ground/gcs/src/libs/eigen/test/geo_transformations.cpp

   1 // This file is part of Eigen, a lightweight C++ template library
   2 // for linear algebra.
   3 //
   4 // Copyright (C) 2008-2009 Gael Guennebaud <gael.guennebaud@inria.fr>
   5 //
   6 // This Source Code Form is subject to the terms of the Mozilla
   7 // Public License v. 2.0. If a copy of the MPL was not distributed
   8 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
   9
  10 #include "main.h"
  11 #include <Eigen/Geometry>
  12 #include <Eigen/LU>
  13 #include <Eigen/SVD>
  14
  15 template<typename T>
  16 Matrix<T,2,1> angleToVec(T a)
  17 {
  18   return Matrix<T,2,1>(std::cos(a), std::sin(a));
  19 }
  20
  21 // This permits to workaround a bug in clang/llvm code generation.
  22 template<typename T>
  23 EIGEN_DONT_INLINE
  24 void dont_over_optimize(T& x) { volatile typename T::Scalar tmp = x(0); x(0) = tmp; }
  25
  26 template<typename Scalar, int Mode, int Options> void non_projective_only()
  27 {
  28     /* this test covers the following files:
  29      Cross.h Quaternion.h, Transform.cpp
  30   */
  31   typedef Matrix<Scalar,3,1> Vector3;
  32   typedef Quaternion<Scalar> Quaternionx;
  33   typedef AngleAxis<Scalar> AngleAxisx;
  34   typedef Transform<Scalar,3,Mode,Options> Transform3;
  35   typedef DiagonalMatrix<Scalar,3> AlignedScaling3;
  36   typedef Translation<Scalar,3> Translation3;
  37
  38   Vector3 v0 = Vector3::Random(),
  39           v1 = Vector3::Random();
  40
  41   Transform3 t0, t1, t2;
  42
  43   Scalar a = internal::random<Scalar>(-Scalar(EIGEN_PI), Scalar(EIGEN_PI));
  44
  45   Quaternionx q1, q2;
  46
  47   q1 = AngleAxisx(a, v0.normalized());
  48
  49   t0 = Transform3::Identity();
  50   VERIFY_IS_APPROX(t0.matrix(), Transform3::MatrixType::Identity());
  51
  52   t0.linear() = q1.toRotationMatrix();
  53
  54   v0 << 50, 2, 1;
  55   t0.scale(v0);
  56
  57   VERIFY_IS_APPROX( (t0 * Vector3(1,0,0)).template head<3>().norm(), v0.x());
  58
  59   t0.setIdentity();
  60   t1.setIdentity();
  61   v1 << 1, 2, 3;
  62   t0.linear() = q1.toRotationMatrix();
  63   t0.pretranslate(v0);
  64   t0.scale(v1);
  65   t1.linear() = q1.conjugate().toRotationMatrix();
  66   t1.prescale(v1.cwiseInverse());
  67   t1.translate(-v0);
  68
  69   VERIFY((t0 * t1).matrix().isIdentity(test_precision<Scalar>()));
  70
  71   t1.fromPositionOrientationScale(v0, q1, v1);
  72   VERIFY_IS_APPROX(t1.matrix(), t0.matrix());
  73   VERIFY_IS_APPROX(t1*v1, t0*v1);
  74
  75   // translation * vector
  76   t0.setIdentity();
  77   t0.translate(v0);
  78   VERIFY_IS_APPROX((t0 * v1).template head<3>(), Translation3(v0) * v1);
  79
  80   // AlignedScaling * vector
  81   t0.setIdentity();
  82   t0.scale(v0);
  83   VERIFY_IS_APPROX((t0 * v1).template head<3>(), AlignedScaling3(v0) * v1);
  84 }
  85
  86 template<typename Scalar, int Mode, int Options> void transformations()
  87 {
  88   /* this test covers the following files:
  89      Cross.h Quaternion.h, Transform.cpp
  90   */
  91   using std::cos;
  92   using std::abs;
  93   typedef Matrix<Scalar,3,3> Matrix3;
  94   typedef Matrix<Scalar,4,4> Matrix4;
  95   typedef Matrix<Scalar,2,1> Vector2;
  96   typedef Matrix<Scalar,3,1> Vector3;
  97   typedef Matrix<Scalar,4,1> Vector4;
  98   typedef Quaternion<Scalar> Quaternionx;
  99   typedef AngleAxis<Scalar> AngleAxisx;
 100   typedef Transform<Scalar,2,Mode,Options> Transform2;
 101   typedef Transform<Scalar,3,Mode,Options> Transform3;
 102   typedef typename Transform3::MatrixType MatrixType;
 103   typedef DiagonalMatrix<Scalar,3> AlignedScaling3;
 104   typedef Translation<Scalar,2> Translation2;
 105   typedef Translation<Scalar,3> Translation3;
 106
 107   Vector3 v0 = Vector3::Random(),
 108           v1 = Vector3::Random();
 109   Matrix3 matrot1, m;
 110
 111   Scalar a = internal::random<Scalar>(-Scalar(EIGEN_PI), Scalar(EIGEN_PI));
 112   Scalar s0 = internal::random<Scalar>(), s1 = internal::random<Scalar>();
 113
 114   while(v0.norm() < test_precision<Scalar>()) v0 = Vector3::Random();
 115   while(v1.norm() < test_precision<Scalar>()) v1 = Vector3::Random();
 116
 117   VERIFY_IS_APPROX(v0, AngleAxisx(a, v0.normalized()) * v0);
 118   VERIFY_IS_APPROX(-v0, AngleAxisx(Scalar(EIGEN_PI), v0.unitOrthogonal()) * v0);
 119   if(abs(cos(a)) > test_precision<Scalar>())
 120   {
 121     VERIFY_IS_APPROX(cos(a)*v0.squaredNorm(), v0.dot(AngleAxisx(a, v0.unitOrthogonal()) * v0));
 122   }
 123   m = AngleAxisx(a, v0.normalized()).toRotationMatrix().adjoint();
 124   VERIFY_IS_APPROX(Matrix3::Identity(), m * AngleAxisx(a, v0.normalized()));
 125   VERIFY_IS_APPROX(Matrix3::Identity(), AngleAxisx(a, v0.normalized()) * m);
 126
 127   Quaternionx q1, q2;
 128   q1 = AngleAxisx(a, v0.normalized());
 129   q2 = AngleAxisx(a, v1.normalized());
 130
 131   // rotation matrix conversion
 132   matrot1 = AngleAxisx(Scalar(0.1), Vector3::UnitX())
 133           * AngleAxisx(Scalar(0.2), Vector3::UnitY())
 134           * AngleAxisx(Scalar(0.3), Vector3::UnitZ());
 135   VERIFY_IS_APPROX(matrot1 * v1,
 136        AngleAxisx(Scalar(0.1), Vector3(1,0,0)).toRotationMatrix()
 137     * (AngleAxisx(Scalar(0.2), Vector3(0,1,0)).toRotationMatrix()
 138     * (AngleAxisx(Scalar(0.3), Vector3(0,0,1)).toRotationMatrix() * v1)));
 139
 140   // angle-axis conversion
 141   AngleAxisx aa = AngleAxisx(q1);
 142   VERIFY_IS_APPROX(q1 * v1, Quaternionx(aa) * v1);
 143
 144   // The following test is stable only if 2*angle != angle and v1 is not colinear with axis
 145   if( (abs(aa.angle()) > test_precision<Scalar>()) && (abs(aa.axis().dot(v1.normalized()))<(Scalar(1)-Scalar(4)*test_precision<Scalar>())) )
 146   {
 147     VERIFY( !(q1 * v1).isApprox(Quaternionx(AngleAxisx(aa.angle()*2,aa.axis())) * v1) );
 148   }
 149
 150   aa.fromRotationMatrix(aa.toRotationMatrix());
 151   VERIFY_IS_APPROX(q1 * v1, Quaternionx(aa) * v1);
 152   // The following test is stable only if 2*angle != angle and v1 is not colinear with axis
 153   if( (abs(aa.angle()) > test_precision<Scalar>()) && (abs(aa.axis().dot(v1.normalized()))<(Scalar(1)-Scalar(4)*test_precision<Scalar>())) )
 154   {
 155     VERIFY( !(q1 * v1).isApprox(Quaternionx(AngleAxisx(aa.angle()*2,aa.axis())) * v1) );
 156   }
 157
 158   // AngleAxis
 159   VERIFY_IS_APPROX(AngleAxisx(a,v1.normalized()).toRotationMatrix(),
 160     Quaternionx(AngleAxisx(a,v1.normalized())).toRotationMatrix());
 161
 162   AngleAxisx aa1;
 163   m = q1.toRotationMatrix();
 164   aa1 = m;
 165   VERIFY_IS_APPROX(AngleAxisx(m).toRotationMatrix(),
 166     Quaternionx(m).toRotationMatrix());
 167
 168   // Transform
 169   // TODO complete the tests !
 170   a = 0;
 171   while (abs(a)<Scalar(0.1))
 172     a = internal::random<Scalar>(-Scalar(0.4)*Scalar(EIGEN_PI), Scalar(0.4)*Scalar(EIGEN_PI));
 173   q1 = AngleAxisx(a, v0.normalized());
 174   Transform3 t0, t1, t2;
 175
 176   // first test setIdentity() and Identity()
 177   t0.setIdentity();
 178   VERIFY_IS_APPROX(t0.matrix(), Transform3::MatrixType::Identity());
 179   t0.matrix().setZero();
 180   t0 = Transform3::Identity();
 181   VERIFY_IS_APPROX(t0.matrix(), Transform3::MatrixType::Identity());
 182
 183   t0.setIdentity();
 184   t1.setIdentity();
 185   v1 << 1, 2, 3;
 186   t0.linear() = q1.toRotationMatrix();
 187   t0.pretranslate(v0);
 188   t0.scale(v1);
 189   t1.linear() = q1.conjugate().toRotationMatrix();
 190   t1.prescale(v1.cwiseInverse());
 191   t1.translate(-v0);
 192
 193   VERIFY((t0 * t1).matrix().isIdentity(test_precision<Scalar>()));
 194
 195   t1.fromPositionOrientationScale(v0, q1, v1);
 196   VERIFY_IS_APPROX(t1.matrix(), t0.matrix());
 197
 198   t0.setIdentity(); t0.scale(v0).rotate(q1.toRotationMatrix());
 199   t1.setIdentity(); t1.scale(v0).rotate(q1);
 200   VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
 201
 202   t0.setIdentity(); t0.scale(v0).rotate(AngleAxisx(q1));
 203   VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
 204
 205   VERIFY_IS_APPROX(t0.scale(a).matrix(), t1.scale(Vector3::Constant(a)).matrix());
 206   VERIFY_IS_APPROX(t0.prescale(a).matrix(), t1.prescale(Vector3::Constant(a)).matrix());
 207
 208   // More transform constructors, operator=, operator*=
 209
 210   Matrix3 mat3 = Matrix3::Random();
 211   Matrix4 mat4;
 212   mat4 << mat3 , Vector3::Zero() , Vector4::Zero().transpose();
 213   Transform3 tmat3(mat3), tmat4(mat4);
 214   if(Mode!=int(AffineCompact))
 215     tmat4.matrix()(3,3) = Scalar(1);
 216   VERIFY_IS_APPROX(tmat3.matrix(), tmat4.matrix());
 217
 218   Scalar a3 = internal::random<Scalar>(-Scalar(EIGEN_PI), Scalar(EIGEN_PI));
 219   Vector3 v3 = Vector3::Random().normalized();
 220   AngleAxisx aa3(a3, v3);
 221   Transform3 t3(aa3);
 222   Transform3 t4;
 223   t4 = aa3;
 224   VERIFY_IS_APPROX(t3.matrix(), t4.matrix());
 225   t4.rotate(AngleAxisx(-a3,v3));
 226   VERIFY_IS_APPROX(t4.matrix(), MatrixType::Identity());
 227   t4 *= aa3;
 228   VERIFY_IS_APPROX(t3.matrix(), t4.matrix());
 229
 230   do {
 231     v3 = Vector3::Random();
 232     dont_over_optimize(v3);
 233   } while (v3.cwiseAbs().minCoeff()<NumTraits<Scalar>::epsilon());
 234   Translation3 tv3(v3);
 235   Transform3 t5(tv3);
 236   t4 = tv3;
 237   VERIFY_IS_APPROX(t5.matrix(), t4.matrix());
 238   t4.translate((-v3).eval());
 239   VERIFY_IS_APPROX(t4.matrix(), MatrixType::Identity());
 240   t4 *= tv3;
 241   VERIFY_IS_APPROX(t5.matrix(), t4.matrix());
 242
 243   AlignedScaling3 sv3(v3);
 244   Transform3 t6(sv3);
 245   t4 = sv3;
 246   VERIFY_IS_APPROX(t6.matrix(), t4.matrix());
 247   t4.scale(v3.cwiseInverse());
 248   VERIFY_IS_APPROX(t4.matrix(), MatrixType::Identity());
 249   t4 *= sv3;
 250   VERIFY_IS_APPROX(t6.matrix(), t4.matrix());
 251
 252   // matrix * transform
 253   VERIFY_IS_APPROX((t3.matrix()*t4).matrix(), (t3*t4).matrix());
 254
 255   // chained Transform product
 256   VERIFY_IS_APPROX(((t3*t4)*t5).matrix(), (t3*(t4*t5)).matrix());
 257
 258   // check that Transform product doesn't have aliasing problems
 259   t5 = t4;
 260   t5 = t5*t5;
 261   VERIFY_IS_APPROX(t5, t4*t4);
 262
 263   // 2D transformation
 264   Transform2 t20, t21;
 265   Vector2 v20 = Vector2::Random();
 266   Vector2 v21 = Vector2::Random();
 267   for (int k=0; k<2; ++k)
 268     if (abs(v21[k])<Scalar(1e-3)) v21[k] = Scalar(1e-3);
 269   t21.setIdentity();
 270   t21.linear() = Rotation2D<Scalar>(a).toRotationMatrix();
 271   VERIFY_IS_APPROX(t20.fromPositionOrientationScale(v20,a,v21).matrix(),
 272     t21.pretranslate(v20).scale(v21).matrix());
 273
 274   t21.setIdentity();
 275   t21.linear() = Rotation2D<Scalar>(-a).toRotationMatrix();
 276   VERIFY( (t20.fromPositionOrientationScale(v20,a,v21)
 277         * (t21.prescale(v21.cwiseInverse()).translate(-v20))).matrix().isIdentity(test_precision<Scalar>()) );
 278
 279   // Transform - new API
 280   // 3D
 281   t0.setIdentity();
 282   t0.rotate(q1).scale(v0).translate(v0);
 283   // mat * aligned scaling and mat * translation
 284   t1 = (Matrix3(q1) * AlignedScaling3(v0)) * Translation3(v0);
 285   VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
 286   t1 = (Matrix3(q1) * Eigen::Scaling(v0)) * Translation3(v0);
 287   VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
 288   t1 = (q1 * Eigen::Scaling(v0)) * Translation3(v0);
 289   VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
 290   // mat * transformation and aligned scaling * translation
 291   t1 = Matrix3(q1) * (AlignedScaling3(v0) * Translation3(v0));
 292   VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
 293
 294
 295   t0.setIdentity();
 296   t0.scale(s0).translate(v0);
 297   t1 = Eigen::Scaling(s0) * Translation3(v0);
 298   VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
 299   t0.prescale(s0);
 300   t1 = Eigen::Scaling(s0) * t1;
 301   VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
 302
 303   t0 = t3;
 304   t0.scale(s0);
 305   t1 = t3 * Eigen::Scaling(s0,s0,s0);
 306   VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
 307   t0.prescale(s0);
 308   t1 = Eigen::Scaling(s0,s0,s0) * t1;
 309   VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
 310
 311   t0 = t3;
 312   t0.scale(s0);
 313   t1 = t3 * Eigen::Scaling(s0);
 314   VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
 315   t0.prescale(s0);
 316   t1 = Eigen::Scaling(s0) * t1;
 317   VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
 318
 319   t0.setIdentity();
 320   t0.prerotate(q1).prescale(v0).pretranslate(v0);
 321   // translation * aligned scaling and transformation * mat
 322   t1 = (Translation3(v0) * AlignedScaling3(v0)) * Transform3(q1);
 323   VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
 324   // scaling * mat and translation * mat
 325   t1 = Translation3(v0) * (AlignedScaling3(v0) * Transform3(q1));
 326   VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
 327
 328   t0.setIdentity();
 329   t0.scale(v0).translate(v0).rotate(q1);
 330   // translation * mat and aligned scaling * transformation
 331   t1 = AlignedScaling3(v0) * (Translation3(v0) * Transform3(q1));
 332   VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
 333   // transformation * aligned scaling
 334   t0.scale(v0);
 335   t1 *= AlignedScaling3(v0);
 336   VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
 337   t1 = AlignedScaling3(v0) * (Translation3(v0) * Transform3(q1));
 338   t1 = t1 * v0.asDiagonal();
 339   VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
 340   // transformation * translation
 341   t0.translate(v0);
 342   t1 = t1 * Translation3(v0);
 343   VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
 344   // translation * transformation
 345   t0.pretranslate(v0);
 346   t1 = Translation3(v0) * t1;
 347   VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
 348
 349   // transform * quaternion
 350   t0.rotate(q1);
 351   t1 = t1 * q1;
 352   VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
 353
 354   // translation * quaternion
 355   t0.translate(v1).rotate(q1);
 356   t1 = t1 * (Translation3(v1) * q1);
 357   VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
 358
 359   // aligned scaling * quaternion
 360   t0.scale(v1).rotate(q1);
 361   t1 = t1 * (AlignedScaling3(v1) * q1);
 362   VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
 363
 364   // quaternion * transform
 365   t0.prerotate(q1);
 366   t1 = q1 * t1;
 367   VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
 368
 369   // quaternion * translation
 370   t0.rotate(q1).translate(v1);
 371   t1 = t1 * (q1 * Translation3(v1));
 372   VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
 373
 374   // quaternion * aligned scaling
 375   t0.rotate(q1).scale(v1);
 376   t1 = t1 * (q1 * AlignedScaling3(v1));
 377   VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
 378
 379   // test transform inversion
 380   t0.setIdentity();
 381   t0.translate(v0);
 382   do {
 383     t0.linear().setRandom();
 384   } while(t0.linear().jacobiSvd().singularValues()(2)<test_precision<Scalar>());
 385   Matrix4 t044 = Matrix4::Zero();
 386   t044(3,3) = 1;
 387   t044.block(0,0,t0.matrix().rows(),4) = t0.matrix();
 388   VERIFY_IS_APPROX(t0.inverse(Affine).matrix(), t044.inverse().block(0,0,t0.matrix().rows(),4));
 389   t0.setIdentity();
 390   t0.translate(v0).rotate(q1);
 391   t044 = Matrix4::Zero();
 392   t044(3,3) = 1;
 393   t044.block(0,0,t0.matrix().rows(),4) = t0.matrix();
 394   VERIFY_IS_APPROX(t0.inverse(Isometry).matrix(), t044.inverse().block(0,0,t0.matrix().rows(),4));
 395
 396   Matrix3 mat_rotation, mat_scaling;
 397   t0.setIdentity();
 398   t0.translate(v0).rotate(q1).scale(v1);
 399   t0.computeRotationScaling(&mat_rotation, &mat_scaling);
 400   VERIFY_IS_APPROX(t0.linear(), mat_rotation * mat_scaling);
 401   VERIFY_IS_APPROX(mat_rotation*mat_rotation.adjoint(), Matrix3::Identity());
 402   VERIFY_IS_APPROX(mat_rotation.determinant(), Scalar(1));
 403   t0.computeScalingRotation(&mat_scaling, &mat_rotation);
 404   VERIFY_IS_APPROX(t0.linear(), mat_scaling * mat_rotation);
 405   VERIFY_IS_APPROX(mat_rotation*mat_rotation.adjoint(), Matrix3::Identity());
 406   VERIFY_IS_APPROX(mat_rotation.determinant(), Scalar(1));
 407
 408   // test casting
 409   Transform<float,3,Mode> t1f = t1.template cast<float>();
 410   VERIFY_IS_APPROX(t1f.template cast<Scalar>(),t1);
 411   Transform<double,3,Mode> t1d = t1.template cast<double>();
 412   VERIFY_IS_APPROX(t1d.template cast<Scalar>(),t1);
 413
 414   Translation3 tr1(v0);
 415   Translation<float,3> tr1f = tr1.template cast<float>();
 416   VERIFY_IS_APPROX(tr1f.template cast<Scalar>(),tr1);
 417   Translation<double,3> tr1d = tr1.template cast<double>();
 418   VERIFY_IS_APPROX(tr1d.template cast<Scalar>(),tr1);
 419
 420   AngleAxis<float> aa1f = aa1.template cast<float>();
 421   VERIFY_IS_APPROX(aa1f.template cast<Scalar>(),aa1);
 422   AngleAxis<double> aa1d = aa1.template cast<double>();
 423   VERIFY_IS_APPROX(aa1d.template cast<Scalar>(),aa1);
 424
 425   Rotation2D<Scalar> r2d1(internal::random<Scalar>());
 426   Rotation2D<float> r2d1f = r2d1.template cast<float>();
 427   VERIFY_IS_APPROX(r2d1f.template cast<Scalar>(),r2d1);
 428   Rotation2D<double> r2d1d = r2d1.template cast<double>();
 429   VERIFY_IS_APPROX(r2d1d.template cast<Scalar>(),r2d1);
 430
 431   for(int k=0; k<100; ++k)
 432   {
 433     Scalar angle = internal::random<Scalar>(-100,100);
 434     Rotation2D<Scalar> rot2(angle);
 435     VERIFY( rot2.smallestPositiveAngle() >= 0 );
 436     VERIFY( rot2.smallestPositiveAngle() <= Scalar(2)*Scalar(EIGEN_PI) );
 437     VERIFY_IS_APPROX( angleToVec(rot2.smallestPositiveAngle()), angleToVec(rot2.angle()) );
 438
 439     VERIFY( rot2.smallestAngle() >= -Scalar(EIGEN_PI) );
 440     VERIFY( rot2.smallestAngle() <=  Scalar(EIGEN_PI) );
 441     VERIFY_IS_APPROX( angleToVec(rot2.smallestAngle()), angleToVec(rot2.angle()) );
 442
 443     Matrix<Scalar,2,2> rot2_as_mat(rot2);
 444     Rotation2D<Scalar> rot3(rot2_as_mat);
 445     VERIFY_IS_APPROX( angleToVec(rot2.smallestAngle()),  angleToVec(rot3.angle()) );
 446   }
 447
 448   s0 = internal::random<Scalar>(-100,100);
 449   s1 = internal::random<Scalar>(-100,100);
 450   Rotation2D<Scalar> R0(s0), R1(s1);
 451
 452   t20 = Translation2(v20) * (R0 * Eigen::Scaling(s0));
 453   t21 = Translation2(v20) * R0 * Eigen::Scaling(s0);
 454   VERIFY_IS_APPROX(t20,t21);
 455
 456   t20 = Translation2(v20) * (R0 * R0.inverse() * Eigen::Scaling(s0));
 457   t21 = Translation2(v20) * Eigen::Scaling(s0);
 458   VERIFY_IS_APPROX(t20,t21);
 459
 460   VERIFY_IS_APPROX(s0, (R0.slerp(0, R1)).angle());
 461   VERIFY_IS_APPROX( angleToVec(R1.smallestPositiveAngle()), angleToVec((R0.slerp(1, R1)).smallestPositiveAngle()) );
 462   VERIFY_IS_APPROX(R0.smallestPositiveAngle(), (R0.slerp(0.5, R0)).smallestPositiveAngle());
 463
 464   if(std::cos(s0)>0)
 465     VERIFY_IS_MUCH_SMALLER_THAN((R0.slerp(0.5, R0.inverse())).smallestAngle(), Scalar(1));
 466   else
 467     VERIFY_IS_APPROX(Scalar(EIGEN_PI), (R0.slerp(0.5, R0.inverse())).smallestPositiveAngle());
 468
 469   // Check path length
 470   Scalar l = 0;
 471   int path_steps = 100;
 472   for(int k=0; k<path_steps; ++k)
 473   {
 474     Scalar a1 = R0.slerp(Scalar(k)/Scalar(path_steps), R1).angle();
 475     Scalar a2 = R0.slerp(Scalar(k+1)/Scalar(path_steps), R1).angle();
 476     l += std::abs(a2-a1);
 477   }
 478   VERIFY(l<=Scalar(EIGEN_PI)*(Scalar(1)+NumTraits<Scalar>::epsilon()*Scalar(path_steps/2)));
 479
 480   // check basic features
 481   {
 482     Rotation2D<Scalar> r1;           // default ctor
 483     r1 = Rotation2D<Scalar>(s0);     // copy assignment
 484     VERIFY_IS_APPROX(r1.angle(),s0);
 485     Rotation2D<Scalar> r2(r1);       // copy ctor
 486     VERIFY_IS_APPROX(r2.angle(),s0);
 487   }
 488
 489   {
 490     Transform3 t32(Matrix4::Random()), t33, t34;
 491     t34 = t33 = t32;
 492     t32.scale(v0);
 493     t33*=AlignedScaling3(v0);
 494     VERIFY_IS_APPROX(t32.matrix(), t33.matrix());
 495     t33 = t34 * AlignedScaling3(v0);
 496     VERIFY_IS_APPROX(t32.matrix(), t33.matrix());
 497   }
 498
 499 }
 500
 501 template<typename A1, typename A2, typename P, typename Q, typename V, typename H>
 502 void transform_associativity_left(const A1& a1, const A2& a2, const P& p, const Q& q, const V& v, const H& h)
 503 {
 504   VERIFY_IS_APPROX( q*(a1*v), (q*a1)*v );
 505   VERIFY_IS_APPROX( q*(a2*v), (q*a2)*v );
 506   VERIFY_IS_APPROX( q*(p*h).hnormalized(),  ((q*p)*h).hnormalized() );
 507 }
 508
 509 template<typename A1, typename A2, typename P, typename Q, typename V, typename H>
 510 void transform_associativity2(const A1& a1, const A2& a2, const P& p, const Q& q, const V& v, const H& h)
 511 {
 512   VERIFY_IS_APPROX( a1*(q*v), (a1*q)*v );
 513   VERIFY_IS_APPROX( a2*(q*v), (a2*q)*v );
 514   VERIFY_IS_APPROX( p *(q*v).homogeneous(), (p *q)*v.homogeneous() );
 515
 516   transform_associativity_left(a1, a2,p, q, v, h);
 517 }
 518
 519 template<typename Scalar, int Dim, int Options,typename RotationType>
 520 void transform_associativity(const RotationType& R)
 521 {
 522   typedef Matrix<Scalar,Dim,1> VectorType;
 523   typedef Matrix<Scalar,Dim+1,1> HVectorType;
 524   typedef Matrix<Scalar,Dim,Dim> LinearType;
 525   typedef Matrix<Scalar,Dim+1,Dim+1> MatrixType;
 526   typedef Transform<Scalar,Dim,AffineCompact,Options> AffineCompactType;
 527   typedef Transform<Scalar,Dim,Affine,Options> AffineType;
 528   typedef Transform<Scalar,Dim,Projective,Options> ProjectiveType;
 529   typedef DiagonalMatrix<Scalar,Dim> ScalingType;
 530   typedef Translation<Scalar,Dim> TranslationType;
 531
 532   AffineCompactType A1c; A1c.matrix().setRandom();
 533   AffineCompactType A2c; A2c.matrix().setRandom();
 534   AffineType A1(A1c);
 535   AffineType A2(A2c);
 536   ProjectiveType P1; P1.matrix().setRandom();
 537   VectorType v1 = VectorType::Random();
 538   VectorType v2 = VectorType::Random();
 539   HVectorType h1 = HVectorType::Random();
 540   Scalar s1 = internal::random<Scalar>();
 541   LinearType L = LinearType::Random();
 542   MatrixType M = MatrixType::Random();
 543
 544   CALL_SUBTEST( transform_associativity2(A1c, A1, P1, A2, v2, h1) );
 545   CALL_SUBTEST( transform_associativity2(A1c, A1, P1, A2c, v2, h1) );
 546   CALL_SUBTEST( transform_associativity2(A1c, A1, P1, v1.asDiagonal(), v2, h1) );
 547   CALL_SUBTEST( transform_associativity2(A1c, A1, P1, ScalingType(v1), v2, h1) );
 548   CALL_SUBTEST( transform_associativity2(A1c, A1, P1, Scaling(v1), v2, h1) );
 549   CALL_SUBTEST( transform_associativity2(A1c, A1, P1, Scaling(s1), v2, h1) );
 550   CALL_SUBTEST( transform_associativity2(A1c, A1, P1, TranslationType(v1), v2, h1) );
 551   CALL_SUBTEST( transform_associativity_left(A1c, A1, P1, L, v2, h1) );
 552   CALL_SUBTEST( transform_associativity2(A1c, A1, P1, R, v2, h1) );
 553
 554   VERIFY_IS_APPROX( A1*(M*h1), (A1*M)*h1 );
 555   VERIFY_IS_APPROX( A1c*(M*h1), (A1c*M)*h1 );
 556   VERIFY_IS_APPROX( P1*(M*h1), (P1*M)*h1 );
 557
 558   VERIFY_IS_APPROX( M*(A1*h1), (M*A1)*h1 );
 559   VERIFY_IS_APPROX( M*(A1c*h1), (M*A1c)*h1 );
 560   VERIFY_IS_APPROX( M*(P1*h1),  ((M*P1)*h1) );
 561 }
 562
 563 template<typename Scalar> void transform_alignment()
 564 {
 565   typedef Transform<Scalar,3,Projective,AutoAlign> Projective3a;
 566   typedef Transform<Scalar,3,Projective,DontAlign> Projective3u;
 567
 568   EIGEN_ALIGN_MAX Scalar array1[16];
 569   EIGEN_ALIGN_MAX Scalar array2[16];
 570   EIGEN_ALIGN_MAX Scalar array3[16+1];
 571   Scalar* array3u = array3+1;
 572
 573   Projective3a *p1 = ::new(reinterpret_cast<void*>(array1)) Projective3a;
 574   Projective3u *p2 = ::new(reinterpret_cast<void*>(array2)) Projective3u;
 575   Projective3u *p3 = ::new(reinterpret_cast<void*>(array3u)) Projective3u;
 576
 577   p1->matrix().setRandom();
 578   *p2 = *p1;
 579   *p3 = *p1;
 580
 581   VERIFY_IS_APPROX(p1->matrix(), p2->matrix());
 582   VERIFY_IS_APPROX(p1->matrix(), p3->matrix());
 583
 584   VERIFY_IS_APPROX( (*p1) * (*p1), (*p2)*(*p3));
 585
 586   #if defined(EIGEN_VECTORIZE) && EIGEN_MAX_STATIC_ALIGN_BYTES>0
 587   if(internal::packet_traits<Scalar>::Vectorizable)
 588     VERIFY_RAISES_ASSERT((::new(reinterpret_cast<void*>(array3u)) Projective3a));
 589   #endif
 590 }
 591
 592 template<typename Scalar, int Dim, int Options> void transform_products()
 593 {
 594   typedef Matrix<Scalar,Dim+1,Dim+1> Mat;
 595   typedef Transform<Scalar,Dim,Projective,Options> Proj;
 596   typedef Transform<Scalar,Dim,Affine,Options> Aff;
 597   typedef Transform<Scalar,Dim,AffineCompact,Options> AffC;
 598
 599   Proj p; p.matrix().setRandom();
 600   Aff a; a.linear().setRandom(); a.translation().setRandom();
 601   AffC ac = a;
 602
 603   Mat p_m(p.matrix()), a_m(a.matrix());
 604
 605   VERIFY_IS_APPROX((p*p).matrix(), p_m*p_m);
 606   VERIFY_IS_APPROX((a*a).matrix(), a_m*a_m);
 607   VERIFY_IS_APPROX((p*a).matrix(), p_m*a_m);
 608   VERIFY_IS_APPROX((a*p).matrix(), a_m*p_m);
 609   VERIFY_IS_APPROX((ac*a).matrix(), a_m*a_m);
 610   VERIFY_IS_APPROX((a*ac).matrix(), a_m*a_m);
 611   VERIFY_IS_APPROX((p*ac).matrix(), p_m*a_m);
 612   VERIFY_IS_APPROX((ac*p).matrix(), a_m*p_m);
 613 }
 614
 615 void test_geo_transformations()
 616 {
 617   for(int i = 0; i < g_repeat; i++) {
 618     CALL_SUBTEST_1(( transformations<double,Affine,AutoAlign>() ));
 619     CALL_SUBTEST_1(( non_projective_only<double,Affine,AutoAlign>() ));
 620
 621     CALL_SUBTEST_2(( transformations<float,AffineCompact,AutoAlign>() ));
 622     CALL_SUBTEST_2(( non_projective_only<float,AffineCompact,AutoAlign>() ));
 623     CALL_SUBTEST_2(( transform_alignment<float>() ));
 624
 625     CALL_SUBTEST_3(( transformations<double,Projective,AutoAlign>() ));
 626     CALL_SUBTEST_3(( transformations<double,Projective,DontAlign>() ));
 627     CALL_SUBTEST_3(( transform_alignment<double>() ));
 628
 629     CALL_SUBTEST_4(( transformations<float,Affine,RowMajor|AutoAlign>() ));
 630     CALL_SUBTEST_4(( non_projective_only<float,Affine,RowMajor>() ));
 631
 632     CALL_SUBTEST_5(( transformations<double,AffineCompact,RowMajor|AutoAlign>() ));
 633     CALL_SUBTEST_5(( non_projective_only<double,AffineCompact,RowMajor>() ));
 634
 635     CALL_SUBTEST_6(( transformations<double,Projective,RowMajor|AutoAlign>() ));
 636     CALL_SUBTEST_6(( transformations<double,Projective,RowMajor|DontAlign>() ));
 637
 638
 639     CALL_SUBTEST_7(( transform_products<double,3,RowMajor|AutoAlign>() ));
 640     CALL_SUBTEST_7(( transform_products<float,2,AutoAlign>() ));
 641
 642     CALL_SUBTEST_8(( transform_associativity<double,2,ColMajor>(Rotation2D<double>(internal::random<double>()*double(EIGEN_PI))) ));
 643     CALL_SUBTEST_8(( transform_associativity<double,3,ColMajor>(Quaterniond::UnitRandom()) ));
 644   }
 645 }