
| |
| /** Build the rotation that transforms a pair of vector into another pair. |
| |
| * <p>Except for possible scale factors, if the instance were applied to |
| * the pair (u<sub>1</sub>, u<sub>2</sub>) it will produce the pair |
| * (v<sub>1</sub>, v<sub>2</sub>).</p> |
| * |
| * <p>If the angular separation between u<sub>1</sub> and u<sub>2</sub> is |
| * not the same as the angular separation between v<sub>1</sub> and |
| * v<sub>2</sub>, then a corrected v'<sub>2</sub> will be used rather than |
| * v<sub>2</sub>, the corrected vector will be in the (v<sub>1</sub>, |
| * v<sub>2</sub>) plane.</p> |
| * |
| * @param u1 first vector of the origin pair |
| * @param u2 second vector of the origin pair |
| * @param v1 desired image of u1 by the rotation |
| * @param v2 desired image of u2 by the rotation |
| * @exception IllegalArgumentException if the norm of one of the vectors is zero |
| */ |
| public Rotation(Vector3D u1, Vector3D u2, Vector3D v1, Vector3D v2) { |
| |
| // norms computation |
| double u1u1 = Vector3D.dotProduct(u1, u1); |
| double u2u2 = Vector3D.dotProduct(u2, u2); |
| double v1v1 = Vector3D.dotProduct(v1, v1); |
| double v2v2 = Vector3D.dotProduct(v2, v2); |
| if ((u1u1 == 0) || (u2u2 == 0) || (v1v1 == 0) || (v2v2 == 0)) { |
| throw MathRuntimeException.createIllegalArgumentException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_DEFINING_VECTOR); |
| } |
| |
| double u1x = u1.getX(); |
| double u1y = u1.getY(); |
| double u1z = u1.getZ(); |
| |
| double u2x = u2.getX(); |
| double u2y = u2.getY(); |
| double u2z = u2.getZ(); |
| |
| // normalize v1 in order to have (v1'|v1') = (u1|u1) |
| double coeff = FastMath.sqrt (u1u1 / v1v1); |
| double v1x = coeff * v1.getX(); |
| double v1y = coeff * v1.getY(); |
| double v1z = coeff * v1.getZ(); |
| v1 = new Vector3D(v1x, v1y, v1z); |
| |
| // adjust v2 in order to have (u1|u2) = (v1|v2) and (v2'|v2') = (u2|u2) |
| double u1u2 = Vector3D.dotProduct(u1, u2); |
| double v1v2 = Vector3D.dotProduct(v1, v2); |
| double coeffU = u1u2 / u1u1; |
| double coeffV = v1v2 / u1u1; |
| double beta = FastMath.sqrt((u2u2 - u1u2 * coeffU) / (v2v2 - v1v2 * coeffV)); |
| double alpha = coeffU - beta * coeffV; |
| double v2x = alpha * v1x + beta * v2.getX(); |
| double v2y = alpha * v1y + beta * v2.getY(); |
| double v2z = alpha * v1z + beta * v2.getZ(); |
| v2 = new Vector3D(v2x, v2y, v2z); |
| |
| // preliminary computation (we use explicit formulation instead |
| // of relying on the Vector3D class in order to avoid building lots |
| // of temporary objects) |
| Vector3D uRef = u1; |
| Vector3D vRef = v1; |
| double dx1 = v1x - u1.getX(); |
| double dy1 = v1y - u1.getY(); |
| double dz1 = v1z - u1.getZ(); |
| double dx2 = v2x - u2.getX(); |
| double dy2 = v2y - u2.getY(); |
| double dz2 = v2z - u2.getZ(); |
| Vector3D k = new Vector3D(dy1 * dz2 - dz1 * dy2, |
| dz1 * dx2 - dx1 * dz2, |
| dx1 * dy2 - dy1 * dx2); |
| double c = k.getX() * (u1y * u2z - u1z * u2y) + |
| k.getY() * (u1z * u2x - u1x * u2z) + |
| k.getZ() * (u1x * u2y - u1y * u2x); |
| |
| if (c == 0) { |
| // the (q1, q2, q3) vector is in the (u1, u2) plane |
| // we try other vectors |
| Vector3D u3 = Vector3D.crossProduct(u1, u2); |
| Vector3D v3 = Vector3D.crossProduct(v1, v2); |
| double u3x = u3.getX(); |
| double u3y = u3.getY(); |
| double u3z = u3.getZ(); |
| double v3x = v3.getX(); |
| double v3y = v3.getY(); |
| double v3z = v3.getZ(); |
| |
| double dx3 = v3x - u3x; |
| double dy3 = v3y - u3y; |
| double dz3 = v3z - u3z; |
| k = new Vector3D(dy1 * dz3 - dz1 * dy3, |
| dz1 * dx3 - dx1 * dz3, |
| dx1 * dy3 - dy1 * dx3); |
| c = k.getX() * (u1y * u3z - u1z * u3y) + |
| k.getY() * (u1z * u3x - u1x * u3z) + |
| k.getZ() * (u1x * u3y - u1y * u3x); |
| |
| if (c == 0) { |
| // the (q1, q2, q3) vector is aligned with u1: |
| // we try (u2, u3) and (v2, v3) |
| k = new Vector3D(dy2 * dz3 - dz2 * dy3, |
| dz2 * dx3 - dx2 * dz3, |
| dx2 * dy3 - dy2 * dx3); |
| c = k.getX() * (u2y * u3z - u2z * u3y) + |
| k.getY() * (u2z * u3x - u2x * u3z) + |
| k.getZ() * (u2x * u3y - u2y * u3x); |
| |
| if (c == 0) { |
| // the (q1, q2, q3) vector is aligned with everything |
| // this is really the identity rotation |
| q0 = 1.0; |
| q1 = 0.0; |
| q2 = 0.0; |
| q3 = 0.0; |
| return; |
| } |
| |
| // we will have to use u2 and v2 to compute the scalar part |
| uRef = u2; |
| vRef = v2; |
| |
| } |
| |
| } |
| |
| // compute the vectorial part |
| c = FastMath.sqrt(c); |
| double inv = 1.0 / (c + c); |
| q1 = inv * k.getX(); |
| q2 = inv * k.getY(); |
| q3 = inv * k.getZ(); |
| |
| // compute the scalar part |
| k = new Vector3D(uRef.getY() * q3 - uRef.getZ() * q2, |
| uRef.getZ() * q1 - uRef.getX() * q3, |
| uRef.getX() * q2 - uRef.getY() * q1); |
| c = Vector3D.dotProduct(k, k); |
| q0 = Vector3D.dotProduct(vRef, k) / (c + c); |
| |
| } |
| |
| /** Build one of the rotations that transform one vector into another one. |
| |
| * <p>Except for a possible scale factor, if the instance were |
| * applied to the vector u it will produce the vector v. There is an |
| * infinite number of such rotations, this constructor choose the |
| * one with the smallest associated angle (i.e. the one whose axis |
| * is orthogonal to the (u, v) plane). If u and v are colinear, an |
| * arbitrary rotation axis is chosen.</p> |
| * |
| * @param u origin vector |
| * @param v desired image of u by the rotation |
| * @exception IllegalArgumentException if the norm of one of the vectors is zero |
| */ |
| public Rotation(Vector3D u, Vector3D v) { |
| |
| double normProduct = u.getNorm() * v.getNorm(); |
| if (normProduct == 0) { |
| throw MathRuntimeException.createIllegalArgumentException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_DEFINING_VECTOR); |
| } |
| |
| double dot = Vector3D.dotProduct(u, 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); |
| q1 = coeff * (v.getY() * u.getZ() - v.getZ() * u.getY()); |
| q2 = coeff * (v.getZ() * u.getX() - v.getX() * u.getZ()); |
| q3 = coeff * (v.getX() * u.getY() - v.getY() * u.getX()); |
| } |
| |
| } |
| |
| /** Build a rotation from three Cardan or Euler elementary rotations. |
| |
| * <p>Cardan rotations are three successive rotations around the |
| * canonical axes X, Y and Z, each axis being used once. There are |
| * 6 such sets of rotations (XYZ, XZY, YXZ, YZX, ZXY and ZYX). Euler |
| * rotations are three successive rotations around the canonical |
| * axes X, Y and Z, the first and last rotations being around the |
| * same axis. There are 6 such sets of rotations (XYX, XZX, YXY, |
| * YZY, ZXZ and ZYZ), the most popular one being ZXZ.</p> |
| * <p>Beware that many people routinely use the term Euler angles even |
| * for what really are Cardan angles (this confusion is especially |
| * widespread in the aerospace business where Roll, Pitch and Yaw angles |
| * are often wrongly tagged as Euler angles).</p> |
| * |
| * @param order order of rotations to use |
| * @param alpha1 angle of the first elementary rotation |
| * @param alpha2 angle of the second elementary rotation |
| * @param alpha3 angle of the third elementary rotation |
| */ |
| public Rotation(RotationOrder order, |
| double alpha1, double alpha2, double alpha3) { |
| Rotation r1 = new Rotation(order.getA1(), alpha1); |
| Rotation r2 = new Rotation(order.getA2(), alpha2); |
| Rotation r3 = new Rotation(order.getA3(), alpha3); |
| Rotation composed = r1.applyTo(r2.applyTo(r3)); |
| q0 = composed.q0; |
| q1 = composed.q1; |
| q2 = composed.q2; |
| q3 = composed.q3; |
| } |
| |
| /** 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. |
| * @return normalized axis of the rotation |
| * @see #Rotation(Vector3D, double) |
| */ |
| public Vector3D getAxis() { |
| double squaredSine = q1 * q1 + q2 * q2 + q3 * q3; |
| if (squaredSine == 0) { |
| return new Vector3D(1, 0, 0); |
| } else if (q0 < 0) { |
| double inverse = 1 / FastMath.sqrt(squaredSine); |
| return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse); |
| } |
| double inverse = -1 / 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 π) |
| * @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>The equations show that each rotation can be defined by two |
| * different values of the Cardan or Euler angles set. For example |
| * if Cardan angles are used, the rotation defined by the angles |
| * a<sub>1</sub>, a<sub>2</sub> and a<sub>3</sub> is the same as |
| * the rotation defined by the angles π + a<sub>1</sub>, π |
| * - a<sub>2</sub> and π + a<sub>3</sub>. This method implements |
| * the following arbitrary choices:</p> |
| * <ul> |
| * <li>for Cardan angles, the chosen set is the one for which the |
| * second angle is between -π/2 and π/2 (i.e its cosine is |
| * positive),</li> |
| * <li>for Euler angles, the chosen set is the one for which the |
| * second angle is between 0 and π (i.e its sine is positive).</li> |
| * </ul> |
| * |
| * <p>Cardan and Euler angle have a very disappointing drawback: all |
| * of them have singularities. This means that if the instance is |
| * too close to the singularities corresponding to the given |
| * rotation order, it will be impossible to retrieve the angles. For |
| * Cardan angles, this is often called gimbal lock. There is |
| * <em>nothing</em> to do to prevent this, it is an intrinsic problem |
| * with Cardan and Euler representation (but not a problem with the |
| * rotation itself, which is perfectly well defined). For Cardan |
| * angles, singularities occur when the second angle is close to |
| * -π/2 or +π/2, for Euler angle singularities occur when the |
| * second angle is close to 0 or π, this implies that the identity |
| * rotation is always singular for Euler angles!</p> |
| * |
| * @param order rotation order to use |
| * @return an array of three angles, in the order specified by the set |
| * @exception CardanEulerSingularityException if the rotation is |
| * singular with respect to the angles set specified |
| */ |
| public double[] getAngles(RotationOrder order) |
| throws CardanEulerSingularityException { |
| |
| 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()) |
| }; |
| |
| } |
| |
| } |
| |
| /** 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 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 instance to another rotation. |
| * Applying the instance to a rotation is computing the composition |
| * in an order compliant with the following rule : let u be any |
| * vector and v its image by r (i.e. r.applyTo(u) = v), let w be the image |
| * of v by the instance (i.e. applyTo(v) = w), then w = comp.applyTo(u), |
| * where comp = applyTo(r). |
| * @param r rotation to apply the rotation to |
| * @return a new rotation which is the composition of r by the instance |
| */ |
| public Rotation applyTo(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. |
| * Applying the inverse of the instance to a rotation is computing |
| * the composition in an order compliant with the following rule : |
| * let u be any vector and v its image by r (i.e. r.applyTo(u) = v), |
| * let w be the inverse image of v by the instance |
| * (i.e. applyInverseTo(v) = w), then w = comp.applyTo(u), where |
| * comp = applyInverseTo(r). |
| * @param r rotation to apply the rotation to |
| * @return a new rotation which is the composition of r by the inverse |
| * of the instance |
| */ |
| public Rotation applyInverseTo(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 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 |
| * @return <i>distance</i> between r1 and r2 |
| */ |
| public static double distance(Rotation r1, Rotation r2) { |
| return r1.applyInverseTo(r2).getAngle(); |
| } |
| |
| } |