2
* Licensed to the Apache Software Foundation (ASF) under one or more
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* contributor license agreements. See the NOTICE file distributed with
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* this work for additional information regarding copyright ownership.
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* The ASF licenses this file to You under the Apache License, Version 2.0
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* (the "License"); you may not use this file except in compliance with
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* the License. You may obtain a copy of the License at
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* http://www.apache.org/licenses/LICENSE-2.0
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS,
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* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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* See the License for the specific language governing permissions and
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* limitations under the License.
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package org.apache.commons.math.geometry;
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import java.io.Serializable;
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* This class implements rotations in a three-dimensional space.
25
* <p>Rotations can be represented by several different mathematical
26
* entities (matrices, axe and angle, Cardan or Euler angles,
27
* quaternions). This class presents an higher level abstraction, more
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* user-oriented and hiding this implementation details. Well, for the
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* curious, we use quaternions for the internal representation. The
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* user can build a rotation from any of these representations, and
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* any of these representations can be retrieved from a
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* <code>Rotation</code> instance (see the various constructors and
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* getters). In addition, a rotation can also be built implicitely
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* from a set of vectors and their image.</p>
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* <p>This implies that this class can be used to convert from one
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* representation to another one. For example, converting a rotation
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* matrix into a set of Cardan angles from can be done using the
38
* followong single line of code:</p>
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* double[] angles = new Rotation(matrix, 1.0e-10).getAngles(RotationOrder.XYZ);
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* <p>Focus is oriented on what a rotation <em>do</em> rather than on its
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* underlying representation. Once it has been built, and regardless of its
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* internal representation, a rotation is an <em>operator</em> which basically
45
* transforms three dimensional {@link Vector3D vectors} into other three
46
* dimensional {@link Vector3D vectors}. Depending on the application, the
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* meaning of these vectors may vary and the semantics of the rotation also.</p>
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* <p>For example in an spacecraft attitude simulation tool, users will often
49
* consider the vectors are fixed (say the Earth direction for example) and the
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* rotation transforms the coordinates coordinates of this vector in inertial
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* frame into the coordinates of the same vector in satellite frame. In this
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* case, the rotation implicitely defines the relation between the two frames.
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* Another example could be a telescope control application, where the rotation
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* would transform the sighting direction at rest into the desired observing
55
* direction when the telescope is pointed towards an object of interest. In this
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* case the rotation transforms the directionf at rest in a topocentric frame
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* into the sighting direction in the same topocentric frame. In many case, both
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* approaches will be combined, in our telescope example, we will probably also
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* need to transform the observing direction in the topocentric frame into the
60
* observing direction in inertial frame taking into account the observatory
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* location and the Earth rotation.</p>
63
* <p>These examples show that a rotation is what the user wants it to be, so this
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* class does not push the user towards one specific definition and hence does not
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* provide methods like <code>projectVectorIntoDestinationFrame</code> or
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* <code>computeTransformedDirection</code>. It provides simpler and more generic
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* methods: {@link #applyTo(Vector3D) applyTo(Vector3D)} and {@link
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* #applyInverseTo(Vector3D) applyInverseTo(Vector3D)}.</p>
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* <p>Since a rotation is basically a vectorial operator, several rotations can be
71
* composed together and the composite operation <code>r = r<sub>1</sub> o
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* r<sub>2</sub></code> (which means that for each vector <code>u</code>,
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* <code>r(u) = r<sub>1</sub>(r<sub>2</sub>(u))</code>) is also a rotation. Hence
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* we can consider that in addition to vectors, a rotation can be applied to other
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* rotations as well (or to itself). With our previous notations, we would say we
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* can apply <code>r<sub>1</sub></code> to <code>r<sub>2</sub></code> and the result
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* we get is <code>r = r<sub>1</sub> o r<sub>2</sub></code>. For this purpose, the
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* class provides the methods: {@link #applyTo(Rotation) applyTo(Rotation)} and
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* {@link #applyInverseTo(Rotation) applyInverseTo(Rotation)}.</p>
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* <p>Rotations are guaranteed to be immutable objects.</p>
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* @version $Revision: 627994 $ $Date: 2008-02-15 03:16:05 -0700 (Fri, 15 Feb 2008) $
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public class Rotation implements Serializable {
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/** Build the identity rotation.
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/** Build a rotation from the quaternion coordinates.
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* <p>A rotation can be built from a <em>normalized</em> quaternion,
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* i.e. a quaternion for which q<sub>0</sub><sup>2</sup> +
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* q<sub>1</sub><sup>2</sup> + q<sub>2</sub><sup>2</sup> +
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* q<sub>3</sub><sup>2</sup> = 1. If the quaternion is not normalized,
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* the constructor can normalize it in a preprocessing step.</p>
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* @param q0 scalar part of the quaternion
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* @param q1 first coordinate of the vectorial part of the quaternion
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* @param q2 second coordinate of the vectorial part of the quaternion
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* @param q3 third coordinate of the vectorial part of the quaternion
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* @param needsNormalization if true, the coordinates are considered
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* not to be normalized, a normalization preprocessing step is performed
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public Rotation(double q0, double q1, double q2, double q3,
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boolean needsNormalization) {
117
if (needsNormalization) {
118
// normalization preprocessing
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double inv = 1.0 / Math.sqrt(q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3);
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/** Build a rotation from an axis and an angle.
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* <p>We use the convention that angles are oriented according to
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* the effect of the rotation on vectors around the axis. That means
136
* that if (i, j, k) is a direct frame and if we first provide +k as
137
* the axis and PI/2 as the angle to this constructor, and then
138
* {@link #applyTo(Vector3D) apply} the instance to +i, we will get
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* @param axis axis around which to rotate
141
* @param angle rotation angle.
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* @exception ArithmeticException if the axis norm is zero
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public Rotation(Vector3D axis, double angle) {
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double norm = axis.getNorm();
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throw new ArithmeticException("zero norm for rotation axis");
151
double halfAngle = -0.5 * angle;
152
double coeff = Math.sin(halfAngle) / norm;
154
q0 = Math.cos (halfAngle);
155
q1 = coeff * axis.getX();
156
q2 = coeff * axis.getY();
157
q3 = coeff * axis.getZ();
161
/** Build a rotation from a 3X3 matrix.
163
* <p>Rotation matrices are orthogonal matrices, i.e. unit matrices
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* (which are matrices for which m.m<sup>T</sup> = I) with real
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* coefficients. The module of the determinant of unit matrices is
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* 1, among the orthogonal 3X3 matrices, only the ones having a
167
* positive determinant (+1) are rotation matrices.</p>
169
* <p>When a rotation is defined by a matrix with truncated values
170
* (typically when it is extracted from a technical sheet where only
171
* four to five significant digits are available), the matrix is not
172
* orthogonal anymore. This constructor handles this case
173
* transparently by using a copy of the given matrix and applying a
174
* correction to the copy in order to perfect its orthogonality. If
175
* the Frobenius norm of the correction needed is above the given
176
* threshold, then the matrix is considered to be too far from a
177
* true rotation matrix and an exception is thrown.<p>
179
* @param m rotation matrix
180
* @param threshold convergence threshold for the iterative
181
* orthogonality correction (convergence is reached when the
182
* difference between two steps of the Frobenius norm of the
183
* correction is below this threshold)
185
* @exception NotARotationMatrixException if the matrix is not a 3X3
186
* matrix, or if it cannot be transformed into an orthogonal matrix
187
* with the given threshold, or if the determinant of the resulting
188
* orthogonal matrix is negative
191
public Rotation(double[][] m, double threshold)
192
throws NotARotationMatrixException {
195
if ((m.length != 3) || (m[0].length != 3) ||
196
(m[1].length != 3) || (m[2].length != 3)) {
197
throw new NotARotationMatrixException("a {0}x{1} matrix" +
198
" cannot be a rotation matrix",
200
Integer.toString(m.length),
201
Integer.toString(m[0].length)
205
// compute a "close" orthogonal matrix
206
double[][] ort = orthogonalizeMatrix(m, threshold);
208
// check the sign of the determinant
209
double det = ort[0][0] * (ort[1][1] * ort[2][2] - ort[2][1] * ort[1][2]) -
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ort[1][0] * (ort[0][1] * ort[2][2] - ort[2][1] * ort[0][2]) +
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ort[2][0] * (ort[0][1] * ort[1][2] - ort[1][1] * ort[0][2]);
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throw new NotARotationMatrixException("the closest orthogonal matrix" +
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" has a negative determinant {0}",
220
// There are different ways to compute the quaternions elements
221
// from the matrix. They all involve computing one element from
222
// the diagonal of the matrix, and computing the three other ones
223
// using a formula involving a division by the first element,
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// which unfortunately can be zero. Since the norm of the
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// quaternion is 1, we know at least one element has an absolute
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// value greater or equal to 0.5, so it is always possible to
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// select the right formula and avoid division by zero and even
228
// numerical inaccuracy. Checking the elements in turn and using
229
// the first one greater than 0.45 is safe (this leads to a simple
230
// test since qi = 0.45 implies 4 qi^2 - 1 = -0.19)
231
double s = ort[0][0] + ort[1][1] + ort[2][2];
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// compute q0 and deduce q1, q2 and q3
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q0 = 0.5 * Math.sqrt(s + 1.0);
235
double inv = 0.25 / q0;
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q1 = inv * (ort[1][2] - ort[2][1]);
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q2 = inv * (ort[2][0] - ort[0][2]);
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q3 = inv * (ort[0][1] - ort[1][0]);
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s = ort[0][0] - ort[1][1] - ort[2][2];
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// compute q1 and deduce q0, q2 and q3
243
q1 = 0.5 * Math.sqrt(s + 1.0);
244
double inv = 0.25 / q1;
245
q0 = inv * (ort[1][2] - ort[2][1]);
246
q2 = inv * (ort[0][1] + ort[1][0]);
247
q3 = inv * (ort[0][2] + ort[2][0]);
249
s = ort[1][1] - ort[0][0] - ort[2][2];
251
// compute q2 and deduce q0, q1 and q3
252
q2 = 0.5 * Math.sqrt(s + 1.0);
253
double inv = 0.25 / q2;
254
q0 = inv * (ort[2][0] - ort[0][2]);
255
q1 = inv * (ort[0][1] + ort[1][0]);
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q3 = inv * (ort[2][1] + ort[1][2]);
258
// compute q3 and deduce q0, q1 and q2
259
s = ort[2][2] - ort[0][0] - ort[1][1];
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q3 = 0.5 * Math.sqrt(s + 1.0);
261
double inv = 0.25 / q3;
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q0 = inv * (ort[0][1] - ort[1][0]);
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q1 = inv * (ort[0][2] + ort[2][0]);
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q2 = inv * (ort[2][1] + ort[1][2]);
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/** Build the rotation that transforms a pair of vector into another pair.
273
* <p>Except for possible scale factors, if the instance were applied to
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* the pair (u<sub>1</sub>, u<sub>2</sub>) it will produce the pair
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* (v<sub>1</sub>, v<sub>2</sub>).</p>
277
* <p>If the angular separation between u<sub>1</sub> and u<sub>2</sub> is
278
* not the same as the angular separation between v<sub>1</sub> and
279
* v<sub>2</sub>, then a corrected v'<sub>2</sub> will be used rather than
280
* v<sub>2</sub>, the corrected vector will be in the (v<sub>1</sub>,
281
* v<sub>2</sub>) plane.</p>
283
* @param u1 first vector of the origin pair
284
* @param u2 second vector of the origin pair
285
* @param v1 desired image of u1 by the rotation
286
* @param v2 desired image of u2 by the rotation
287
* @exception IllegalArgumentException if the norm of one of the vectors is zero
289
public Rotation(Vector3D u1, Vector3D u2, Vector3D v1, Vector3D v2) {
292
double u1u1 = Vector3D.dotProduct(u1, u1);
293
double u2u2 = Vector3D.dotProduct(u2, u2);
294
double v1v1 = Vector3D.dotProduct(v1, v1);
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double v2v2 = Vector3D.dotProduct(v2, v2);
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if ((u1u1 == 0) || (u2u2 == 0) || (v1v1 == 0) || (v2v2 == 0)) {
297
throw new IllegalArgumentException("zero norm for rotation defining vector");
300
double u1x = u1.getX();
301
double u1y = u1.getY();
302
double u1z = u1.getZ();
304
double u2x = u2.getX();
305
double u2y = u2.getY();
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double u2z = u2.getZ();
308
// normalize v1 in order to have (v1'|v1') = (u1|u1)
309
double coeff = Math.sqrt (u1u1 / v1v1);
310
double v1x = coeff * v1.getX();
311
double v1y = coeff * v1.getY();
312
double v1z = coeff * v1.getZ();
313
v1 = new Vector3D(v1x, v1y, v1z);
315
// adjust v2 in order to have (u1|u2) = (v1|v2) and (v2'|v2') = (u2|u2)
316
double u1u2 = Vector3D.dotProduct(u1, u2);
317
double v1v2 = Vector3D.dotProduct(v1, v2);
318
double coeffU = u1u2 / u1u1;
319
double coeffV = v1v2 / u1u1;
320
double beta = Math.sqrt((u2u2 - u1u2 * coeffU) / (v2v2 - v1v2 * coeffV));
321
double alpha = coeffU - beta * coeffV;
322
double v2x = alpha * v1x + beta * v2.getX();
323
double v2y = alpha * v1y + beta * v2.getY();
324
double v2z = alpha * v1z + beta * v2.getZ();
325
v2 = new Vector3D(v2x, v2y, v2z);
327
// preliminary computation (we use explicit formulation instead
328
// of relying on the Vector3D class in order to avoid building lots
329
// of temporary objects)
332
double dx1 = v1x - u1.getX();
333
double dy1 = v1y - u1.getY();
334
double dz1 = v1z - u1.getZ();
335
double dx2 = v2x - u2.getX();
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double dy2 = v2y - u2.getY();
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double dz2 = v2z - u2.getZ();
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Vector3D k = new Vector3D(dy1 * dz2 - dz1 * dy2,
339
dz1 * dx2 - dx1 * dz2,
340
dx1 * dy2 - dy1 * dx2);
341
double c = k.getX() * (u1y * u2z - u1z * u2y) +
342
k.getY() * (u1z * u2x - u1x * u2z) +
343
k.getZ() * (u1x * u2y - u1y * u2x);
346
// the (q1, q2, q3) vector is in the (u1, u2) plane
347
// we try other vectors
348
Vector3D u3 = Vector3D.crossProduct(u1, u2);
349
Vector3D v3 = Vector3D.crossProduct(v1, v2);
350
double u3x = u3.getX();
351
double u3y = u3.getY();
352
double u3z = u3.getZ();
353
double v3x = v3.getX();
354
double v3y = v3.getY();
355
double v3z = v3.getZ();
357
double dx3 = v3x - u3x;
358
double dy3 = v3y - u3y;
359
double dz3 = v3z - u3z;
360
k = new Vector3D(dy1 * dz3 - dz1 * dy3,
361
dz1 * dx3 - dx1 * dz3,
362
dx1 * dy3 - dy1 * dx3);
363
c = k.getX() * (u1y * u3z - u1z * u3y) +
364
k.getY() * (u1z * u3x - u1x * u3z) +
365
k.getZ() * (u1x * u3y - u1y * u3x);
368
// the (q1, q2, q3) vector is aligned with u1:
369
// we try (u2, u3) and (v2, v3)
370
k = new Vector3D(dy2 * dz3 - dz2 * dy3,
371
dz2 * dx3 - dx2 * dz3,
372
dx2 * dy3 - dy2 * dx3);
373
c = k.getX() * (u2y * u3z - u2z * u3y) +
374
k.getY() * (u2z * u3x - u2x * u3z) +
375
k.getZ() * (u2x * u3y - u2y * u3x);
378
// the (q1, q2, q3) vector is aligned with everything
379
// this is really the identity rotation
387
// we will have to use u2 and v2 to compute the scalar part
395
// compute the vectorial part
397
double inv = 1.0 / (c + c);
402
// compute the scalar part
403
k = new Vector3D(uRef.getY() * q3 - uRef.getZ() * q2,
404
uRef.getZ() * q1 - uRef.getX() * q3,
405
uRef.getX() * q2 - uRef.getY() * q1);
406
c = Vector3D.dotProduct(k, k);
407
q0 = Vector3D.dotProduct(vRef, k) / (c + c);
411
/** Build one of the rotations that transform one vector into another one.
413
* <p>Except for a possible scale factor, if the instance were
414
* applied to the vector u it will produce the vector v. There is an
415
* infinite number of such rotations, this constructor choose the
416
* one with the smallest associated angle (i.e. the one whose axis
417
* is orthogonal to the (u, v) plane). If u and v are colinear, an
418
* arbitrary rotation axis is chosen.</p>
420
* @param u origin vector
421
* @param v desired image of u by the rotation
422
* @exception IllegalArgumentException if the norm of one of the vectors is zero
424
public Rotation(Vector3D u, Vector3D v) {
426
double normProduct = u.getNorm() * v.getNorm();
427
if (normProduct == 0) {
428
throw new IllegalArgumentException("zero norm for rotation defining vector");
431
double dot = Vector3D.dotProduct(u, v);
433
if (dot < ((2.0e-15 - 1.0) * normProduct)) {
434
// special case u = -v: we select a PI angle rotation around
435
// an arbitrary vector orthogonal to u
436
Vector3D w = u.orthogonal();
442
// general case: (u, v) defines a plane, we select
443
// the shortest possible rotation: axis orthogonal to this plane
444
q0 = Math.sqrt(0.5 * (1.0 + dot / normProduct));
445
double coeff = 1.0 / (2.0 * q0 * normProduct);
446
q1 = coeff * (v.getY() * u.getZ() - v.getZ() * u.getY());
447
q2 = coeff * (v.getZ() * u.getX() - v.getX() * u.getZ());
448
q3 = coeff * (v.getX() * u.getY() - v.getY() * u.getX());
453
/** Build a rotation from three Cardan or Euler elementary rotations.
455
* <p>Cardan rotations are three successive rotations around the
456
* canonical axes X, Y and Z, each axis beeing used once. There are
457
* 6 such sets of rotations (XYZ, XZY, YXZ, YZX, ZXY and ZYX). Euler
458
* rotations are three successive rotations around the canonical
459
* axes X, Y and Z, the first and last rotations beeing around the
460
* same axis. There are 6 such sets of rotations (XYX, XZX, YXY,
461
* YZY, ZXZ and ZYZ), the most popular one being ZXZ.</p>
462
* <p>Beware that many people routinely use the term Euler angles even
463
* for what really are Cardan angles (this confusion is especially
464
* widespread in the aerospace business where Roll, Pitch and Yaw angles
465
* are often wrongly tagged as Euler angles).</p>
467
* @param order order of rotations to use
468
* @param alpha1 angle of the first elementary rotation
469
* @param alpha2 angle of the second elementary rotation
470
* @param alpha3 angle of the third elementary rotation
472
public Rotation(RotationOrder order,
473
double alpha1, double alpha2, double alpha3) {
474
Rotation r1 = new Rotation(order.getA1(), alpha1);
475
Rotation r2 = new Rotation(order.getA2(), alpha2);
476
Rotation r3 = new Rotation(order.getA3(), alpha3);
477
Rotation composed = r1.applyTo(r2.applyTo(r3));
484
/** Revert a rotation.
485
* Build a rotation which reverse the effect of another
486
* rotation. This means that if r(u) = v, then r.revert(v) = u. The
487
* instance is not changed.
488
* @return a new rotation whose effect is the reverse of the effect
491
public Rotation revert() {
492
return new Rotation(-q0, q1, q2, q3, false);
495
/** Get the scalar coordinate of the quaternion.
496
* @return scalar coordinate of the quaternion
498
public double getQ0() {
502
/** Get the first coordinate of the vectorial part of the quaternion.
503
* @return first coordinate of the vectorial part of the quaternion
505
public double getQ1() {
509
/** Get the second coordinate of the vectorial part of the quaternion.
510
* @return second coordinate of the vectorial part of the quaternion
512
public double getQ2() {
516
/** Get the third coordinate of the vectorial part of the quaternion.
517
* @return third coordinate of the vectorial part of the quaternion
519
public double getQ3() {
523
/** Get the normalized axis of the rotation.
524
* @return normalized axis of the rotation
526
public Vector3D getAxis() {
527
double squaredSine = q1 * q1 + q2 * q2 + q3 * q3;
528
if (squaredSine == 0) {
529
return new Vector3D(1, 0, 0);
531
double inverse = 1 / Math.sqrt(squaredSine);
532
return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
534
double inverse = -1 / Math.sqrt(squaredSine);
535
return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
538
/** Get the angle of the rotation.
539
* @return angle of the rotation (between 0 and π)
541
public double getAngle() {
542
if ((q0 < -0.1) || (q0 > 0.1)) {
543
return 2 * Math.asin(Math.sqrt(q1 * q1 + q2 * q2 + q3 * q3));
545
return 2 * Math.acos(-q0);
547
return 2 * Math.acos(q0);
550
/** Get the Cardan or Euler angles corresponding to the instance.
552
* <p>The equations show that each rotation can be defined by two
553
* different values of the Cardan or Euler angles set. For example
554
* if Cardan angles are used, the rotation defined by the angles
555
* a<sub>1</sub>, a<sub>2</sub> and a<sub>3</sub> is the same as
556
* the rotation defined by the angles π + a<sub>1</sub>, π
557
* - a<sub>2</sub> and π + a<sub>3</sub>. This method implements
558
* the following arbitrary choices:</p>
560
* <li>for Cardan angles, the chosen set is the one for which the
561
* second angle is between -π/2 and π/2 (i.e its cosine is
563
* <li>for Euler angles, the chosen set is the one for which the
564
* second angle is between 0 and π (i.e its sine is positive).</li>
567
* <p>Cardan and Euler angle have a very disappointing drawback: all
568
* of them have singularities. This means that if the instance is
569
* too close to the singularities corresponding to the given
570
* rotation order, it will be impossible to retrieve the angles. For
571
* Cardan angles, this is often called gimbal lock. There is
572
* <em>nothing</em> to do to prevent this, it is an intrinsic problem
573
* with Cardan and Euler representation (but not a problem with the
574
* rotation itself, which is perfectly well defined). For Cardan
575
* angles, singularities occur when the second angle is close to
576
* -π/2 or +π/2, for Euler angle singularities occur when the
577
* second angle is close to 0 or π, this implies that the identity
578
* rotation is always singular for Euler angles!</p>
580
* @param order rotation order to use
581
* @return an array of three angles, in the order specified by the set
582
* @exception CardanEulerSingularityException if the rotation is
583
* singular with respect to the angles set specified
585
public double[] getAngles(RotationOrder order)
586
throws CardanEulerSingularityException {
588
if (order == RotationOrder.XYZ) {
590
// r (Vector3D.plusK) coordinates are :
591
// sin (theta), -cos (theta) sin (phi), cos (theta) cos (phi)
592
// (-r) (Vector3D.plusI) coordinates are :
593
// cos (psi) cos (theta), -sin (psi) cos (theta), sin (theta)
594
// and we can choose to have theta in the interval [-PI/2 ; +PI/2]
595
Vector3D v1 = applyTo(Vector3D.plusK);
596
Vector3D v2 = applyInverseTo(Vector3D.plusI);
597
if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
598
throw new CardanEulerSingularityException(true);
600
return new double[] {
601
Math.atan2(-(v1.getY()), v1.getZ()),
602
Math.asin(v2.getZ()),
603
Math.atan2(-(v2.getY()), v2.getX())
606
} else if (order == RotationOrder.XZY) {
608
// r (Vector3D.plusJ) coordinates are :
609
// -sin (psi), cos (psi) cos (phi), cos (psi) sin (phi)
610
// (-r) (Vector3D.plusI) coordinates are :
611
// cos (theta) cos (psi), -sin (psi), sin (theta) cos (psi)
612
// and we can choose to have psi in the interval [-PI/2 ; +PI/2]
613
Vector3D v1 = applyTo(Vector3D.plusJ);
614
Vector3D v2 = applyInverseTo(Vector3D.plusI);
615
if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
616
throw new CardanEulerSingularityException(true);
618
return new double[] {
619
Math.atan2(v1.getZ(), v1.getY()),
620
-Math.asin(v2.getY()),
621
Math.atan2(v2.getZ(), v2.getX())
624
} else if (order == RotationOrder.YXZ) {
626
// r (Vector3D.plusK) coordinates are :
627
// cos (phi) sin (theta), -sin (phi), cos (phi) cos (theta)
628
// (-r) (Vector3D.plusJ) coordinates are :
629
// sin (psi) cos (phi), cos (psi) cos (phi), -sin (phi)
630
// and we can choose to have phi in the interval [-PI/2 ; +PI/2]
631
Vector3D v1 = applyTo(Vector3D.plusK);
632
Vector3D v2 = applyInverseTo(Vector3D.plusJ);
633
if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
634
throw new CardanEulerSingularityException(true);
636
return new double[] {
637
Math.atan2(v1.getX(), v1.getZ()),
638
-Math.asin(v2.getZ()),
639
Math.atan2(v2.getX(), v2.getY())
642
} else if (order == RotationOrder.YZX) {
644
// r (Vector3D.plusI) coordinates are :
645
// cos (psi) cos (theta), sin (psi), -cos (psi) sin (theta)
646
// (-r) (Vector3D.plusJ) coordinates are :
647
// sin (psi), cos (phi) cos (psi), -sin (phi) cos (psi)
648
// and we can choose to have psi in the interval [-PI/2 ; +PI/2]
649
Vector3D v1 = applyTo(Vector3D.plusI);
650
Vector3D v2 = applyInverseTo(Vector3D.plusJ);
651
if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
652
throw new CardanEulerSingularityException(true);
654
return new double[] {
655
Math.atan2(-(v1.getZ()), v1.getX()),
656
Math.asin(v2.getX()),
657
Math.atan2(-(v2.getZ()), v2.getY())
660
} else if (order == RotationOrder.ZXY) {
662
// r (Vector3D.plusJ) coordinates are :
663
// -cos (phi) sin (psi), cos (phi) cos (psi), sin (phi)
664
// (-r) (Vector3D.plusK) coordinates are :
665
// -sin (theta) cos (phi), sin (phi), cos (theta) cos (phi)
666
// and we can choose to have phi in the interval [-PI/2 ; +PI/2]
667
Vector3D v1 = applyTo(Vector3D.plusJ);
668
Vector3D v2 = applyInverseTo(Vector3D.plusK);
669
if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
670
throw new CardanEulerSingularityException(true);
672
return new double[] {
673
Math.atan2(-(v1.getX()), v1.getY()),
674
Math.asin(v2.getY()),
675
Math.atan2(-(v2.getX()), v2.getZ())
678
} else if (order == RotationOrder.ZYX) {
680
// r (Vector3D.plusI) coordinates are :
681
// cos (theta) cos (psi), cos (theta) sin (psi), -sin (theta)
682
// (-r) (Vector3D.plusK) coordinates are :
683
// -sin (theta), sin (phi) cos (theta), cos (phi) cos (theta)
684
// and we can choose to have theta in the interval [-PI/2 ; +PI/2]
685
Vector3D v1 = applyTo(Vector3D.plusI);
686
Vector3D v2 = applyInverseTo(Vector3D.plusK);
687
if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
688
throw new CardanEulerSingularityException(true);
690
return new double[] {
691
Math.atan2(v1.getY(), v1.getX()),
692
-Math.asin(v2.getX()),
693
Math.atan2(v2.getY(), v2.getZ())
696
} else if (order == RotationOrder.XYX) {
698
// r (Vector3D.plusI) coordinates are :
699
// cos (theta), sin (phi1) sin (theta), -cos (phi1) sin (theta)
700
// (-r) (Vector3D.plusI) coordinates are :
701
// cos (theta), sin (theta) sin (phi2), sin (theta) cos (phi2)
702
// and we can choose to have theta in the interval [0 ; PI]
703
Vector3D v1 = applyTo(Vector3D.plusI);
704
Vector3D v2 = applyInverseTo(Vector3D.plusI);
705
if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
706
throw new CardanEulerSingularityException(false);
708
return new double[] {
709
Math.atan2(v1.getY(), -v1.getZ()),
710
Math.acos(v2.getX()),
711
Math.atan2(v2.getY(), v2.getZ())
714
} else if (order == RotationOrder.XZX) {
716
// r (Vector3D.plusI) coordinates are :
717
// cos (psi), cos (phi1) sin (psi), sin (phi1) sin (psi)
718
// (-r) (Vector3D.plusI) coordinates are :
719
// cos (psi), -sin (psi) cos (phi2), sin (psi) sin (phi2)
720
// and we can choose to have psi in the interval [0 ; PI]
721
Vector3D v1 = applyTo(Vector3D.plusI);
722
Vector3D v2 = applyInverseTo(Vector3D.plusI);
723
if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
724
throw new CardanEulerSingularityException(false);
726
return new double[] {
727
Math.atan2(v1.getZ(), v1.getY()),
728
Math.acos(v2.getX()),
729
Math.atan2(v2.getZ(), -v2.getY())
732
} else if (order == RotationOrder.YXY) {
734
// r (Vector3D.plusJ) coordinates are :
735
// sin (theta1) sin (phi), cos (phi), cos (theta1) sin (phi)
736
// (-r) (Vector3D.plusJ) coordinates are :
737
// sin (phi) sin (theta2), cos (phi), -sin (phi) cos (theta2)
738
// and we can choose to have phi in the interval [0 ; PI]
739
Vector3D v1 = applyTo(Vector3D.plusJ);
740
Vector3D v2 = applyInverseTo(Vector3D.plusJ);
741
if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
742
throw new CardanEulerSingularityException(false);
744
return new double[] {
745
Math.atan2(v1.getX(), v1.getZ()),
746
Math.acos(v2.getY()),
747
Math.atan2(v2.getX(), -v2.getZ())
750
} else if (order == RotationOrder.YZY) {
752
// r (Vector3D.plusJ) coordinates are :
753
// -cos (theta1) sin (psi), cos (psi), sin (theta1) sin (psi)
754
// (-r) (Vector3D.plusJ) coordinates are :
755
// sin (psi) cos (theta2), cos (psi), sin (psi) sin (theta2)
756
// and we can choose to have psi in the interval [0 ; PI]
757
Vector3D v1 = applyTo(Vector3D.plusJ);
758
Vector3D v2 = applyInverseTo(Vector3D.plusJ);
759
if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
760
throw new CardanEulerSingularityException(false);
762
return new double[] {
763
Math.atan2(v1.getZ(), -v1.getX()),
764
Math.acos(v2.getY()),
765
Math.atan2(v2.getZ(), v2.getX())
768
} else if (order == RotationOrder.ZXZ) {
770
// r (Vector3D.plusK) coordinates are :
771
// sin (psi1) sin (phi), -cos (psi1) sin (phi), cos (phi)
772
// (-r) (Vector3D.plusK) coordinates are :
773
// sin (phi) sin (psi2), sin (phi) cos (psi2), cos (phi)
774
// and we can choose to have phi in the interval [0 ; PI]
775
Vector3D v1 = applyTo(Vector3D.plusK);
776
Vector3D v2 = applyInverseTo(Vector3D.plusK);
777
if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
778
throw new CardanEulerSingularityException(false);
780
return new double[] {
781
Math.atan2(v1.getX(), -v1.getY()),
782
Math.acos(v2.getZ()),
783
Math.atan2(v2.getX(), v2.getY())
786
} else { // last possibility is ZYZ
788
// r (Vector3D.plusK) coordinates are :
789
// cos (psi1) sin (theta), sin (psi1) sin (theta), cos (theta)
790
// (-r) (Vector3D.plusK) coordinates are :
791
// -sin (theta) cos (psi2), sin (theta) sin (psi2), cos (theta)
792
// and we can choose to have theta in the interval [0 ; PI]
793
Vector3D v1 = applyTo(Vector3D.plusK);
794
Vector3D v2 = applyInverseTo(Vector3D.plusK);
795
if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
796
throw new CardanEulerSingularityException(false);
798
return new double[] {
799
Math.atan2(v1.getY(), v1.getX()),
800
Math.acos(v2.getZ()),
801
Math.atan2(v2.getY(), -v2.getX())
808
/** Get the 3X3 matrix corresponding to the instance
809
* @return the matrix corresponding to the instance
811
public double[][] getMatrix() {
814
double q0q0 = q0 * q0;
815
double q0q1 = q0 * q1;
816
double q0q2 = q0 * q2;
817
double q0q3 = q0 * q3;
818
double q1q1 = q1 * q1;
819
double q1q2 = q1 * q2;
820
double q1q3 = q1 * q3;
821
double q2q2 = q2 * q2;
822
double q2q3 = q2 * q3;
823
double q3q3 = q3 * q3;
826
double[][] m = new double[3][];
827
m[0] = new double[3];
828
m[1] = new double[3];
829
m[2] = new double[3];
831
m [0][0] = 2.0 * (q0q0 + q1q1) - 1.0;
832
m [1][0] = 2.0 * (q1q2 - q0q3);
833
m [2][0] = 2.0 * (q1q3 + q0q2);
835
m [0][1] = 2.0 * (q1q2 + q0q3);
836
m [1][1] = 2.0 * (q0q0 + q2q2) - 1.0;
837
m [2][1] = 2.0 * (q2q3 - q0q1);
839
m [0][2] = 2.0 * (q1q3 - q0q2);
840
m [1][2] = 2.0 * (q2q3 + q0q1);
841
m [2][2] = 2.0 * (q0q0 + q3q3) - 1.0;
847
/** Apply the rotation to a vector.
848
* @param u vector to apply the rotation to
849
* @return a new vector which is the image of u by the rotation
851
public Vector3D applyTo(Vector3D u) {
857
double s = q1 * x + q2 * y + q3 * z;
859
return new Vector3D(2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x,
860
2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y,
861
2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - z);
865
/** Apply the inverse of the rotation to a vector.
866
* @param u vector to apply the inverse of the rotation to
867
* @return a new vector which such that u is its image by the rotation
869
public Vector3D applyInverseTo(Vector3D u) {
875
double s = q1 * x + q2 * y + q3 * z;
878
return new Vector3D(2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x,
879
2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y,
880
2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - z);
884
/** Apply the instance to another rotation.
885
* Applying the instance to a rotation is computing the composition
886
* in an order compliant with the following rule : let u be any
887
* vector and v its image by r (i.e. r.applyTo(u) = v), let w be the image
888
* of v by the instance (i.e. applyTo(v) = w), then w = comp.applyTo(u),
889
* where comp = applyTo(r).
890
* @param r rotation to apply the rotation to
891
* @return a new rotation which is the composition of r by the instance
893
public Rotation applyTo(Rotation r) {
894
return new Rotation(r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),
895
r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),
896
r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),
897
r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),
901
/** Apply the inverse of the instance to another rotation.
902
* Applying the inverse of the instance to a rotation is computing
903
* the composition in an order compliant with the following rule :
904
* let u be any vector and v its image by r (i.e. r.applyTo(u) = v),
905
* let w be the inverse image of v by the instance
906
* (i.e. applyInverseTo(v) = w), then w = comp.applyTo(u), where
907
* comp = applyInverseTo(r).
908
* @param r rotation to apply the rotation to
909
* @return a new rotation which is the composition of r by the inverse
912
public Rotation applyInverseTo(Rotation r) {
913
return new Rotation(-r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),
914
-r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),
915
-r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),
916
-r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),
920
/** Perfect orthogonality on a 3X3 matrix.
921
* @param m initial matrix (not exactly orthogonal)
922
* @param threshold convergence threshold for the iterative
923
* orthogonality correction (convergence is reached when the
924
* difference between two steps of the Frobenius norm of the
925
* correction is below this threshold)
926
* @return an orthogonal matrix close to m
927
* @exception NotARotationMatrixException if the matrix cannot be
928
* orthogonalized with the given threshold after 10 iterations
930
private double[][] orthogonalizeMatrix(double[][] m, double threshold)
931
throws NotARotationMatrixException {
947
double[][] o = new double[3][3];
952
// iterative correction: Xn+1 = Xn - 0.5 * (Xn.Mt.Xn - M)
957
double mx00 = m0[0] * x00 + m1[0] * x10 + m2[0] * x20;
958
double mx10 = m0[1] * x00 + m1[1] * x10 + m2[1] * x20;
959
double mx20 = m0[2] * x00 + m1[2] * x10 + m2[2] * x20;
960
double mx01 = m0[0] * x01 + m1[0] * x11 + m2[0] * x21;
961
double mx11 = m0[1] * x01 + m1[1] * x11 + m2[1] * x21;
962
double mx21 = m0[2] * x01 + m1[2] * x11 + m2[2] * x21;
963
double mx02 = m0[0] * x02 + m1[0] * x12 + m2[0] * x22;
964
double mx12 = m0[1] * x02 + m1[1] * x12 + m2[1] * x22;
965
double mx22 = m0[2] * x02 + m1[2] * x12 + m2[2] * x22;
968
o0[0] = x00 - 0.5 * (x00 * mx00 + x01 * mx10 + x02 * mx20 - m0[0]);
969
o0[1] = x01 - 0.5 * (x00 * mx01 + x01 * mx11 + x02 * mx21 - m0[1]);
970
o0[2] = x02 - 0.5 * (x00 * mx02 + x01 * mx12 + x02 * mx22 - m0[2]);
971
o1[0] = x10 - 0.5 * (x10 * mx00 + x11 * mx10 + x12 * mx20 - m1[0]);
972
o1[1] = x11 - 0.5 * (x10 * mx01 + x11 * mx11 + x12 * mx21 - m1[1]);
973
o1[2] = x12 - 0.5 * (x10 * mx02 + x11 * mx12 + x12 * mx22 - m1[2]);
974
o2[0] = x20 - 0.5 * (x20 * mx00 + x21 * mx10 + x22 * mx20 - m2[0]);
975
o2[1] = x21 - 0.5 * (x20 * mx01 + x21 * mx11 + x22 * mx21 - m2[1]);
976
o2[2] = x22 - 0.5 * (x20 * mx02 + x21 * mx12 + x22 * mx22 - m2[2]);
978
// correction on each elements
979
double corr00 = o0[0] - m0[0];
980
double corr01 = o0[1] - m0[1];
981
double corr02 = o0[2] - m0[2];
982
double corr10 = o1[0] - m1[0];
983
double corr11 = o1[1] - m1[1];
984
double corr12 = o1[2] - m1[2];
985
double corr20 = o2[0] - m2[0];
986
double corr21 = o2[1] - m2[1];
987
double corr22 = o2[2] - m2[2];
989
// Frobenius norm of the correction
990
fn1 = corr00 * corr00 + corr01 * corr01 + corr02 * corr02 +
991
corr10 * corr10 + corr11 * corr11 + corr12 * corr12 +
992
corr20 * corr20 + corr21 * corr21 + corr22 * corr22;
995
if (Math.abs(fn1 - fn) <= threshold)
998
// prepare next iteration
1012
// the algorithm did not converge after 10 iterations
1013
throw new NotARotationMatrixException("unable to orthogonalize matrix" +
1014
" in {0} iterations",
1016
Integer.toString(i - 1)
1020
/** Scalar coordinate of the quaternion. */
1021
private final double q0;
1023
/** First coordinate of the vectorial part of the quaternion. */
1024
private final double q1;
1026
/** Second coordinate of the vectorial part of the quaternion. */
1027
private final double q2;
1029
/** Third coordinate of the vectorial part of the quaternion. */
1030
private final double q3;
1032
/** Serializable version identifier */
1033
private static final long serialVersionUID = 8225864499430109352L;
added
removed
src/java/org/apache/commons/math/geometry/Rotation.java
