001 /* 002 * Licensed to the Apache Software Foundation (ASF) under one or more 003 * contributor license agreements. See the NOTICE file distributed with 004 * this work for additional information regarding copyright ownership. 005 * The ASF licenses this file to You under the Apache License, Version 2.0 006 * (the "License"); you may not use this file except in compliance with 007 * the License. You may obtain a copy of the License at 008 * 009 * http://www.apache.org/licenses/LICENSE-2.0 010 * 011 * Unless required by applicable law or agreed to in writing, software 012 * distributed under the License is distributed on an "AS IS" BASIS, 013 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 014 * See the License for the specific language governing permissions and 015 * limitations under the License. 016 */ 017 018 package org.apache.commons.math.geometry; 019 020 import java.io.Serializable; 021 022 import org.apache.commons.math.MathRuntimeException; 023 024 /** 025 * This class implements rotations in a three-dimensional space. 026 * 027 * <p>Rotations can be represented by several different mathematical 028 * entities (matrices, axe and angle, Cardan or Euler angles, 029 * quaternions). This class presents an higher level abstraction, more 030 * user-oriented and hiding this implementation details. Well, for the 031 * curious, we use quaternions for the internal representation. The 032 * user can build a rotation from any of these representations, and 033 * any of these representations can be retrieved from a 034 * <code>Rotation</code> instance (see the various constructors and 035 * getters). In addition, a rotation can also be built implicitely 036 * from a set of vectors and their image.</p> 037 * <p>This implies that this class can be used to convert from one 038 * representation to another one. For example, converting a rotation 039 * matrix into a set of Cardan angles from can be done using the 040 * followong single line of code:</p> 041 * <pre> 042 * double[] angles = new Rotation(matrix, 1.0e-10).getAngles(RotationOrder.XYZ); 043 * </pre> 044 * <p>Focus is oriented on what a rotation <em>do</em> rather than on its 045 * underlying representation. Once it has been built, and regardless of its 046 * internal representation, a rotation is an <em>operator</em> which basically 047 * transforms three dimensional {@link Vector3D vectors} into other three 048 * dimensional {@link Vector3D vectors}. Depending on the application, the 049 * meaning of these vectors may vary and the semantics of the rotation also.</p> 050 * <p>For example in an spacecraft attitude simulation tool, users will often 051 * consider the vectors are fixed (say the Earth direction for example) and the 052 * rotation transforms the coordinates coordinates of this vector in inertial 053 * frame into the coordinates of the same vector in satellite frame. In this 054 * case, the rotation implicitely defines the relation between the two frames. 055 * Another example could be a telescope control application, where the rotation 056 * would transform the sighting direction at rest into the desired observing 057 * direction when the telescope is pointed towards an object of interest. In this 058 * case the rotation transforms the directionf at rest in a topocentric frame 059 * into the sighting direction in the same topocentric frame. In many case, both 060 * approaches will be combined, in our telescope example, we will probably also 061 * need to transform the observing direction in the topocentric frame into the 062 * observing direction in inertial frame taking into account the observatory 063 * location and the Earth rotation.</p> 064 * 065 * <p>These examples show that a rotation is what the user wants it to be, so this 066 * class does not push the user towards one specific definition and hence does not 067 * provide methods like <code>projectVectorIntoDestinationFrame</code> or 068 * <code>computeTransformedDirection</code>. It provides simpler and more generic 069 * methods: {@link #applyTo(Vector3D) applyTo(Vector3D)} and {@link 070 * #applyInverseTo(Vector3D) applyInverseTo(Vector3D)}.</p> 071 * 072 * <p>Since a rotation is basically a vectorial operator, several rotations can be 073 * composed together and the composite operation <code>r = r<sub>1</sub> o 074 * r<sub>2</sub></code> (which means that for each vector <code>u</code>, 075 * <code>r(u) = r<sub>1</sub>(r<sub>2</sub>(u))</code>) is also a rotation. Hence 076 * we can consider that in addition to vectors, a rotation can be applied to other 077 * rotations as well (or to itself). With our previous notations, we would say we 078 * can apply <code>r<sub>1</sub></code> to <code>r<sub>2</sub></code> and the result 079 * we get is <code>r = r<sub>1</sub> o r<sub>2</sub></code>. For this purpose, the 080 * class provides the methods: {@link #applyTo(Rotation) applyTo(Rotation)} and 081 * {@link #applyInverseTo(Rotation) applyInverseTo(Rotation)}.</p> 082 * 083 * <p>Rotations are guaranteed to be immutable objects.</p> 084 * 085 * @version $Revision: 772119 $ $Date: 2009-05-06 05:43:28 -0400 (Wed, 06 May 2009) $ 086 * @see Vector3D 087 * @see RotationOrder 088 * @since 1.2 089 */ 090 091 public class Rotation implements Serializable { 092 093 /** Identity rotation. */ 094 public static final Rotation IDENTITY = new Rotation(1.0, 0.0, 0.0, 0.0, false); 095 096 /** Serializable version identifier */ 097 private static final long serialVersionUID = -2153622329907944313L; 098 099 /** Scalar coordinate of the quaternion. */ 100 private final double q0; 101 102 /** First coordinate of the vectorial part of the quaternion. */ 103 private final double q1; 104 105 /** Second coordinate of the vectorial part of the quaternion. */ 106 private final double q2; 107 108 /** Third coordinate of the vectorial part of the quaternion. */ 109 private final double q3; 110 111 /** Build a rotation from the quaternion coordinates. 112 * <p>A rotation can be built from a <em>normalized</em> quaternion, 113 * i.e. a quaternion for which q<sub>0</sub><sup>2</sup> + 114 * q<sub>1</sub><sup>2</sup> + q<sub>2</sub><sup>2</sup> + 115 * q<sub>3</sub><sup>2</sup> = 1. If the quaternion is not normalized, 116 * the constructor can normalize it in a preprocessing step.</p> 117 * @param q0 scalar part of the quaternion 118 * @param q1 first coordinate of the vectorial part of the quaternion 119 * @param q2 second coordinate of the vectorial part of the quaternion 120 * @param q3 third coordinate of the vectorial part of the quaternion 121 * @param needsNormalization if true, the coordinates are considered 122 * not to be normalized, a normalization preprocessing step is performed 123 * before using them 124 */ 125 public Rotation(double q0, double q1, double q2, double q3, 126 boolean needsNormalization) { 127 128 if (needsNormalization) { 129 // normalization preprocessing 130 double inv = 1.0 / Math.sqrt(q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3); 131 q0 *= inv; 132 q1 *= inv; 133 q2 *= inv; 134 q3 *= inv; 135 } 136 137 this.q0 = q0; 138 this.q1 = q1; 139 this.q2 = q2; 140 this.q3 = q3; 141 142 } 143 144 /** Build a rotation from an axis and an angle. 145 * <p>We use the convention that angles are oriented according to 146 * the effect of the rotation on vectors around the axis. That means 147 * that if (i, j, k) is a direct frame and if we first provide +k as 148 * the axis and PI/2 as the angle to this constructor, and then 149 * {@link #applyTo(Vector3D) apply} the instance to +i, we will get 150 * +j.</p> 151 * @param axis axis around which to rotate 152 * @param angle rotation angle. 153 * @exception ArithmeticException if the axis norm is zero 154 */ 155 public Rotation(Vector3D axis, double angle) { 156 157 double norm = axis.getNorm(); 158 if (norm == 0) { 159 throw MathRuntimeException.createArithmeticException("zero norm for rotation axis"); 160 } 161 162 double halfAngle = -0.5 * angle; 163 double coeff = Math.sin(halfAngle) / norm; 164 165 q0 = Math.cos (halfAngle); 166 q1 = coeff * axis.getX(); 167 q2 = coeff * axis.getY(); 168 q3 = coeff * axis.getZ(); 169 170 } 171 172 /** Build a rotation from a 3X3 matrix. 173 174 * <p>Rotation matrices are orthogonal matrices, i.e. unit matrices 175 * (which are matrices for which m.m<sup>T</sup> = I) with real 176 * coefficients. The module of the determinant of unit matrices is 177 * 1, among the orthogonal 3X3 matrices, only the ones having a 178 * positive determinant (+1) are rotation matrices.</p> 179 180 * <p>When a rotation is defined by a matrix with truncated values 181 * (typically when it is extracted from a technical sheet where only 182 * four to five significant digits are available), the matrix is not 183 * orthogonal anymore. This constructor handles this case 184 * transparently by using a copy of the given matrix and applying a 185 * correction to the copy in order to perfect its orthogonality. If 186 * the Frobenius norm of the correction needed is above the given 187 * threshold, then the matrix is considered to be too far from a 188 * true rotation matrix and an exception is thrown.<p> 189 190 * @param m rotation matrix 191 * @param threshold convergence threshold for the iterative 192 * orthogonality correction (convergence is reached when the 193 * difference between two steps of the Frobenius norm of the 194 * correction is below this threshold) 195 196 * @exception NotARotationMatrixException if the matrix is not a 3X3 197 * matrix, or if it cannot be transformed into an orthogonal matrix 198 * with the given threshold, or if the determinant of the resulting 199 * orthogonal matrix is negative 200 201 */ 202 public Rotation(double[][] m, double threshold) 203 throws NotARotationMatrixException { 204 205 // dimension check 206 if ((m.length != 3) || (m[0].length != 3) || 207 (m[1].length != 3) || (m[2].length != 3)) { 208 throw new NotARotationMatrixException( 209 "a {0}x{1} matrix cannot be a rotation matrix", 210 m.length, m[0].length); 211 } 212 213 // compute a "close" orthogonal matrix 214 double[][] ort = orthogonalizeMatrix(m, threshold); 215 216 // check the sign of the determinant 217 double det = ort[0][0] * (ort[1][1] * ort[2][2] - ort[2][1] * ort[1][2]) - 218 ort[1][0] * (ort[0][1] * ort[2][2] - ort[2][1] * ort[0][2]) + 219 ort[2][0] * (ort[0][1] * ort[1][2] - ort[1][1] * ort[0][2]); 220 if (det < 0.0) { 221 throw new NotARotationMatrixException( 222 "the closest orthogonal matrix has a negative determinant {0}", 223 det); 224 } 225 226 // There are different ways to compute the quaternions elements 227 // from the matrix. They all involve computing one element from 228 // the diagonal of the matrix, and computing the three other ones 229 // using a formula involving a division by the first element, 230 // which unfortunately can be zero. Since the norm of the 231 // quaternion is 1, we know at least one element has an absolute 232 // value greater or equal to 0.5, so it is always possible to 233 // select the right formula and avoid division by zero and even 234 // numerical inaccuracy. Checking the elements in turn and using 235 // the first one greater than 0.45 is safe (this leads to a simple 236 // test since qi = 0.45 implies 4 qi^2 - 1 = -0.19) 237 double s = ort[0][0] + ort[1][1] + ort[2][2]; 238 if (s > -0.19) { 239 // compute q0 and deduce q1, q2 and q3 240 q0 = 0.5 * Math.sqrt(s + 1.0); 241 double inv = 0.25 / q0; 242 q1 = inv * (ort[1][2] - ort[2][1]); 243 q2 = inv * (ort[2][0] - ort[0][2]); 244 q3 = inv * (ort[0][1] - ort[1][0]); 245 } else { 246 s = ort[0][0] - ort[1][1] - ort[2][2]; 247 if (s > -0.19) { 248 // compute q1 and deduce q0, q2 and q3 249 q1 = 0.5 * Math.sqrt(s + 1.0); 250 double inv = 0.25 / q1; 251 q0 = inv * (ort[1][2] - ort[2][1]); 252 q2 = inv * (ort[0][1] + ort[1][0]); 253 q3 = inv * (ort[0][2] + ort[2][0]); 254 } else { 255 s = ort[1][1] - ort[0][0] - ort[2][2]; 256 if (s > -0.19) { 257 // compute q2 and deduce q0, q1 and q3 258 q2 = 0.5 * Math.sqrt(s + 1.0); 259 double inv = 0.25 / q2; 260 q0 = inv * (ort[2][0] - ort[0][2]); 261 q1 = inv * (ort[0][1] + ort[1][0]); 262 q3 = inv * (ort[2][1] + ort[1][2]); 263 } else { 264 // compute q3 and deduce q0, q1 and q2 265 s = ort[2][2] - ort[0][0] - ort[1][1]; 266 q3 = 0.5 * Math.sqrt(s + 1.0); 267 double inv = 0.25 / q3; 268 q0 = inv * (ort[0][1] - ort[1][0]); 269 q1 = inv * (ort[0][2] + ort[2][0]); 270 q2 = inv * (ort[2][1] + ort[1][2]); 271 } 272 } 273 } 274 275 } 276 277 /** Build the rotation that transforms a pair of vector into another pair. 278 279 * <p>Except for possible scale factors, if the instance were applied to 280 * the pair (u<sub>1</sub>, u<sub>2</sub>) it will produce the pair 281 * (v<sub>1</sub>, v<sub>2</sub>).</p> 282 283 * <p>If the angular separation between u<sub>1</sub> and u<sub>2</sub> is 284 * not the same as the angular separation between v<sub>1</sub> and 285 * v<sub>2</sub>, then a corrected v'<sub>2</sub> will be used rather than 286 * v<sub>2</sub>, the corrected vector will be in the (v<sub>1</sub>, 287 * v<sub>2</sub>) plane.</p> 288 289 * @param u1 first vector of the origin pair 290 * @param u2 second vector of the origin pair 291 * @param v1 desired image of u1 by the rotation 292 * @param v2 desired image of u2 by the rotation 293 * @exception IllegalArgumentException if the norm of one of the vectors is zero 294 */ 295 public Rotation(Vector3D u1, Vector3D u2, Vector3D v1, Vector3D v2) { 296 297 // norms computation 298 double u1u1 = Vector3D.dotProduct(u1, u1); 299 double u2u2 = Vector3D.dotProduct(u2, u2); 300 double v1v1 = Vector3D.dotProduct(v1, v1); 301 double v2v2 = Vector3D.dotProduct(v2, v2); 302 if ((u1u1 == 0) || (u2u2 == 0) || (v1v1 == 0) || (v2v2 == 0)) { 303 throw MathRuntimeException.createIllegalArgumentException("zero norm for rotation defining vector"); 304 } 305 306 double u1x = u1.getX(); 307 double u1y = u1.getY(); 308 double u1z = u1.getZ(); 309 310 double u2x = u2.getX(); 311 double u2y = u2.getY(); 312 double u2z = u2.getZ(); 313 314 // normalize v1 in order to have (v1'|v1') = (u1|u1) 315 double coeff = Math.sqrt (u1u1 / v1v1); 316 double v1x = coeff * v1.getX(); 317 double v1y = coeff * v1.getY(); 318 double v1z = coeff * v1.getZ(); 319 v1 = new Vector3D(v1x, v1y, v1z); 320 321 // adjust v2 in order to have (u1|u2) = (v1|v2) and (v2'|v2') = (u2|u2) 322 double u1u2 = Vector3D.dotProduct(u1, u2); 323 double v1v2 = Vector3D.dotProduct(v1, v2); 324 double coeffU = u1u2 / u1u1; 325 double coeffV = v1v2 / u1u1; 326 double beta = Math.sqrt((u2u2 - u1u2 * coeffU) / (v2v2 - v1v2 * coeffV)); 327 double alpha = coeffU - beta * coeffV; 328 double v2x = alpha * v1x + beta * v2.getX(); 329 double v2y = alpha * v1y + beta * v2.getY(); 330 double v2z = alpha * v1z + beta * v2.getZ(); 331 v2 = new Vector3D(v2x, v2y, v2z); 332 333 // preliminary computation (we use explicit formulation instead 334 // of relying on the Vector3D class in order to avoid building lots 335 // of temporary objects) 336 Vector3D uRef = u1; 337 Vector3D vRef = v1; 338 double dx1 = v1x - u1.getX(); 339 double dy1 = v1y - u1.getY(); 340 double dz1 = v1z - u1.getZ(); 341 double dx2 = v2x - u2.getX(); 342 double dy2 = v2y - u2.getY(); 343 double dz2 = v2z - u2.getZ(); 344 Vector3D k = new Vector3D(dy1 * dz2 - dz1 * dy2, 345 dz1 * dx2 - dx1 * dz2, 346 dx1 * dy2 - dy1 * dx2); 347 double c = k.getX() * (u1y * u2z - u1z * u2y) + 348 k.getY() * (u1z * u2x - u1x * u2z) + 349 k.getZ() * (u1x * u2y - u1y * u2x); 350 351 if (c == 0) { 352 // the (q1, q2, q3) vector is in the (u1, u2) plane 353 // we try other vectors 354 Vector3D u3 = Vector3D.crossProduct(u1, u2); 355 Vector3D v3 = Vector3D.crossProduct(v1, v2); 356 double u3x = u3.getX(); 357 double u3y = u3.getY(); 358 double u3z = u3.getZ(); 359 double v3x = v3.getX(); 360 double v3y = v3.getY(); 361 double v3z = v3.getZ(); 362 363 double dx3 = v3x - u3x; 364 double dy3 = v3y - u3y; 365 double dz3 = v3z - u3z; 366 k = new Vector3D(dy1 * dz3 - dz1 * dy3, 367 dz1 * dx3 - dx1 * dz3, 368 dx1 * dy3 - dy1 * dx3); 369 c = k.getX() * (u1y * u3z - u1z * u3y) + 370 k.getY() * (u1z * u3x - u1x * u3z) + 371 k.getZ() * (u1x * u3y - u1y * u3x); 372 373 if (c == 0) { 374 // the (q1, q2, q3) vector is aligned with u1: 375 // we try (u2, u3) and (v2, v3) 376 k = new Vector3D(dy2 * dz3 - dz2 * dy3, 377 dz2 * dx3 - dx2 * dz3, 378 dx2 * dy3 - dy2 * dx3); 379 c = k.getX() * (u2y * u3z - u2z * u3y) + 380 k.getY() * (u2z * u3x - u2x * u3z) + 381 k.getZ() * (u2x * u3y - u2y * u3x); 382 383 if (c == 0) { 384 // the (q1, q2, q3) vector is aligned with everything 385 // this is really the identity rotation 386 q0 = 1.0; 387 q1 = 0.0; 388 q2 = 0.0; 389 q3 = 0.0; 390 return; 391 } 392 393 // we will have to use u2 and v2 to compute the scalar part 394 uRef = u2; 395 vRef = v2; 396 397 } 398 399 } 400 401 // compute the vectorial part 402 c = Math.sqrt(c); 403 double inv = 1.0 / (c + c); 404 q1 = inv * k.getX(); 405 q2 = inv * k.getY(); 406 q3 = inv * k.getZ(); 407 408 // compute the scalar part 409 k = new Vector3D(uRef.getY() * q3 - uRef.getZ() * q2, 410 uRef.getZ() * q1 - uRef.getX() * q3, 411 uRef.getX() * q2 - uRef.getY() * q1); 412 c = Vector3D.dotProduct(k, k); 413 q0 = Vector3D.dotProduct(vRef, k) / (c + c); 414 415 } 416 417 /** Build one of the rotations that transform one vector into another one. 418 419 * <p>Except for a possible scale factor, if the instance were 420 * applied to the vector u it will produce the vector v. There is an 421 * infinite number of such rotations, this constructor choose the 422 * one with the smallest associated angle (i.e. the one whose axis 423 * is orthogonal to the (u, v) plane). If u and v are colinear, an 424 * arbitrary rotation axis is chosen.</p> 425 426 * @param u origin vector 427 * @param v desired image of u by the rotation 428 * @exception IllegalArgumentException if the norm of one of the vectors is zero 429 */ 430 public Rotation(Vector3D u, Vector3D v) { 431 432 double normProduct = u.getNorm() * v.getNorm(); 433 if (normProduct == 0) { 434 throw MathRuntimeException.createIllegalArgumentException("zero norm for rotation defining vector"); 435 } 436 437 double dot = Vector3D.dotProduct(u, v); 438 439 if (dot < ((2.0e-15 - 1.0) * normProduct)) { 440 // special case u = -v: we select a PI angle rotation around 441 // an arbitrary vector orthogonal to u 442 Vector3D w = u.orthogonal(); 443 q0 = 0.0; 444 q1 = -w.getX(); 445 q2 = -w.getY(); 446 q3 = -w.getZ(); 447 } else { 448 // general case: (u, v) defines a plane, we select 449 // the shortest possible rotation: axis orthogonal to this plane 450 q0 = Math.sqrt(0.5 * (1.0 + dot / normProduct)); 451 double coeff = 1.0 / (2.0 * q0 * normProduct); 452 q1 = coeff * (v.getY() * u.getZ() - v.getZ() * u.getY()); 453 q2 = coeff * (v.getZ() * u.getX() - v.getX() * u.getZ()); 454 q3 = coeff * (v.getX() * u.getY() - v.getY() * u.getX()); 455 } 456 457 } 458 459 /** Build a rotation from three Cardan or Euler elementary rotations. 460 461 * <p>Cardan rotations are three successive rotations around the 462 * canonical axes X, Y and Z, each axis being used once. There are 463 * 6 such sets of rotations (XYZ, XZY, YXZ, YZX, ZXY and ZYX). Euler 464 * rotations are three successive rotations around the canonical 465 * axes X, Y and Z, the first and last rotations being around the 466 * same axis. There are 6 such sets of rotations (XYX, XZX, YXY, 467 * YZY, ZXZ and ZYZ), the most popular one being ZXZ.</p> 468 * <p>Beware that many people routinely use the term Euler angles even 469 * for what really are Cardan angles (this confusion is especially 470 * widespread in the aerospace business where Roll, Pitch and Yaw angles 471 * are often wrongly tagged as Euler angles).</p> 472 473 * @param order order of rotations to use 474 * @param alpha1 angle of the first elementary rotation 475 * @param alpha2 angle of the second elementary rotation 476 * @param alpha3 angle of the third elementary rotation 477 */ 478 public Rotation(RotationOrder order, 479 double alpha1, double alpha2, double alpha3) { 480 Rotation r1 = new Rotation(order.getA1(), alpha1); 481 Rotation r2 = new Rotation(order.getA2(), alpha2); 482 Rotation r3 = new Rotation(order.getA3(), alpha3); 483 Rotation composed = r1.applyTo(r2.applyTo(r3)); 484 q0 = composed.q0; 485 q1 = composed.q1; 486 q2 = composed.q2; 487 q3 = composed.q3; 488 } 489 490 /** Revert a rotation. 491 * Build a rotation which reverse the effect of another 492 * rotation. This means that if r(u) = v, then r.revert(v) = u. The 493 * instance is not changed. 494 * @return a new rotation whose effect is the reverse of the effect 495 * of the instance 496 */ 497 public Rotation revert() { 498 return new Rotation(-q0, q1, q2, q3, false); 499 } 500 501 /** Get the scalar coordinate of the quaternion. 502 * @return scalar coordinate of the quaternion 503 */ 504 public double getQ0() { 505 return q0; 506 } 507 508 /** Get the first coordinate of the vectorial part of the quaternion. 509 * @return first coordinate of the vectorial part of the quaternion 510 */ 511 public double getQ1() { 512 return q1; 513 } 514 515 /** Get the second coordinate of the vectorial part of the quaternion. 516 * @return second coordinate of the vectorial part of the quaternion 517 */ 518 public double getQ2() { 519 return q2; 520 } 521 522 /** Get the third coordinate of the vectorial part of the quaternion. 523 * @return third coordinate of the vectorial part of the quaternion 524 */ 525 public double getQ3() { 526 return q3; 527 } 528 529 /** Get the normalized axis of the rotation. 530 * @return normalized axis of the rotation 531 */ 532 public Vector3D getAxis() { 533 double squaredSine = q1 * q1 + q2 * q2 + q3 * q3; 534 if (squaredSine == 0) { 535 return new Vector3D(1, 0, 0); 536 } else if (q0 < 0) { 537 double inverse = 1 / Math.sqrt(squaredSine); 538 return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse); 539 } 540 double inverse = -1 / Math.sqrt(squaredSine); 541 return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse); 542 } 543 544 /** Get the angle of the rotation. 545 * @return angle of the rotation (between 0 and π) 546 */ 547 public double getAngle() { 548 if ((q0 < -0.1) || (q0 > 0.1)) { 549 return 2 * Math.asin(Math.sqrt(q1 * q1 + q2 * q2 + q3 * q3)); 550 } else if (q0 < 0) { 551 return 2 * Math.acos(-q0); 552 } 553 return 2 * Math.acos(q0); 554 } 555 556 /** Get the Cardan or Euler angles corresponding to the instance. 557 558 * <p>The equations show that each rotation can be defined by two 559 * different values of the Cardan or Euler angles set. For example 560 * if Cardan angles are used, the rotation defined by the angles 561 * a<sub>1</sub>, a<sub>2</sub> and a<sub>3</sub> is the same as 562 * the rotation defined by the angles π + a<sub>1</sub>, π 563 * - a<sub>2</sub> and π + a<sub>3</sub>. This method implements 564 * the following arbitrary choices:</p> 565 * <ul> 566 * <li>for Cardan angles, the chosen set is the one for which the 567 * second angle is between -π/2 and π/2 (i.e its cosine is 568 * positive),</li> 569 * <li>for Euler angles, the chosen set is the one for which the 570 * second angle is between 0 and π (i.e its sine is positive).</li> 571 * </ul> 572 573 * <p>Cardan and Euler angle have a very disappointing drawback: all 574 * of them have singularities. This means that if the instance is 575 * too close to the singularities corresponding to the given 576 * rotation order, it will be impossible to retrieve the angles. For 577 * Cardan angles, this is often called gimbal lock. There is 578 * <em>nothing</em> to do to prevent this, it is an intrinsic problem 579 * with Cardan and Euler representation (but not a problem with the 580 * rotation itself, which is perfectly well defined). For Cardan 581 * angles, singularities occur when the second angle is close to 582 * -π/2 or +π/2, for Euler angle singularities occur when the 583 * second angle is close to 0 or π, this implies that the identity 584 * rotation is always singular for Euler angles!</p> 585 586 * @param order rotation order to use 587 * @return an array of three angles, in the order specified by the set 588 * @exception CardanEulerSingularityException if the rotation is 589 * singular with respect to the angles set specified 590 */ 591 public double[] getAngles(RotationOrder order) 592 throws CardanEulerSingularityException { 593 594 if (order == RotationOrder.XYZ) { 595 596 // r (Vector3D.plusK) coordinates are : 597 // sin (theta), -cos (theta) sin (phi), cos (theta) cos (phi) 598 // (-r) (Vector3D.plusI) coordinates are : 599 // cos (psi) cos (theta), -sin (psi) cos (theta), sin (theta) 600 // and we can choose to have theta in the interval [-PI/2 ; +PI/2] 601 Vector3D v1 = applyTo(Vector3D.PLUS_K); 602 Vector3D v2 = applyInverseTo(Vector3D.PLUS_I); 603 if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) { 604 throw new CardanEulerSingularityException(true); 605 } 606 return new double[] { 607 Math.atan2(-(v1.getY()), v1.getZ()), 608 Math.asin(v2.getZ()), 609 Math.atan2(-(v2.getY()), v2.getX()) 610 }; 611 612 } else if (order == RotationOrder.XZY) { 613 614 // r (Vector3D.plusJ) coordinates are : 615 // -sin (psi), cos (psi) cos (phi), cos (psi) sin (phi) 616 // (-r) (Vector3D.plusI) coordinates are : 617 // cos (theta) cos (psi), -sin (psi), sin (theta) cos (psi) 618 // and we can choose to have psi in the interval [-PI/2 ; +PI/2] 619 Vector3D v1 = applyTo(Vector3D.PLUS_J); 620 Vector3D v2 = applyInverseTo(Vector3D.PLUS_I); 621 if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) { 622 throw new CardanEulerSingularityException(true); 623 } 624 return new double[] { 625 Math.atan2(v1.getZ(), v1.getY()), 626 -Math.asin(v2.getY()), 627 Math.atan2(v2.getZ(), v2.getX()) 628 }; 629 630 } else if (order == RotationOrder.YXZ) { 631 632 // r (Vector3D.plusK) coordinates are : 633 // cos (phi) sin (theta), -sin (phi), cos (phi) cos (theta) 634 // (-r) (Vector3D.plusJ) coordinates are : 635 // sin (psi) cos (phi), cos (psi) cos (phi), -sin (phi) 636 // and we can choose to have phi in the interval [-PI/2 ; +PI/2] 637 Vector3D v1 = applyTo(Vector3D.PLUS_K); 638 Vector3D v2 = applyInverseTo(Vector3D.PLUS_J); 639 if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) { 640 throw new CardanEulerSingularityException(true); 641 } 642 return new double[] { 643 Math.atan2(v1.getX(), v1.getZ()), 644 -Math.asin(v2.getZ()), 645 Math.atan2(v2.getX(), v2.getY()) 646 }; 647 648 } else if (order == RotationOrder.YZX) { 649 650 // r (Vector3D.plusI) coordinates are : 651 // cos (psi) cos (theta), sin (psi), -cos (psi) sin (theta) 652 // (-r) (Vector3D.plusJ) coordinates are : 653 // sin (psi), cos (phi) cos (psi), -sin (phi) cos (psi) 654 // and we can choose to have psi in the interval [-PI/2 ; +PI/2] 655 Vector3D v1 = applyTo(Vector3D.PLUS_I); 656 Vector3D v2 = applyInverseTo(Vector3D.PLUS_J); 657 if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) { 658 throw new CardanEulerSingularityException(true); 659 } 660 return new double[] { 661 Math.atan2(-(v1.getZ()), v1.getX()), 662 Math.asin(v2.getX()), 663 Math.atan2(-(v2.getZ()), v2.getY()) 664 }; 665 666 } else if (order == RotationOrder.ZXY) { 667 668 // r (Vector3D.plusJ) coordinates are : 669 // -cos (phi) sin (psi), cos (phi) cos (psi), sin (phi) 670 // (-r) (Vector3D.plusK) coordinates are : 671 // -sin (theta) cos (phi), sin (phi), cos (theta) cos (phi) 672 // and we can choose to have phi in the interval [-PI/2 ; +PI/2] 673 Vector3D v1 = applyTo(Vector3D.PLUS_J); 674 Vector3D v2 = applyInverseTo(Vector3D.PLUS_K); 675 if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) { 676 throw new CardanEulerSingularityException(true); 677 } 678 return new double[] { 679 Math.atan2(-(v1.getX()), v1.getY()), 680 Math.asin(v2.getY()), 681 Math.atan2(-(v2.getX()), v2.getZ()) 682 }; 683 684 } else if (order == RotationOrder.ZYX) { 685 686 // r (Vector3D.plusI) coordinates are : 687 // cos (theta) cos (psi), cos (theta) sin (psi), -sin (theta) 688 // (-r) (Vector3D.plusK) coordinates are : 689 // -sin (theta), sin (phi) cos (theta), cos (phi) cos (theta) 690 // and we can choose to have theta in the interval [-PI/2 ; +PI/2] 691 Vector3D v1 = applyTo(Vector3D.PLUS_I); 692 Vector3D v2 = applyInverseTo(Vector3D.PLUS_K); 693 if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) { 694 throw new CardanEulerSingularityException(true); 695 } 696 return new double[] { 697 Math.atan2(v1.getY(), v1.getX()), 698 -Math.asin(v2.getX()), 699 Math.atan2(v2.getY(), v2.getZ()) 700 }; 701 702 } else if (order == RotationOrder.XYX) { 703 704 // r (Vector3D.plusI) coordinates are : 705 // cos (theta), sin (phi1) sin (theta), -cos (phi1) sin (theta) 706 // (-r) (Vector3D.plusI) coordinates are : 707 // cos (theta), sin (theta) sin (phi2), sin (theta) cos (phi2) 708 // and we can choose to have theta in the interval [0 ; PI] 709 Vector3D v1 = applyTo(Vector3D.PLUS_I); 710 Vector3D v2 = applyInverseTo(Vector3D.PLUS_I); 711 if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) { 712 throw new CardanEulerSingularityException(false); 713 } 714 return new double[] { 715 Math.atan2(v1.getY(), -v1.getZ()), 716 Math.acos(v2.getX()), 717 Math.atan2(v2.getY(), v2.getZ()) 718 }; 719 720 } else if (order == RotationOrder.XZX) { 721 722 // r (Vector3D.plusI) coordinates are : 723 // cos (psi), cos (phi1) sin (psi), sin (phi1) sin (psi) 724 // (-r) (Vector3D.plusI) coordinates are : 725 // cos (psi), -sin (psi) cos (phi2), sin (psi) sin (phi2) 726 // and we can choose to have psi in the interval [0 ; PI] 727 Vector3D v1 = applyTo(Vector3D.PLUS_I); 728 Vector3D v2 = applyInverseTo(Vector3D.PLUS_I); 729 if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) { 730 throw new CardanEulerSingularityException(false); 731 } 732 return new double[] { 733 Math.atan2(v1.getZ(), v1.getY()), 734 Math.acos(v2.getX()), 735 Math.atan2(v2.getZ(), -v2.getY()) 736 }; 737 738 } else if (order == RotationOrder.YXY) { 739 740 // r (Vector3D.plusJ) coordinates are : 741 // sin (theta1) sin (phi), cos (phi), cos (theta1) sin (phi) 742 // (-r) (Vector3D.plusJ) coordinates are : 743 // sin (phi) sin (theta2), cos (phi), -sin (phi) cos (theta2) 744 // and we can choose to have phi in the interval [0 ; PI] 745 Vector3D v1 = applyTo(Vector3D.PLUS_J); 746 Vector3D v2 = applyInverseTo(Vector3D.PLUS_J); 747 if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) { 748 throw new CardanEulerSingularityException(false); 749 } 750 return new double[] { 751 Math.atan2(v1.getX(), v1.getZ()), 752 Math.acos(v2.getY()), 753 Math.atan2(v2.getX(), -v2.getZ()) 754 }; 755 756 } else if (order == RotationOrder.YZY) { 757 758 // r (Vector3D.plusJ) coordinates are : 759 // -cos (theta1) sin (psi), cos (psi), sin (theta1) sin (psi) 760 // (-r) (Vector3D.plusJ) coordinates are : 761 // sin (psi) cos (theta2), cos (psi), sin (psi) sin (theta2) 762 // and we can choose to have psi in the interval [0 ; PI] 763 Vector3D v1 = applyTo(Vector3D.PLUS_J); 764 Vector3D v2 = applyInverseTo(Vector3D.PLUS_J); 765 if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) { 766 throw new CardanEulerSingularityException(false); 767 } 768 return new double[] { 769 Math.atan2(v1.getZ(), -v1.getX()), 770 Math.acos(v2.getY()), 771 Math.atan2(v2.getZ(), v2.getX()) 772 }; 773 774 } else if (order == RotationOrder.ZXZ) { 775 776 // r (Vector3D.plusK) coordinates are : 777 // sin (psi1) sin (phi), -cos (psi1) sin (phi), cos (phi) 778 // (-r) (Vector3D.plusK) coordinates are : 779 // sin (phi) sin (psi2), sin (phi) cos (psi2), cos (phi) 780 // and we can choose to have phi in the interval [0 ; PI] 781 Vector3D v1 = applyTo(Vector3D.PLUS_K); 782 Vector3D v2 = applyInverseTo(Vector3D.PLUS_K); 783 if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) { 784 throw new CardanEulerSingularityException(false); 785 } 786 return new double[] { 787 Math.atan2(v1.getX(), -v1.getY()), 788 Math.acos(v2.getZ()), 789 Math.atan2(v2.getX(), v2.getY()) 790 }; 791 792 } else { // last possibility is ZYZ 793 794 // r (Vector3D.plusK) coordinates are : 795 // cos (psi1) sin (theta), sin (psi1) sin (theta), cos (theta) 796 // (-r) (Vector3D.plusK) coordinates are : 797 // -sin (theta) cos (psi2), sin (theta) sin (psi2), cos (theta) 798 // and we can choose to have theta in the interval [0 ; PI] 799 Vector3D v1 = applyTo(Vector3D.PLUS_K); 800 Vector3D v2 = applyInverseTo(Vector3D.PLUS_K); 801 if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) { 802 throw new CardanEulerSingularityException(false); 803 } 804 return new double[] { 805 Math.atan2(v1.getY(), v1.getX()), 806 Math.acos(v2.getZ()), 807 Math.atan2(v2.getY(), -v2.getX()) 808 }; 809 810 } 811 812 } 813 814 /** Get the 3X3 matrix corresponding to the instance 815 * @return the matrix corresponding to the instance 816 */ 817 public double[][] getMatrix() { 818 819 // products 820 double q0q0 = q0 * q0; 821 double q0q1 = q0 * q1; 822 double q0q2 = q0 * q2; 823 double q0q3 = q0 * q3; 824 double q1q1 = q1 * q1; 825 double q1q2 = q1 * q2; 826 double q1q3 = q1 * q3; 827 double q2q2 = q2 * q2; 828 double q2q3 = q2 * q3; 829 double q3q3 = q3 * q3; 830 831 // create the matrix 832 double[][] m = new double[3][]; 833 m[0] = new double[3]; 834 m[1] = new double[3]; 835 m[2] = new double[3]; 836 837 m [0][0] = 2.0 * (q0q0 + q1q1) - 1.0; 838 m [1][0] = 2.0 * (q1q2 - q0q3); 839 m [2][0] = 2.0 * (q1q3 + q0q2); 840 841 m [0][1] = 2.0 * (q1q2 + q0q3); 842 m [1][1] = 2.0 * (q0q0 + q2q2) - 1.0; 843 m [2][1] = 2.0 * (q2q3 - q0q1); 844 845 m [0][2] = 2.0 * (q1q3 - q0q2); 846 m [1][2] = 2.0 * (q2q3 + q0q1); 847 m [2][2] = 2.0 * (q0q0 + q3q3) - 1.0; 848 849 return m; 850 851 } 852 853 /** Apply the rotation to a vector. 854 * @param u vector to apply the rotation to 855 * @return a new vector which is the image of u by the rotation 856 */ 857 public Vector3D applyTo(Vector3D u) { 858 859 double x = u.getX(); 860 double y = u.getY(); 861 double z = u.getZ(); 862 863 double s = q1 * x + q2 * y + q3 * z; 864 865 return new Vector3D(2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x, 866 2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y, 867 2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - z); 868 869 } 870 871 /** Apply the inverse of the rotation to a vector. 872 * @param u vector to apply the inverse of the rotation to 873 * @return a new vector which such that u is its image by the rotation 874 */ 875 public Vector3D applyInverseTo(Vector3D u) { 876 877 double x = u.getX(); 878 double y = u.getY(); 879 double z = u.getZ(); 880 881 double s = q1 * x + q2 * y + q3 * z; 882 double m0 = -q0; 883 884 return new Vector3D(2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x, 885 2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y, 886 2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - z); 887 888 } 889 890 /** Apply the instance to another rotation. 891 * Applying the instance to a rotation is computing the composition 892 * in an order compliant with the following rule : let u be any 893 * vector and v its image by r (i.e. r.applyTo(u) = v), let w be the image 894 * of v by the instance (i.e. applyTo(v) = w), then w = comp.applyTo(u), 895 * where comp = applyTo(r). 896 * @param r rotation to apply the rotation to 897 * @return a new rotation which is the composition of r by the instance 898 */ 899 public Rotation applyTo(Rotation r) { 900 return new Rotation(r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3), 901 r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2), 902 r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3), 903 r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1), 904 false); 905 } 906 907 /** Apply the inverse of the instance to another rotation. 908 * Applying the inverse of the instance to a rotation is computing 909 * the composition in an order compliant with the following rule : 910 * let u be any vector and v its image by r (i.e. r.applyTo(u) = v), 911 * let w be the inverse image of v by the instance 912 * (i.e. applyInverseTo(v) = w), then w = comp.applyTo(u), where 913 * comp = applyInverseTo(r). 914 * @param r rotation to apply the rotation to 915 * @return a new rotation which is the composition of r by the inverse 916 * of the instance 917 */ 918 public Rotation applyInverseTo(Rotation r) { 919 return new Rotation(-r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3), 920 -r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2), 921 -r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3), 922 -r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1), 923 false); 924 } 925 926 /** Perfect orthogonality on a 3X3 matrix. 927 * @param m initial matrix (not exactly orthogonal) 928 * @param threshold convergence threshold for the iterative 929 * orthogonality correction (convergence is reached when the 930 * difference between two steps of the Frobenius norm of the 931 * correction is below this threshold) 932 * @return an orthogonal matrix close to m 933 * @exception NotARotationMatrixException if the matrix cannot be 934 * orthogonalized with the given threshold after 10 iterations 935 */ 936 private double[][] orthogonalizeMatrix(double[][] m, double threshold) 937 throws NotARotationMatrixException { 938 double[] m0 = m[0]; 939 double[] m1 = m[1]; 940 double[] m2 = m[2]; 941 double x00 = m0[0]; 942 double x01 = m0[1]; 943 double x02 = m0[2]; 944 double x10 = m1[0]; 945 double x11 = m1[1]; 946 double x12 = m1[2]; 947 double x20 = m2[0]; 948 double x21 = m2[1]; 949 double x22 = m2[2]; 950 double fn = 0; 951 double fn1; 952 953 double[][] o = new double[3][3]; 954 double[] o0 = o[0]; 955 double[] o1 = o[1]; 956 double[] o2 = o[2]; 957 958 // iterative correction: Xn+1 = Xn - 0.5 * (Xn.Mt.Xn - M) 959 int i = 0; 960 while (++i < 11) { 961 962 // Mt.Xn 963 double mx00 = m0[0] * x00 + m1[0] * x10 + m2[0] * x20; 964 double mx10 = m0[1] * x00 + m1[1] * x10 + m2[1] * x20; 965 double mx20 = m0[2] * x00 + m1[2] * x10 + m2[2] * x20; 966 double mx01 = m0[0] * x01 + m1[0] * x11 + m2[0] * x21; 967 double mx11 = m0[1] * x01 + m1[1] * x11 + m2[1] * x21; 968 double mx21 = m0[2] * x01 + m1[2] * x11 + m2[2] * x21; 969 double mx02 = m0[0] * x02 + m1[0] * x12 + m2[0] * x22; 970 double mx12 = m0[1] * x02 + m1[1] * x12 + m2[1] * x22; 971 double mx22 = m0[2] * x02 + m1[2] * x12 + m2[2] * x22; 972 973 // Xn+1 974 o0[0] = x00 - 0.5 * (x00 * mx00 + x01 * mx10 + x02 * mx20 - m0[0]); 975 o0[1] = x01 - 0.5 * (x00 * mx01 + x01 * mx11 + x02 * mx21 - m0[1]); 976 o0[2] = x02 - 0.5 * (x00 * mx02 + x01 * mx12 + x02 * mx22 - m0[2]); 977 o1[0] = x10 - 0.5 * (x10 * mx00 + x11 * mx10 + x12 * mx20 - m1[0]); 978 o1[1] = x11 - 0.5 * (x10 * mx01 + x11 * mx11 + x12 * mx21 - m1[1]); 979 o1[2] = x12 - 0.5 * (x10 * mx02 + x11 * mx12 + x12 * mx22 - m1[2]); 980 o2[0] = x20 - 0.5 * (x20 * mx00 + x21 * mx10 + x22 * mx20 - m2[0]); 981 o2[1] = x21 - 0.5 * (x20 * mx01 + x21 * mx11 + x22 * mx21 - m2[1]); 982 o2[2] = x22 - 0.5 * (x20 * mx02 + x21 * mx12 + x22 * mx22 - m2[2]); 983 984 // correction on each elements 985 double corr00 = o0[0] - m0[0]; 986 double corr01 = o0[1] - m0[1]; 987 double corr02 = o0[2] - m0[2]; 988 double corr10 = o1[0] - m1[0]; 989 double corr11 = o1[1] - m1[1]; 990 double corr12 = o1[2] - m1[2]; 991 double corr20 = o2[0] - m2[0]; 992 double corr21 = o2[1] - m2[1]; 993 double corr22 = o2[2] - m2[2]; 994 995 // Frobenius norm of the correction 996 fn1 = corr00 * corr00 + corr01 * corr01 + corr02 * corr02 + 997 corr10 * corr10 + corr11 * corr11 + corr12 * corr12 + 998 corr20 * corr20 + corr21 * corr21 + corr22 * corr22; 999 1000 // convergence test 1001 if (Math.abs(fn1 - fn) <= threshold) 1002 return o; 1003 1004 // prepare next iteration 1005 x00 = o0[0]; 1006 x01 = o0[1]; 1007 x02 = o0[2]; 1008 x10 = o1[0]; 1009 x11 = o1[1]; 1010 x12 = o1[2]; 1011 x20 = o2[0]; 1012 x21 = o2[1]; 1013 x22 = o2[2]; 1014 fn = fn1; 1015 1016 } 1017 1018 // the algorithm did not converge after 10 iterations 1019 throw new NotARotationMatrixException( 1020 "unable to orthogonalize matrix in {0} iterations", 1021 i - 1); 1022 } 1023 1024 /** Compute the <i>distance</i> between two rotations. 1025 * <p>The <i>distance</i> is intended here as a way to check if two 1026 * rotations are almost similar (i.e. they transform vectors the same way) 1027 * or very different. It is mathematically defined as the angle of 1028 * the rotation r that prepended to one of the rotations gives the other 1029 * one:</p> 1030 * <pre> 1031 * r<sub>1</sub>(r) = r<sub>2</sub> 1032 * </pre> 1033 * <p>This distance is an angle between 0 and π. Its value is the smallest 1034 * possible upper bound of the angle in radians between r<sub>1</sub>(v) 1035 * and r<sub>2</sub>(v) for all possible vectors v. This upper bound is 1036 * reached for some v. The distance is equal to 0 if and only if the two 1037 * rotations are identical.</p> 1038 * <p>Comparing two rotations should always be done using this value rather 1039 * than for example comparing the components of the quaternions. It is much 1040 * more stable, and has a geometric meaning. Also comparing quaternions 1041 * components is error prone since for example quaternions (0.36, 0.48, -0.48, -0.64) 1042 * and (-0.36, -0.48, 0.48, 0.64) represent exactly the same rotation despite 1043 * their components are different (they are exact opposites).</p> 1044 * @param r1 first rotation 1045 * @param r2 second rotation 1046 * @return <i>distance</i> between r1 and r2 1047 */ 1048 public static double distance(Rotation r1, Rotation r2) { 1049 return r1.applyInverseTo(r2).getAngle(); 1050 } 1051 1052 }