xref: /aosp_15_r20/external/eigen/Eigen/src/Geometry/Quaternion.h (revision bf2c37156dfe67e5dfebd6d394bad8b2ab5804d4)
1 // This file is part of Eigen, a lightweight C++ template library
2 // for linear algebra.
3 //
4 // Copyright (C) 2008-2010 Gael Guennebaud <[email protected]>
5 // Copyright (C) 2009 Mathieu Gautier <[email protected]>
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 */
x()66   EIGEN_DEVICE_FUNC inline CoeffReturnType x() const { return this->derived().coeffs().coeff(0); }
67   /** \returns the \c y coefficient */
y()68   EIGEN_DEVICE_FUNC inline CoeffReturnType y() const { return this->derived().coeffs().coeff(1); }
69   /** \returns the \c z coefficient */
z()70   EIGEN_DEVICE_FUNC inline CoeffReturnType z() const { return this->derived().coeffs().coeff(2); }
71   /** \returns the \c w coefficient */
w()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) */
x()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) */
y()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) */
z()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) */
w()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) */
vec()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) */
vec()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) */
coeffs()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) */
coeffs()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     */
Identity()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     */
setIdentity()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     */
squaredNorm()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     */
norm()125   EIGEN_DEVICE_FUNC inline Scalar norm() const { return coeffs().norm(); }
126 
127   /** Normalizes the quaternion \c *this
128     * \sa normalized(), MatrixBase::normalize() */
normalize()129   EIGEN_DEVICE_FUNC inline void normalize() { coeffs().normalize(); }
130   /** \returns a normalized copy of \c *this
131     * \sa normalize(), MatrixBase::normalized() */
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     */
dot(const QuaternionBase<OtherDerived> & other)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
cast()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
cast()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
871