Quaternion.h (34367B)
1 // This file is part of Eigen, a lightweight C++ template library 2 // for linear algebra. 3 // 4 // Copyright (C) 2008-2010 Gael Guennebaud <gael.guennebaud@inria.fr> 5 // Copyright (C) 2009 Mathieu Gautier <mathieu.gautier@cea.fr> 6 // 7 // This Source Code Form is subject to the terms of the Mozilla 8 // Public License v. 2.0. If a copy of the MPL was not distributed 9 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. 10 11 #ifndef EIGEN_QUATERNION_H 12 #define EIGEN_QUATERNION_H 13 namespace Eigen { 14 15 16 /*************************************************************************** 17 * Definition of QuaternionBase<Derived> 18 * The implementation is at the end of the file 19 ***************************************************************************/ 20 21 namespace internal { 22 template<typename Other, 23 int OtherRows=Other::RowsAtCompileTime, 24 int OtherCols=Other::ColsAtCompileTime> 25 struct quaternionbase_assign_impl; 26 } 27 28 /** \geometry_module \ingroup Geometry_Module 29 * \class QuaternionBase 30 * \brief Base class for quaternion expressions 31 * \tparam Derived derived type (CRTP) 32 * \sa class Quaternion 33 */ 34 template<class Derived> 35 class QuaternionBase : public RotationBase<Derived, 3> 36 { 37 public: 38 typedef RotationBase<Derived, 3> Base; 39 40 using Base::operator*; 41 using Base::derived; 42 43 typedef typename internal::traits<Derived>::Scalar Scalar; 44 typedef typename NumTraits<Scalar>::Real RealScalar; 45 typedef typename internal::traits<Derived>::Coefficients Coefficients; 46 typedef typename Coefficients::CoeffReturnType CoeffReturnType; 47 typedef typename internal::conditional<bool(internal::traits<Derived>::Flags&LvalueBit), 48 Scalar&, CoeffReturnType>::type NonConstCoeffReturnType; 49 50 51 enum { 52 Flags = Eigen::internal::traits<Derived>::Flags 53 }; 54 55 // typedef typename Matrix<Scalar,4,1> Coefficients; 56 /** the type of a 3D vector */ 57 typedef Matrix<Scalar,3,1> Vector3; 58 /** the equivalent rotation matrix type */ 59 typedef Matrix<Scalar,3,3> Matrix3; 60 /** the equivalent angle-axis type */ 61 typedef AngleAxis<Scalar> AngleAxisType; 62 63 64 65 /** \returns the \c x coefficient */ 66 EIGEN_DEVICE_FUNC inline CoeffReturnType x() const { return this->derived().coeffs().coeff(0); } 67 /** \returns the \c y coefficient */ 68 EIGEN_DEVICE_FUNC inline CoeffReturnType y() const { return this->derived().coeffs().coeff(1); } 69 /** \returns the \c z coefficient */ 70 EIGEN_DEVICE_FUNC inline CoeffReturnType z() const { return this->derived().coeffs().coeff(2); } 71 /** \returns the \c w coefficient */ 72 EIGEN_DEVICE_FUNC inline CoeffReturnType w() const { return this->derived().coeffs().coeff(3); } 73 74 /** \returns a reference to the \c x coefficient (if Derived is a non-const lvalue) */ 75 EIGEN_DEVICE_FUNC inline NonConstCoeffReturnType x() { return this->derived().coeffs().x(); } 76 /** \returns a reference to the \c y coefficient (if Derived is a non-const lvalue) */ 77 EIGEN_DEVICE_FUNC inline NonConstCoeffReturnType y() { return this->derived().coeffs().y(); } 78 /** \returns a reference to the \c z coefficient (if Derived is a non-const lvalue) */ 79 EIGEN_DEVICE_FUNC inline NonConstCoeffReturnType z() { return this->derived().coeffs().z(); } 80 /** \returns a reference to the \c w coefficient (if Derived is a non-const lvalue) */ 81 EIGEN_DEVICE_FUNC inline NonConstCoeffReturnType w() { return this->derived().coeffs().w(); } 82 83 /** \returns a read-only vector expression of the imaginary part (x,y,z) */ 84 EIGEN_DEVICE_FUNC inline const VectorBlock<const Coefficients,3> vec() const { return coeffs().template head<3>(); } 85 86 /** \returns a vector expression of the imaginary part (x,y,z) */ 87 EIGEN_DEVICE_FUNC inline VectorBlock<Coefficients,3> vec() { return coeffs().template head<3>(); } 88 89 /** \returns a read-only vector expression of the coefficients (x,y,z,w) */ 90 EIGEN_DEVICE_FUNC inline const typename internal::traits<Derived>::Coefficients& coeffs() const { return derived().coeffs(); } 91 92 /** \returns a vector expression of the coefficients (x,y,z,w) */ 93 EIGEN_DEVICE_FUNC inline typename internal::traits<Derived>::Coefficients& coeffs() { return derived().coeffs(); } 94 95 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE QuaternionBase<Derived>& operator=(const QuaternionBase<Derived>& other); 96 template<class OtherDerived> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& operator=(const QuaternionBase<OtherDerived>& other); 97 98 // disabled this copy operator as it is giving very strange compilation errors when compiling 99 // test_stdvector with GCC 4.4.2. This looks like a GCC bug though, so feel free to re-enable it if it's 100 // useful; however notice that we already have the templated operator= above and e.g. in MatrixBase 101 // we didn't have to add, in addition to templated operator=, such a non-templated copy operator. 102 // Derived& operator=(const QuaternionBase& other) 103 // { return operator=<Derived>(other); } 104 105 EIGEN_DEVICE_FUNC Derived& operator=(const AngleAxisType& aa); 106 template<class OtherDerived> EIGEN_DEVICE_FUNC Derived& operator=(const MatrixBase<OtherDerived>& m); 107 108 /** \returns a quaternion representing an identity rotation 109 * \sa MatrixBase::Identity() 110 */ 111 EIGEN_DEVICE_FUNC static inline Quaternion<Scalar> Identity() { return Quaternion<Scalar>(Scalar(1), Scalar(0), Scalar(0), Scalar(0)); } 112 113 /** \sa QuaternionBase::Identity(), MatrixBase::setIdentity() 114 */ 115 EIGEN_DEVICE_FUNC inline QuaternionBase& setIdentity() { coeffs() << Scalar(0), Scalar(0), Scalar(0), Scalar(1); return *this; } 116 117 /** \returns the squared norm of the quaternion's coefficients 118 * \sa QuaternionBase::norm(), MatrixBase::squaredNorm() 119 */ 120 EIGEN_DEVICE_FUNC inline Scalar squaredNorm() const { return coeffs().squaredNorm(); } 121 122 /** \returns the norm of the quaternion's coefficients 123 * \sa QuaternionBase::squaredNorm(), MatrixBase::norm() 124 */ 125 EIGEN_DEVICE_FUNC inline Scalar norm() const { return coeffs().norm(); } 126 127 /** Normalizes the quaternion \c *this 128 * \sa normalized(), MatrixBase::normalize() */ 129 EIGEN_DEVICE_FUNC inline void normalize() { coeffs().normalize(); } 130 /** \returns a normalized copy of \c *this 131 * \sa normalize(), MatrixBase::normalized() */ 132 EIGEN_DEVICE_FUNC inline Quaternion<Scalar> normalized() const { return Quaternion<Scalar>(coeffs().normalized()); } 133 134 /** \returns the dot product of \c *this and \a other 135 * Geometrically speaking, the dot product of two unit quaternions 136 * corresponds to the cosine of half the angle between the two rotations. 137 * \sa angularDistance() 138 */ 139 template<class OtherDerived> EIGEN_DEVICE_FUNC inline Scalar dot(const QuaternionBase<OtherDerived>& other) const { return coeffs().dot(other.coeffs()); } 140 141 template<class OtherDerived> EIGEN_DEVICE_FUNC Scalar angularDistance(const QuaternionBase<OtherDerived>& other) const; 142 143 /** \returns an equivalent 3x3 rotation matrix */ 144 EIGEN_DEVICE_FUNC inline Matrix3 toRotationMatrix() const; 145 146 /** \returns the quaternion which transform \a a into \a b through a rotation */ 147 template<typename Derived1, typename Derived2> 148 EIGEN_DEVICE_FUNC Derived& setFromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b); 149 150 template<class OtherDerived> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Quaternion<Scalar> operator* (const QuaternionBase<OtherDerived>& q) const; 151 template<class OtherDerived> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& operator*= (const QuaternionBase<OtherDerived>& q); 152 153 /** \returns the quaternion describing the inverse rotation */ 154 EIGEN_DEVICE_FUNC Quaternion<Scalar> inverse() const; 155 156 /** \returns the conjugated quaternion */ 157 EIGEN_DEVICE_FUNC Quaternion<Scalar> conjugate() const; 158 159 template<class OtherDerived> EIGEN_DEVICE_FUNC Quaternion<Scalar> slerp(const Scalar& t, const QuaternionBase<OtherDerived>& other) const; 160 161 /** \returns true if each coefficients of \c *this and \a other are all exactly equal. 162 * \warning When using floating point scalar values you probably should rather use a 163 * fuzzy comparison such as isApprox() 164 * \sa isApprox(), operator!= */ 165 template<class OtherDerived> 166 EIGEN_DEVICE_FUNC inline bool operator==(const QuaternionBase<OtherDerived>& other) const 167 { return coeffs() == other.coeffs(); } 168 169 /** \returns true if at least one pair of coefficients of \c *this and \a other are not exactly equal to each other. 170 * \warning When using floating point scalar values you probably should rather use a 171 * fuzzy comparison such as isApprox() 172 * \sa isApprox(), operator== */ 173 template<class OtherDerived> 174 EIGEN_DEVICE_FUNC inline bool operator!=(const QuaternionBase<OtherDerived>& other) const 175 { return coeffs() != other.coeffs(); } 176 177 /** \returns \c true if \c *this is approximately equal to \a other, within the precision 178 * determined by \a prec. 179 * 180 * \sa MatrixBase::isApprox() */ 181 template<class OtherDerived> 182 EIGEN_DEVICE_FUNC bool isApprox(const QuaternionBase<OtherDerived>& other, const RealScalar& prec = NumTraits<Scalar>::dummy_precision()) const 183 { return coeffs().isApprox(other.coeffs(), prec); } 184 185 /** return the result vector of \a v through the rotation*/ 186 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Vector3 _transformVector(const Vector3& v) const; 187 188 #ifdef EIGEN_PARSED_BY_DOXYGEN 189 /** \returns \c *this with scalar type casted to \a NewScalarType 190 * 191 * Note that if \a NewScalarType is equal to the current scalar type of \c *this 192 * then this function smartly returns a const reference to \c *this. 193 */ 194 template<typename NewScalarType> 195 EIGEN_DEVICE_FUNC inline typename internal::cast_return_type<Derived,Quaternion<NewScalarType> >::type cast() const; 196 197 #else 198 199 template<typename NewScalarType> 200 EIGEN_DEVICE_FUNC inline 201 typename internal::enable_if<internal::is_same<Scalar,NewScalarType>::value,const Derived&>::type cast() const 202 { 203 return derived(); 204 } 205 206 template<typename NewScalarType> 207 EIGEN_DEVICE_FUNC inline 208 typename internal::enable_if<!internal::is_same<Scalar,NewScalarType>::value,Quaternion<NewScalarType> >::type cast() const 209 { 210 return Quaternion<NewScalarType>(coeffs().template cast<NewScalarType>()); 211 } 212 #endif 213 214 #ifndef EIGEN_NO_IO 215 friend std::ostream& operator<<(std::ostream& s, const QuaternionBase<Derived>& q) { 216 s << q.x() << "i + " << q.y() << "j + " << q.z() << "k" << " + " << q.w(); 217 return s; 218 } 219 #endif 220 221 #ifdef EIGEN_QUATERNIONBASE_PLUGIN 222 # include EIGEN_QUATERNIONBASE_PLUGIN 223 #endif 224 protected: 225 EIGEN_DEFAULT_COPY_CONSTRUCTOR(QuaternionBase) 226 EIGEN_DEFAULT_EMPTY_CONSTRUCTOR_AND_DESTRUCTOR(QuaternionBase) 227 }; 228 229 /*************************************************************************** 230 * Definition/implementation of Quaternion<Scalar> 231 ***************************************************************************/ 232 233 /** \geometry_module \ingroup Geometry_Module 234 * 235 * \class Quaternion 236 * 237 * \brief The quaternion class used to represent 3D orientations and rotations 238 * 239 * \tparam _Scalar the scalar type, i.e., the type of the coefficients 240 * \tparam _Options controls the memory alignment of the coefficients. Can be \# AutoAlign or \# DontAlign. Default is AutoAlign. 241 * 242 * This class represents a quaternion \f$ w+xi+yj+zk \f$ that is a convenient representation of 243 * orientations and rotations of objects in three dimensions. Compared to other representations 244 * like Euler angles or 3x3 matrices, quaternions offer the following advantages: 245 * \li \b compact storage (4 scalars) 246 * \li \b efficient to compose (28 flops), 247 * \li \b stable spherical interpolation 248 * 249 * The following two typedefs are provided for convenience: 250 * \li \c Quaternionf for \c float 251 * \li \c Quaterniond for \c double 252 * 253 * \warning Operations interpreting the quaternion as rotation have undefined behavior if the quaternion is not normalized. 254 * 255 * \sa class AngleAxis, class Transform 256 */ 257 258 namespace internal { 259 template<typename _Scalar,int _Options> 260 struct traits<Quaternion<_Scalar,_Options> > 261 { 262 typedef Quaternion<_Scalar,_Options> PlainObject; 263 typedef _Scalar Scalar; 264 typedef Matrix<_Scalar,4,1,_Options> Coefficients; 265 enum{ 266 Alignment = internal::traits<Coefficients>::Alignment, 267 Flags = LvalueBit 268 }; 269 }; 270 } 271 272 template<typename _Scalar, int _Options> 273 class Quaternion : public QuaternionBase<Quaternion<_Scalar,_Options> > 274 { 275 public: 276 typedef QuaternionBase<Quaternion<_Scalar,_Options> > Base; 277 enum { NeedsAlignment = internal::traits<Quaternion>::Alignment>0 }; 278 279 typedef _Scalar Scalar; 280 281 EIGEN_INHERIT_ASSIGNMENT_OPERATORS(Quaternion) 282 using Base::operator*=; 283 284 typedef typename internal::traits<Quaternion>::Coefficients Coefficients; 285 typedef typename Base::AngleAxisType AngleAxisType; 286 287 /** Default constructor leaving the quaternion uninitialized. */ 288 EIGEN_DEVICE_FUNC inline Quaternion() {} 289 290 /** Constructs and initializes the quaternion \f$ w+xi+yj+zk \f$ from 291 * its four coefficients \a w, \a x, \a y and \a z. 292 * 293 * \warning Note the order of the arguments: the real \a w coefficient first, 294 * while internally the coefficients are stored in the following order: 295 * [\c x, \c y, \c z, \c w] 296 */ 297 EIGEN_DEVICE_FUNC inline Quaternion(const Scalar& w, const Scalar& x, const Scalar& y, const Scalar& z) : m_coeffs(x, y, z, w){} 298 299 /** Constructs and initialize a quaternion from the array data */ 300 EIGEN_DEVICE_FUNC explicit inline Quaternion(const Scalar* data) : m_coeffs(data) {} 301 302 /** Copy constructor */ 303 template<class Derived> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Quaternion(const QuaternionBase<Derived>& other) { this->Base::operator=(other); } 304 305 /** Constructs and initializes a quaternion from the angle-axis \a aa */ 306 EIGEN_DEVICE_FUNC explicit inline Quaternion(const AngleAxisType& aa) { *this = aa; } 307 308 /** Constructs and initializes a quaternion from either: 309 * - a rotation matrix expression, 310 * - a 4D vector expression representing quaternion coefficients. 311 */ 312 template<typename Derived> 313 EIGEN_DEVICE_FUNC explicit inline Quaternion(const MatrixBase<Derived>& other) { *this = other; } 314 315 /** Explicit copy constructor with scalar conversion */ 316 template<typename OtherScalar, int OtherOptions> 317 EIGEN_DEVICE_FUNC explicit inline Quaternion(const Quaternion<OtherScalar, OtherOptions>& other) 318 { m_coeffs = other.coeffs().template cast<Scalar>(); } 319 320 #if EIGEN_HAS_RVALUE_REFERENCES 321 // We define a copy constructor, which means we don't get an implicit move constructor or assignment operator. 322 /** Default move constructor */ 323 EIGEN_DEVICE_FUNC inline Quaternion(Quaternion&& other) EIGEN_NOEXCEPT_IF(std::is_nothrow_move_constructible<Scalar>::value) 324 : m_coeffs(std::move(other.coeffs())) 325 {} 326 327 /** Default move assignment operator */ 328 EIGEN_DEVICE_FUNC Quaternion& operator=(Quaternion&& other) EIGEN_NOEXCEPT_IF(std::is_nothrow_move_assignable<Scalar>::value) 329 { 330 m_coeffs = std::move(other.coeffs()); 331 return *this; 332 } 333 #endif 334 335 EIGEN_DEVICE_FUNC static Quaternion UnitRandom(); 336 337 template<typename Derived1, typename Derived2> 338 EIGEN_DEVICE_FUNC static Quaternion FromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b); 339 340 EIGEN_DEVICE_FUNC inline Coefficients& coeffs() { return m_coeffs;} 341 EIGEN_DEVICE_FUNC inline const Coefficients& coeffs() const { return m_coeffs;} 342 343 EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF(bool(NeedsAlignment)) 344 345 #ifdef EIGEN_QUATERNION_PLUGIN 346 # include EIGEN_QUATERNION_PLUGIN 347 #endif 348 349 protected: 350 Coefficients m_coeffs; 351 352 #ifndef EIGEN_PARSED_BY_DOXYGEN 353 static EIGEN_STRONG_INLINE void _check_template_params() 354 { 355 EIGEN_STATIC_ASSERT( (_Options & DontAlign) == _Options, 356 INVALID_MATRIX_TEMPLATE_PARAMETERS) 357 } 358 #endif 359 }; 360 361 /** \ingroup Geometry_Module 362 * single precision quaternion type */ 363 typedef Quaternion<float> Quaternionf; 364 /** \ingroup Geometry_Module 365 * double precision quaternion type */ 366 typedef Quaternion<double> Quaterniond; 367 368 /*************************************************************************** 369 * Specialization of Map<Quaternion<Scalar>> 370 ***************************************************************************/ 371 372 namespace internal { 373 template<typename _Scalar, int _Options> 374 struct traits<Map<Quaternion<_Scalar>, _Options> > : traits<Quaternion<_Scalar, (int(_Options)&Aligned)==Aligned ? AutoAlign : DontAlign> > 375 { 376 typedef Map<Matrix<_Scalar,4,1>, _Options> Coefficients; 377 }; 378 } 379 380 namespace internal { 381 template<typename _Scalar, int _Options> 382 struct traits<Map<const Quaternion<_Scalar>, _Options> > : traits<Quaternion<_Scalar, (int(_Options)&Aligned)==Aligned ? AutoAlign : DontAlign> > 383 { 384 typedef Map<const Matrix<_Scalar,4,1>, _Options> Coefficients; 385 typedef traits<Quaternion<_Scalar, (int(_Options)&Aligned)==Aligned ? AutoAlign : DontAlign> > TraitsBase; 386 enum { 387 Flags = TraitsBase::Flags & ~LvalueBit 388 }; 389 }; 390 } 391 392 /** \ingroup Geometry_Module 393 * \brief Quaternion expression mapping a constant memory buffer 394 * 395 * \tparam _Scalar the type of the Quaternion coefficients 396 * \tparam _Options see class Map 397 * 398 * This is a specialization of class Map for Quaternion. This class allows to view 399 * a 4 scalar memory buffer as an Eigen's Quaternion object. 400 * 401 * \sa class Map, class Quaternion, class QuaternionBase 402 */ 403 template<typename _Scalar, int _Options> 404 class Map<const Quaternion<_Scalar>, _Options > 405 : public QuaternionBase<Map<const Quaternion<_Scalar>, _Options> > 406 { 407 public: 408 typedef QuaternionBase<Map<const Quaternion<_Scalar>, _Options> > Base; 409 410 typedef _Scalar Scalar; 411 typedef typename internal::traits<Map>::Coefficients Coefficients; 412 EIGEN_INHERIT_ASSIGNMENT_OPERATORS(Map) 413 using Base::operator*=; 414 415 /** Constructs a Mapped Quaternion object from the pointer \a coeffs 416 * 417 * The pointer \a coeffs must reference the four coefficients of Quaternion in the following order: 418 * \code *coeffs == {x, y, z, w} \endcode 419 * 420 * If the template parameter _Options is set to #Aligned, then the pointer coeffs must be aligned. */ 421 EIGEN_DEVICE_FUNC explicit EIGEN_STRONG_INLINE Map(const Scalar* coeffs) : m_coeffs(coeffs) {} 422 423 EIGEN_DEVICE_FUNC inline const Coefficients& coeffs() const { return m_coeffs;} 424 425 protected: 426 const Coefficients m_coeffs; 427 }; 428 429 /** \ingroup Geometry_Module 430 * \brief Expression of a quaternion from a memory buffer 431 * 432 * \tparam _Scalar the type of the Quaternion coefficients 433 * \tparam _Options see class Map 434 * 435 * This is a specialization of class Map for Quaternion. This class allows to view 436 * a 4 scalar memory buffer as an Eigen's Quaternion object. 437 * 438 * \sa class Map, class Quaternion, class QuaternionBase 439 */ 440 template<typename _Scalar, int _Options> 441 class Map<Quaternion<_Scalar>, _Options > 442 : public QuaternionBase<Map<Quaternion<_Scalar>, _Options> > 443 { 444 public: 445 typedef QuaternionBase<Map<Quaternion<_Scalar>, _Options> > Base; 446 447 typedef _Scalar Scalar; 448 typedef typename internal::traits<Map>::Coefficients Coefficients; 449 EIGEN_INHERIT_ASSIGNMENT_OPERATORS(Map) 450 using Base::operator*=; 451 452 /** Constructs a Mapped Quaternion object from the pointer \a coeffs 453 * 454 * The pointer \a coeffs must reference the four coefficients of Quaternion in the following order: 455 * \code *coeffs == {x, y, z, w} \endcode 456 * 457 * If the template parameter _Options is set to #Aligned, then the pointer coeffs must be aligned. */ 458 EIGEN_DEVICE_FUNC explicit EIGEN_STRONG_INLINE Map(Scalar* coeffs) : m_coeffs(coeffs) {} 459 460 EIGEN_DEVICE_FUNC inline Coefficients& coeffs() { return m_coeffs; } 461 EIGEN_DEVICE_FUNC inline const Coefficients& coeffs() const { return m_coeffs; } 462 463 protected: 464 Coefficients m_coeffs; 465 }; 466 467 /** \ingroup Geometry_Module 468 * Map an unaligned array of single precision scalars as a quaternion */ 469 typedef Map<Quaternion<float>, 0> QuaternionMapf; 470 /** \ingroup Geometry_Module 471 * Map an unaligned array of double precision scalars as a quaternion */ 472 typedef Map<Quaternion<double>, 0> QuaternionMapd; 473 /** \ingroup Geometry_Module 474 * Map a 16-byte aligned array of single precision scalars as a quaternion */ 475 typedef Map<Quaternion<float>, Aligned> QuaternionMapAlignedf; 476 /** \ingroup Geometry_Module 477 * Map a 16-byte aligned array of double precision scalars as a quaternion */ 478 typedef Map<Quaternion<double>, Aligned> QuaternionMapAlignedd; 479 480 /*************************************************************************** 481 * Implementation of QuaternionBase methods 482 ***************************************************************************/ 483 484 // Generic Quaternion * Quaternion product 485 // This product can be specialized for a given architecture via the Arch template argument. 486 namespace internal { 487 template<int Arch, class Derived1, class Derived2, typename Scalar> struct quat_product 488 { 489 EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE Quaternion<Scalar> run(const QuaternionBase<Derived1>& a, const QuaternionBase<Derived2>& b){ 490 return Quaternion<Scalar> 491 ( 492 a.w() * b.w() - a.x() * b.x() - a.y() * b.y() - a.z() * b.z(), 493 a.w() * b.x() + a.x() * b.w() + a.y() * b.z() - a.z() * b.y(), 494 a.w() * b.y() + a.y() * b.w() + a.z() * b.x() - a.x() * b.z(), 495 a.w() * b.z() + a.z() * b.w() + a.x() * b.y() - a.y() * b.x() 496 ); 497 } 498 }; 499 } 500 501 /** \returns the concatenation of two rotations as a quaternion-quaternion product */ 502 template <class Derived> 503 template <class OtherDerived> 504 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Quaternion<typename internal::traits<Derived>::Scalar> 505 QuaternionBase<Derived>::operator* (const QuaternionBase<OtherDerived>& other) const 506 { 507 EIGEN_STATIC_ASSERT((internal::is_same<typename Derived::Scalar, typename OtherDerived::Scalar>::value), 508 YOU_MIXED_DIFFERENT_NUMERIC_TYPES__YOU_NEED_TO_USE_THE_CAST_METHOD_OF_MATRIXBASE_TO_CAST_NUMERIC_TYPES_EXPLICITLY) 509 return internal::quat_product<Architecture::Target, Derived, OtherDerived, 510 typename internal::traits<Derived>::Scalar>::run(*this, other); 511 } 512 513 /** \sa operator*(Quaternion) */ 514 template <class Derived> 515 template <class OtherDerived> 516 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& QuaternionBase<Derived>::operator*= (const QuaternionBase<OtherDerived>& other) 517 { 518 derived() = derived() * other.derived(); 519 return derived(); 520 } 521 522 /** Rotation of a vector by a quaternion. 523 * \remarks If the quaternion is used to rotate several points (>1) 524 * then it is much more efficient to first convert it to a 3x3 Matrix. 525 * Comparison of the operation cost for n transformations: 526 * - Quaternion2: 30n 527 * - Via a Matrix3: 24 + 15n 528 */ 529 template <class Derived> 530 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE typename QuaternionBase<Derived>::Vector3 531 QuaternionBase<Derived>::_transformVector(const Vector3& v) const 532 { 533 // Note that this algorithm comes from the optimization by hand 534 // of the conversion to a Matrix followed by a Matrix/Vector product. 535 // It appears to be much faster than the common algorithm found 536 // in the literature (30 versus 39 flops). It also requires two 537 // Vector3 as temporaries. 538 Vector3 uv = this->vec().cross(v); 539 uv += uv; 540 return v + this->w() * uv + this->vec().cross(uv); 541 } 542 543 template<class Derived> 544 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE QuaternionBase<Derived>& QuaternionBase<Derived>::operator=(const QuaternionBase<Derived>& other) 545 { 546 coeffs() = other.coeffs(); 547 return derived(); 548 } 549 550 template<class Derived> 551 template<class OtherDerived> 552 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& QuaternionBase<Derived>::operator=(const QuaternionBase<OtherDerived>& other) 553 { 554 coeffs() = other.coeffs(); 555 return derived(); 556 } 557 558 /** Set \c *this from an angle-axis \a aa and returns a reference to \c *this 559 */ 560 template<class Derived> 561 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& QuaternionBase<Derived>::operator=(const AngleAxisType& aa) 562 { 563 EIGEN_USING_STD(cos) 564 EIGEN_USING_STD(sin) 565 Scalar ha = Scalar(0.5)*aa.angle(); // Scalar(0.5) to suppress precision loss warnings 566 this->w() = cos(ha); 567 this->vec() = sin(ha) * aa.axis(); 568 return derived(); 569 } 570 571 /** Set \c *this from the expression \a xpr: 572 * - if \a xpr is a 4x1 vector, then \a xpr is assumed to be a quaternion 573 * - if \a xpr is a 3x3 matrix, then \a xpr is assumed to be rotation matrix 574 * and \a xpr is converted to a quaternion 575 */ 576 577 template<class Derived> 578 template<class MatrixDerived> 579 EIGEN_DEVICE_FUNC inline Derived& QuaternionBase<Derived>::operator=(const MatrixBase<MatrixDerived>& xpr) 580 { 581 EIGEN_STATIC_ASSERT((internal::is_same<typename Derived::Scalar, typename MatrixDerived::Scalar>::value), 582 YOU_MIXED_DIFFERENT_NUMERIC_TYPES__YOU_NEED_TO_USE_THE_CAST_METHOD_OF_MATRIXBASE_TO_CAST_NUMERIC_TYPES_EXPLICITLY) 583 internal::quaternionbase_assign_impl<MatrixDerived>::run(*this, xpr.derived()); 584 return derived(); 585 } 586 587 /** Convert the quaternion to a 3x3 rotation matrix. The quaternion is required to 588 * be normalized, otherwise the result is undefined. 589 */ 590 template<class Derived> 591 EIGEN_DEVICE_FUNC inline typename QuaternionBase<Derived>::Matrix3 592 QuaternionBase<Derived>::toRotationMatrix(void) const 593 { 594 // NOTE if inlined, then gcc 4.2 and 4.4 get rid of the temporary (not gcc 4.3 !!) 595 // if not inlined then the cost of the return by value is huge ~ +35%, 596 // however, not inlining this function is an order of magnitude slower, so 597 // it has to be inlined, and so the return by value is not an issue 598 Matrix3 res; 599 600 const Scalar tx = Scalar(2)*this->x(); 601 const Scalar ty = Scalar(2)*this->y(); 602 const Scalar tz = Scalar(2)*this->z(); 603 const Scalar twx = tx*this->w(); 604 const Scalar twy = ty*this->w(); 605 const Scalar twz = tz*this->w(); 606 const Scalar txx = tx*this->x(); 607 const Scalar txy = ty*this->x(); 608 const Scalar txz = tz*this->x(); 609 const Scalar tyy = ty*this->y(); 610 const Scalar tyz = tz*this->y(); 611 const Scalar tzz = tz*this->z(); 612 613 res.coeffRef(0,0) = Scalar(1)-(tyy+tzz); 614 res.coeffRef(0,1) = txy-twz; 615 res.coeffRef(0,2) = txz+twy; 616 res.coeffRef(1,0) = txy+twz; 617 res.coeffRef(1,1) = Scalar(1)-(txx+tzz); 618 res.coeffRef(1,2) = tyz-twx; 619 res.coeffRef(2,0) = txz-twy; 620 res.coeffRef(2,1) = tyz+twx; 621 res.coeffRef(2,2) = Scalar(1)-(txx+tyy); 622 623 return res; 624 } 625 626 /** Sets \c *this to be a quaternion representing a rotation between 627 * the two arbitrary vectors \a a and \a b. In other words, the built 628 * rotation represent a rotation sending the line of direction \a a 629 * to the line of direction \a b, both lines passing through the origin. 630 * 631 * \returns a reference to \c *this. 632 * 633 * Note that the two input vectors do \b not have to be normalized, and 634 * do not need to have the same norm. 635 */ 636 template<class Derived> 637 template<typename Derived1, typename Derived2> 638 EIGEN_DEVICE_FUNC inline Derived& QuaternionBase<Derived>::setFromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b) 639 { 640 EIGEN_USING_STD(sqrt) 641 Vector3 v0 = a.normalized(); 642 Vector3 v1 = b.normalized(); 643 Scalar c = v1.dot(v0); 644 645 // if dot == -1, vectors are nearly opposites 646 // => accurately compute the rotation axis by computing the 647 // intersection of the two planes. This is done by solving: 648 // x^T v0 = 0 649 // x^T v1 = 0 650 // under the constraint: 651 // ||x|| = 1 652 // which yields a singular value problem 653 if (c < Scalar(-1)+NumTraits<Scalar>::dummy_precision()) 654 { 655 c = numext::maxi(c,Scalar(-1)); 656 Matrix<Scalar,2,3> m; m << v0.transpose(), v1.transpose(); 657 JacobiSVD<Matrix<Scalar,2,3> > svd(m, ComputeFullV); 658 Vector3 axis = svd.matrixV().col(2); 659 660 Scalar w2 = (Scalar(1)+c)*Scalar(0.5); 661 this->w() = sqrt(w2); 662 this->vec() = axis * sqrt(Scalar(1) - w2); 663 return derived(); 664 } 665 Vector3 axis = v0.cross(v1); 666 Scalar s = sqrt((Scalar(1)+c)*Scalar(2)); 667 Scalar invs = Scalar(1)/s; 668 this->vec() = axis * invs; 669 this->w() = s * Scalar(0.5); 670 671 return derived(); 672 } 673 674 /** \returns a random unit quaternion following a uniform distribution law on SO(3) 675 * 676 * \note The implementation is based on http://planning.cs.uiuc.edu/node198.html 677 */ 678 template<typename Scalar, int Options> 679 EIGEN_DEVICE_FUNC Quaternion<Scalar,Options> Quaternion<Scalar,Options>::UnitRandom() 680 { 681 EIGEN_USING_STD(sqrt) 682 EIGEN_USING_STD(sin) 683 EIGEN_USING_STD(cos) 684 const Scalar u1 = internal::random<Scalar>(0, 1), 685 u2 = internal::random<Scalar>(0, 2*EIGEN_PI), 686 u3 = internal::random<Scalar>(0, 2*EIGEN_PI); 687 const Scalar a = sqrt(Scalar(1) - u1), 688 b = sqrt(u1); 689 return Quaternion (a * sin(u2), a * cos(u2), b * sin(u3), b * cos(u3)); 690 } 691 692 693 /** Returns a quaternion representing a rotation between 694 * the two arbitrary vectors \a a and \a b. In other words, the built 695 * rotation represent a rotation sending the line of direction \a a 696 * to the line of direction \a b, both lines passing through the origin. 697 * 698 * \returns resulting quaternion 699 * 700 * Note that the two input vectors do \b not have to be normalized, and 701 * do not need to have the same norm. 702 */ 703 template<typename Scalar, int Options> 704 template<typename Derived1, typename Derived2> 705 EIGEN_DEVICE_FUNC Quaternion<Scalar,Options> Quaternion<Scalar,Options>::FromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b) 706 { 707 Quaternion quat; 708 quat.setFromTwoVectors(a, b); 709 return quat; 710 } 711 712 713 /** \returns the multiplicative inverse of \c *this 714 * Note that in most cases, i.e., if you simply want the opposite rotation, 715 * and/or the quaternion is normalized, then it is enough to use the conjugate. 716 * 717 * \sa QuaternionBase::conjugate() 718 */ 719 template <class Derived> 720 EIGEN_DEVICE_FUNC inline Quaternion<typename internal::traits<Derived>::Scalar> QuaternionBase<Derived>::inverse() const 721 { 722 // FIXME should this function be called multiplicativeInverse and conjugate() be called inverse() or opposite() ?? 723 Scalar n2 = this->squaredNorm(); 724 if (n2 > Scalar(0)) 725 return Quaternion<Scalar>(conjugate().coeffs() / n2); 726 else 727 { 728 // return an invalid result to flag the error 729 return Quaternion<Scalar>(Coefficients::Zero()); 730 } 731 } 732 733 // Generic conjugate of a Quaternion 734 namespace internal { 735 template<int Arch, class Derived, typename Scalar> struct quat_conj 736 { 737 EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE Quaternion<Scalar> run(const QuaternionBase<Derived>& q){ 738 return Quaternion<Scalar>(q.w(),-q.x(),-q.y(),-q.z()); 739 } 740 }; 741 } 742 743 /** \returns the conjugate of the \c *this which is equal to the multiplicative inverse 744 * if the quaternion is normalized. 745 * The conjugate of a quaternion represents the opposite rotation. 746 * 747 * \sa Quaternion2::inverse() 748 */ 749 template <class Derived> 750 EIGEN_DEVICE_FUNC inline Quaternion<typename internal::traits<Derived>::Scalar> 751 QuaternionBase<Derived>::conjugate() const 752 { 753 return internal::quat_conj<Architecture::Target, Derived, 754 typename internal::traits<Derived>::Scalar>::run(*this); 755 756 } 757 758 /** \returns the angle (in radian) between two rotations 759 * \sa dot() 760 */ 761 template <class Derived> 762 template <class OtherDerived> 763 EIGEN_DEVICE_FUNC inline typename internal::traits<Derived>::Scalar 764 QuaternionBase<Derived>::angularDistance(const QuaternionBase<OtherDerived>& other) const 765 { 766 EIGEN_USING_STD(atan2) 767 Quaternion<Scalar> d = (*this) * other.conjugate(); 768 return Scalar(2) * atan2( d.vec().norm(), numext::abs(d.w()) ); 769 } 770 771 772 773 /** \returns the spherical linear interpolation between the two quaternions 774 * \c *this and \a other at the parameter \a t in [0;1]. 775 * 776 * This represents an interpolation for a constant motion between \c *this and \a other, 777 * see also http://en.wikipedia.org/wiki/Slerp. 778 */ 779 template <class Derived> 780 template <class OtherDerived> 781 EIGEN_DEVICE_FUNC Quaternion<typename internal::traits<Derived>::Scalar> 782 QuaternionBase<Derived>::slerp(const Scalar& t, const QuaternionBase<OtherDerived>& other) const 783 { 784 EIGEN_USING_STD(acos) 785 EIGEN_USING_STD(sin) 786 const Scalar one = Scalar(1) - NumTraits<Scalar>::epsilon(); 787 Scalar d = this->dot(other); 788 Scalar absD = numext::abs(d); 789 790 Scalar scale0; 791 Scalar scale1; 792 793 if(absD>=one) 794 { 795 scale0 = Scalar(1) - t; 796 scale1 = t; 797 } 798 else 799 { 800 // theta is the angle between the 2 quaternions 801 Scalar theta = acos(absD); 802 Scalar sinTheta = sin(theta); 803 804 scale0 = sin( ( Scalar(1) - t ) * theta) / sinTheta; 805 scale1 = sin( ( t * theta) ) / sinTheta; 806 } 807 if(d<Scalar(0)) scale1 = -scale1; 808 809 return Quaternion<Scalar>(scale0 * coeffs() + scale1 * other.coeffs()); 810 } 811 812 namespace internal { 813 814 // set from a rotation matrix 815 template<typename Other> 816 struct quaternionbase_assign_impl<Other,3,3> 817 { 818 typedef typename Other::Scalar Scalar; 819 template<class Derived> EIGEN_DEVICE_FUNC static inline void run(QuaternionBase<Derived>& q, const Other& a_mat) 820 { 821 const typename internal::nested_eval<Other,2>::type mat(a_mat); 822 EIGEN_USING_STD(sqrt) 823 // This algorithm comes from "Quaternion Calculus and Fast Animation", 824 // Ken Shoemake, 1987 SIGGRAPH course notes 825 Scalar t = mat.trace(); 826 if (t > Scalar(0)) 827 { 828 t = sqrt(t + Scalar(1.0)); 829 q.w() = Scalar(0.5)*t; 830 t = Scalar(0.5)/t; 831 q.x() = (mat.coeff(2,1) - mat.coeff(1,2)) * t; 832 q.y() = (mat.coeff(0,2) - mat.coeff(2,0)) * t; 833 q.z() = (mat.coeff(1,0) - mat.coeff(0,1)) * t; 834 } 835 else 836 { 837 Index i = 0; 838 if (mat.coeff(1,1) > mat.coeff(0,0)) 839 i = 1; 840 if (mat.coeff(2,2) > mat.coeff(i,i)) 841 i = 2; 842 Index j = (i+1)%3; 843 Index k = (j+1)%3; 844 845 t = sqrt(mat.coeff(i,i)-mat.coeff(j,j)-mat.coeff(k,k) + Scalar(1.0)); 846 q.coeffs().coeffRef(i) = Scalar(0.5) * t; 847 t = Scalar(0.5)/t; 848 q.w() = (mat.coeff(k,j)-mat.coeff(j,k))*t; 849 q.coeffs().coeffRef(j) = (mat.coeff(j,i)+mat.coeff(i,j))*t; 850 q.coeffs().coeffRef(k) = (mat.coeff(k,i)+mat.coeff(i,k))*t; 851 } 852 } 853 }; 854 855 // set from a vector of coefficients assumed to be a quaternion 856 template<typename Other> 857 struct quaternionbase_assign_impl<Other,4,1> 858 { 859 typedef typename Other::Scalar Scalar; 860 template<class Derived> EIGEN_DEVICE_FUNC static inline void run(QuaternionBase<Derived>& q, const Other& vec) 861 { 862 q.coeffs() = vec; 863 } 864 }; 865 866 } // end namespace internal 867 868 } // end namespace Eigen 869 870 #endif // EIGEN_QUATERNION_H