Eigen  3.3.4
Quaternion.h
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 
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;
62 
63 
64 
66  EIGEN_DEVICE_FUNC inline CoeffReturnType x() const { return this->derived().coeffs().coeff(0); }
68  EIGEN_DEVICE_FUNC inline CoeffReturnType y() const { return this->derived().coeffs().coeff(1); }
70  EIGEN_DEVICE_FUNC inline CoeffReturnType z() const { return this->derived().coeffs().coeff(2); }
72  EIGEN_DEVICE_FUNC inline CoeffReturnType w() const { return this->derived().coeffs().coeff(3); }
73 
75  EIGEN_DEVICE_FUNC inline NonConstCoeffReturnType x() { return this->derived().coeffs().x(); }
77  EIGEN_DEVICE_FUNC inline NonConstCoeffReturnType y() { return this->derived().coeffs().y(); }
79  EIGEN_DEVICE_FUNC inline NonConstCoeffReturnType z() { return this->derived().coeffs().z(); }
81  EIGEN_DEVICE_FUNC inline NonConstCoeffReturnType w() { return this->derived().coeffs().w(); }
82 
84  EIGEN_DEVICE_FUNC inline const VectorBlock<const Coefficients,3> vec() const { return coeffs().template head<3>(); }
85 
87  EIGEN_DEVICE_FUNC inline VectorBlock<Coefficients,3> vec() { return coeffs().template head<3>(); }
88 
90  EIGEN_DEVICE_FUNC inline const typename internal::traits<Derived>::Coefficients& coeffs() const { return derived().coeffs(); }
91 
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 
111  EIGEN_DEVICE_FUNC static inline Quaternion<Scalar> Identity() { return Quaternion<Scalar>(Scalar(1), Scalar(0), Scalar(0), Scalar(0)); }
112 
115  EIGEN_DEVICE_FUNC inline QuaternionBase& setIdentity() { coeffs() << Scalar(0), Scalar(0), Scalar(0), Scalar(1); return *this; }
116 
120  EIGEN_DEVICE_FUNC inline Scalar squaredNorm() const { return coeffs().squaredNorm(); }
121 
125  EIGEN_DEVICE_FUNC inline Scalar norm() const { return coeffs().norm(); }
126 
129  EIGEN_DEVICE_FUNC inline void normalize() { coeffs().normalize(); }
132  EIGEN_DEVICE_FUNC inline Quaternion<Scalar> normalized() const { return Quaternion<Scalar>(coeffs().normalized()); }
133 
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 
144  EIGEN_DEVICE_FUNC Matrix3 toRotationMatrix() const;
145 
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 
154  EIGEN_DEVICE_FUNC Quaternion<Scalar> inverse() const;
155 
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 
165  template<class OtherDerived>
166  EIGEN_DEVICE_FUNC bool isApprox(const QuaternionBase<OtherDerived>& other, const RealScalar& prec = NumTraits<Scalar>::dummy_precision()) const
167  { return coeffs().isApprox(other.coeffs(), prec); }
168 
170  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Vector3 _transformVector(const Vector3& v) const;
171 
177  template<typename NewScalarType>
178  EIGEN_DEVICE_FUNC inline typename internal::cast_return_type<Derived,Quaternion<NewScalarType> >::type cast() const
179  {
180  return typename internal::cast_return_type<Derived,Quaternion<NewScalarType> >::type(derived());
181  }
182 
183 #ifdef EIGEN_QUATERNIONBASE_PLUGIN
184 # include EIGEN_QUATERNIONBASE_PLUGIN
185 #endif
186 };
187 
188 /***************************************************************************
189 * Definition/implementation of Quaternion<Scalar>
190 ***************************************************************************/
191 
217 namespace internal {
218 template<typename _Scalar,int _Options>
219 struct traits<Quaternion<_Scalar,_Options> >
220 {
221  typedef Quaternion<_Scalar,_Options> PlainObject;
222  typedef _Scalar Scalar;
223  typedef Matrix<_Scalar,4,1,_Options> Coefficients;
224  enum{
225  Alignment = internal::traits<Coefficients>::Alignment,
226  Flags = LvalueBit
227  };
228 };
229 }
230 
231 template<typename _Scalar, int _Options>
232 class Quaternion : public QuaternionBase<Quaternion<_Scalar,_Options> >
233 {
234 public:
236  enum { NeedsAlignment = internal::traits<Quaternion>::Alignment>0 };
237 
238  typedef _Scalar Scalar;
239 
240  EIGEN_INHERIT_ASSIGNMENT_OPERATORS(Quaternion)
241  using Base::operator*=;
242 
243  typedef typename internal::traits<Quaternion>::Coefficients Coefficients;
244  typedef typename Base::AngleAxisType AngleAxisType;
245 
247  EIGEN_DEVICE_FUNC inline Quaternion() {}
248 
256  EIGEN_DEVICE_FUNC inline Quaternion(const Scalar& w, const Scalar& x, const Scalar& y, const Scalar& z) : m_coeffs(x, y, z, w){}
257 
259  EIGEN_DEVICE_FUNC explicit inline Quaternion(const Scalar* data) : m_coeffs(data) {}
260 
262  template<class Derived> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Quaternion(const QuaternionBase<Derived>& other) { this->Base::operator=(other); }
263 
265  EIGEN_DEVICE_FUNC explicit inline Quaternion(const AngleAxisType& aa) { *this = aa; }
266 
271  template<typename Derived>
272  EIGEN_DEVICE_FUNC explicit inline Quaternion(const MatrixBase<Derived>& other) { *this = other; }
273 
275  template<typename OtherScalar, int OtherOptions>
276  EIGEN_DEVICE_FUNC explicit inline Quaternion(const Quaternion<OtherScalar, OtherOptions>& other)
277  { m_coeffs = other.coeffs().template cast<Scalar>(); }
278 
279  EIGEN_DEVICE_FUNC static Quaternion UnitRandom();
280 
281  template<typename Derived1, typename Derived2>
282  EIGEN_DEVICE_FUNC static Quaternion FromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b);
283 
284  EIGEN_DEVICE_FUNC inline Coefficients& coeffs() { return m_coeffs;}
285  EIGEN_DEVICE_FUNC inline const Coefficients& coeffs() const { return m_coeffs;}
286 
287  EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF(bool(NeedsAlignment))
288 
289 #ifdef EIGEN_QUATERNION_PLUGIN
290 # include EIGEN_QUATERNION_PLUGIN
291 #endif
292 
293 protected:
294  Coefficients m_coeffs;
295 
296 #ifndef EIGEN_PARSED_BY_DOXYGEN
297  static EIGEN_STRONG_INLINE void _check_template_params()
298  {
299  EIGEN_STATIC_ASSERT( (_Options & DontAlign) == _Options,
300  INVALID_MATRIX_TEMPLATE_PARAMETERS)
301  }
302 #endif
303 };
304 
311 
312 /***************************************************************************
313 * Specialization of Map<Quaternion<Scalar>>
314 ***************************************************************************/
315 
316 namespace internal {
317  template<typename _Scalar, int _Options>
318  struct traits<Map<Quaternion<_Scalar>, _Options> > : traits<Quaternion<_Scalar, (int(_Options)&Aligned)==Aligned ? AutoAlign : DontAlign> >
319  {
320  typedef Map<Matrix<_Scalar,4,1>, _Options> Coefficients;
321  };
322 }
323 
324 namespace internal {
325  template<typename _Scalar, int _Options>
326  struct traits<Map<const Quaternion<_Scalar>, _Options> > : traits<Quaternion<_Scalar, (int(_Options)&Aligned)==Aligned ? AutoAlign : DontAlign> >
327  {
328  typedef Map<const Matrix<_Scalar,4,1>, _Options> Coefficients;
329  typedef traits<Quaternion<_Scalar, (int(_Options)&Aligned)==Aligned ? AutoAlign : DontAlign> > TraitsBase;
330  enum {
331  Flags = TraitsBase::Flags & ~LvalueBit
332  };
333  };
334 }
335 
347 template<typename _Scalar, int _Options>
348 class Map<const Quaternion<_Scalar>, _Options >
349  : public QuaternionBase<Map<const Quaternion<_Scalar>, _Options> >
350 {
351  public:
352  typedef QuaternionBase<Map<const Quaternion<_Scalar>, _Options> > Base;
353 
354  typedef _Scalar Scalar;
355  typedef typename internal::traits<Map>::Coefficients Coefficients;
356  EIGEN_INHERIT_ASSIGNMENT_OPERATORS(Map)
357  using Base::operator*=;
358 
365  EIGEN_DEVICE_FUNC explicit EIGEN_STRONG_INLINE Map(const Scalar* coeffs) : m_coeffs(coeffs) {}
366 
367  EIGEN_DEVICE_FUNC inline const Coefficients& coeffs() const { return m_coeffs;}
368 
369  protected:
370  const Coefficients m_coeffs;
371 };
372 
384 template<typename _Scalar, int _Options>
385 class Map<Quaternion<_Scalar>, _Options >
386  : public QuaternionBase<Map<Quaternion<_Scalar>, _Options> >
387 {
388  public:
389  typedef QuaternionBase<Map<Quaternion<_Scalar>, _Options> > Base;
390 
391  typedef _Scalar Scalar;
392  typedef typename internal::traits<Map>::Coefficients Coefficients;
393  EIGEN_INHERIT_ASSIGNMENT_OPERATORS(Map)
394  using Base::operator*=;
395 
402  EIGEN_DEVICE_FUNC explicit EIGEN_STRONG_INLINE Map(Scalar* coeffs) : m_coeffs(coeffs) {}
403 
404  EIGEN_DEVICE_FUNC inline Coefficients& coeffs() { return m_coeffs; }
405  EIGEN_DEVICE_FUNC inline const Coefficients& coeffs() const { return m_coeffs; }
406 
407  protected:
408  Coefficients m_coeffs;
409 };
410 
419 typedef Map<Quaternion<float>, Aligned> QuaternionMapAlignedf;
422 typedef Map<Quaternion<double>, Aligned> QuaternionMapAlignedd;
423 
424 /***************************************************************************
425 * Implementation of QuaternionBase methods
426 ***************************************************************************/
427 
428 // Generic Quaternion * Quaternion product
429 // This product can be specialized for a given architecture via the Arch template argument.
430 namespace internal {
431 template<int Arch, class Derived1, class Derived2, typename Scalar> struct quat_product
432 {
433  EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE Quaternion<Scalar> run(const QuaternionBase<Derived1>& a, const QuaternionBase<Derived2>& b){
434  return Quaternion<Scalar>
435  (
436  a.w() * b.w() - a.x() * b.x() - a.y() * b.y() - a.z() * b.z(),
437  a.w() * b.x() + a.x() * b.w() + a.y() * b.z() - a.z() * b.y(),
438  a.w() * b.y() + a.y() * b.w() + a.z() * b.x() - a.x() * b.z(),
439  a.w() * b.z() + a.z() * b.w() + a.x() * b.y() - a.y() * b.x()
440  );
441  }
442 };
443 }
444 
446 template <class Derived>
447 template <class OtherDerived>
448 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Quaternion<typename internal::traits<Derived>::Scalar>
450 {
451  EIGEN_STATIC_ASSERT((internal::is_same<typename Derived::Scalar, typename OtherDerived::Scalar>::value),
452  YOU_MIXED_DIFFERENT_NUMERIC_TYPES__YOU_NEED_TO_USE_THE_CAST_METHOD_OF_MATRIXBASE_TO_CAST_NUMERIC_TYPES_EXPLICITLY)
453  return internal::quat_product<Architecture::Target, Derived, OtherDerived,
454  typename internal::traits<Derived>::Scalar>::run(*this, other);
455 }
456 
458 template <class Derived>
459 template <class OtherDerived>
460 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& QuaternionBase<Derived>::operator*= (const QuaternionBase<OtherDerived>& other)
461 {
462  derived() = derived() * other.derived();
463  return derived();
464 }
465 
473 template <class Derived>
474 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE typename QuaternionBase<Derived>::Vector3
476 {
477  // Note that this algorithm comes from the optimization by hand
478  // of the conversion to a Matrix followed by a Matrix/Vector product.
479  // It appears to be much faster than the common algorithm found
480  // in the literature (30 versus 39 flops). It also requires two
481  // Vector3 as temporaries.
482  Vector3 uv = this->vec().cross(v);
483  uv += uv;
484  return v + this->w() * uv + this->vec().cross(uv);
485 }
486 
487 template<class Derived>
488 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE QuaternionBase<Derived>& QuaternionBase<Derived>::operator=(const QuaternionBase<Derived>& other)
489 {
490  coeffs() = other.coeffs();
491  return derived();
492 }
493 
494 template<class Derived>
495 template<class OtherDerived>
496 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& QuaternionBase<Derived>::operator=(const QuaternionBase<OtherDerived>& other)
497 {
498  coeffs() = other.coeffs();
499  return derived();
500 }
501 
504 template<class Derived>
505 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& QuaternionBase<Derived>::operator=(const AngleAxisType& aa)
506 {
507  EIGEN_USING_STD_MATH(cos)
508  EIGEN_USING_STD_MATH(sin)
509  Scalar ha = Scalar(0.5)*aa.angle(); // Scalar(0.5) to suppress precision loss warnings
510  this->w() = cos(ha);
511  this->vec() = sin(ha) * aa.axis();
512  return derived();
513 }
514 
521 template<class Derived>
522 template<class MatrixDerived>
523 EIGEN_DEVICE_FUNC inline Derived& QuaternionBase<Derived>::operator=(const MatrixBase<MatrixDerived>& xpr)
524 {
525  EIGEN_STATIC_ASSERT((internal::is_same<typename Derived::Scalar, typename MatrixDerived::Scalar>::value),
526  YOU_MIXED_DIFFERENT_NUMERIC_TYPES__YOU_NEED_TO_USE_THE_CAST_METHOD_OF_MATRIXBASE_TO_CAST_NUMERIC_TYPES_EXPLICITLY)
527  internal::quaternionbase_assign_impl<MatrixDerived>::run(*this, xpr.derived());
528  return derived();
529 }
530 
534 template<class Derived>
535 EIGEN_DEVICE_FUNC inline typename QuaternionBase<Derived>::Matrix3
537 {
538  // NOTE if inlined, then gcc 4.2 and 4.4 get rid of the temporary (not gcc 4.3 !!)
539  // if not inlined then the cost of the return by value is huge ~ +35%,
540  // however, not inlining this function is an order of magnitude slower, so
541  // it has to be inlined, and so the return by value is not an issue
542  Matrix3 res;
543 
544  const Scalar tx = Scalar(2)*this->x();
545  const Scalar ty = Scalar(2)*this->y();
546  const Scalar tz = Scalar(2)*this->z();
547  const Scalar twx = tx*this->w();
548  const Scalar twy = ty*this->w();
549  const Scalar twz = tz*this->w();
550  const Scalar txx = tx*this->x();
551  const Scalar txy = ty*this->x();
552  const Scalar txz = tz*this->x();
553  const Scalar tyy = ty*this->y();
554  const Scalar tyz = tz*this->y();
555  const Scalar tzz = tz*this->z();
556 
557  res.coeffRef(0,0) = Scalar(1)-(tyy+tzz);
558  res.coeffRef(0,1) = txy-twz;
559  res.coeffRef(0,2) = txz+twy;
560  res.coeffRef(1,0) = txy+twz;
561  res.coeffRef(1,1) = Scalar(1)-(txx+tzz);
562  res.coeffRef(1,2) = tyz-twx;
563  res.coeffRef(2,0) = txz-twy;
564  res.coeffRef(2,1) = tyz+twx;
565  res.coeffRef(2,2) = Scalar(1)-(txx+tyy);
566 
567  return res;
568 }
569 
580 template<class Derived>
581 template<typename Derived1, typename Derived2>
582 EIGEN_DEVICE_FUNC inline Derived& QuaternionBase<Derived>::setFromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b)
583 {
584  EIGEN_USING_STD_MATH(sqrt)
585  Vector3 v0 = a.normalized();
586  Vector3 v1 = b.normalized();
587  Scalar c = v1.dot(v0);
588 
589  // if dot == -1, vectors are nearly opposites
590  // => accurately compute the rotation axis by computing the
591  // intersection of the two planes. This is done by solving:
592  // x^T v0 = 0
593  // x^T v1 = 0
594  // under the constraint:
595  // ||x|| = 1
596  // which yields a singular value problem
597  if (c < Scalar(-1)+NumTraits<Scalar>::dummy_precision())
598  {
599  c = numext::maxi(c,Scalar(-1));
600  Matrix<Scalar,2,3> m; m << v0.transpose(), v1.transpose();
602  Vector3 axis = svd.matrixV().col(2);
603 
604  Scalar w2 = (Scalar(1)+c)*Scalar(0.5);
605  this->w() = sqrt(w2);
606  this->vec() = axis * sqrt(Scalar(1) - w2);
607  return derived();
608  }
609  Vector3 axis = v0.cross(v1);
610  Scalar s = sqrt((Scalar(1)+c)*Scalar(2));
611  Scalar invs = Scalar(1)/s;
612  this->vec() = axis * invs;
613  this->w() = s * Scalar(0.5);
614 
615  return derived();
616 }
617 
622 template<typename Scalar, int Options>
624 {
625  EIGEN_USING_STD_MATH(sqrt)
626  EIGEN_USING_STD_MATH(sin)
627  EIGEN_USING_STD_MATH(cos)
628  const Scalar u1 = internal::random<Scalar>(0, 1),
629  u2 = internal::random<Scalar>(0, 2*EIGEN_PI),
630  u3 = internal::random<Scalar>(0, 2*EIGEN_PI);
631  const Scalar a = sqrt(1 - u1),
632  b = sqrt(u1);
633  return Quaternion (a * sin(u2), a * cos(u2), b * sin(u3), b * cos(u3));
634 }
635 
636 
647 template<typename Scalar, int Options>
648 template<typename Derived1, typename Derived2>
650 {
651  Quaternion quat;
652  quat.setFromTwoVectors(a, b);
653  return quat;
654 }
655 
656 
663 template <class Derived>
665 {
666  // FIXME should this function be called multiplicativeInverse and conjugate() be called inverse() or opposite() ??
667  Scalar n2 = this->squaredNorm();
668  if (n2 > Scalar(0))
669  return Quaternion<Scalar>(conjugate().coeffs() / n2);
670  else
671  {
672  // return an invalid result to flag the error
673  return Quaternion<Scalar>(Coefficients::Zero());
674  }
675 }
676 
677 // Generic conjugate of a Quaternion
678 namespace internal {
679 template<int Arch, class Derived, typename Scalar> struct quat_conj
680 {
681  EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE Quaternion<Scalar> run(const QuaternionBase<Derived>& q){
682  return Quaternion<Scalar>(q.w(),-q.x(),-q.y(),-q.z());
683  }
684 };
685 }
686 
693 template <class Derived>
696 {
697  return internal::quat_conj<Architecture::Target, Derived,
698  typename internal::traits<Derived>::Scalar>::run(*this);
699 
700 }
701 
705 template <class Derived>
706 template <class OtherDerived>
707 EIGEN_DEVICE_FUNC inline typename internal::traits<Derived>::Scalar
709 {
710  EIGEN_USING_STD_MATH(atan2)
711  Quaternion<Scalar> d = (*this) * other.conjugate();
712  return Scalar(2) * atan2( d.vec().norm(), numext::abs(d.w()) );
713 }
714 
715 
716 
723 template <class Derived>
724 template <class OtherDerived>
727 {
728  EIGEN_USING_STD_MATH(acos)
729  EIGEN_USING_STD_MATH(sin)
730  const Scalar one = Scalar(1) - NumTraits<Scalar>::epsilon();
731  Scalar d = this->dot(other);
732  Scalar absD = numext::abs(d);
733 
734  Scalar scale0;
735  Scalar scale1;
736 
737  if(absD>=one)
738  {
739  scale0 = Scalar(1) - t;
740  scale1 = t;
741  }
742  else
743  {
744  // theta is the angle between the 2 quaternions
745  Scalar theta = acos(absD);
746  Scalar sinTheta = sin(theta);
747 
748  scale0 = sin( ( Scalar(1) - t ) * theta) / sinTheta;
749  scale1 = sin( ( t * theta) ) / sinTheta;
750  }
751  if(d<Scalar(0)) scale1 = -scale1;
752 
753  return Quaternion<Scalar>(scale0 * coeffs() + scale1 * other.coeffs());
754 }
755 
756 namespace internal {
757 
758 // set from a rotation matrix
759 template<typename Other>
760 struct quaternionbase_assign_impl<Other,3,3>
761 {
762  typedef typename Other::Scalar Scalar;
763  template<class Derived> EIGEN_DEVICE_FUNC static inline void run(QuaternionBase<Derived>& q, const Other& a_mat)
764  {
765  const typename internal::nested_eval<Other,2>::type mat(a_mat);
766  EIGEN_USING_STD_MATH(sqrt)
767  // This algorithm comes from "Quaternion Calculus and Fast Animation",
768  // Ken Shoemake, 1987 SIGGRAPH course notes
769  Scalar t = mat.trace();
770  if (t > Scalar(0))
771  {
772  t = sqrt(t + Scalar(1.0));
773  q.w() = Scalar(0.5)*t;
774  t = Scalar(0.5)/t;
775  q.x() = (mat.coeff(2,1) - mat.coeff(1,2)) * t;
776  q.y() = (mat.coeff(0,2) - mat.coeff(2,0)) * t;
777  q.z() = (mat.coeff(1,0) - mat.coeff(0,1)) * t;
778  }
779  else
780  {
781  Index i = 0;
782  if (mat.coeff(1,1) > mat.coeff(0,0))
783  i = 1;
784  if (mat.coeff(2,2) > mat.coeff(i,i))
785  i = 2;
786  Index j = (i+1)%3;
787  Index k = (j+1)%3;
788 
789  t = sqrt(mat.coeff(i,i)-mat.coeff(j,j)-mat.coeff(k,k) + Scalar(1.0));
790  q.coeffs().coeffRef(i) = Scalar(0.5) * t;
791  t = Scalar(0.5)/t;
792  q.w() = (mat.coeff(k,j)-mat.coeff(j,k))*t;
793  q.coeffs().coeffRef(j) = (mat.coeff(j,i)+mat.coeff(i,j))*t;
794  q.coeffs().coeffRef(k) = (mat.coeff(k,i)+mat.coeff(i,k))*t;
795  }
796  }
797 };
798 
799 // set from a vector of coefficients assumed to be a quaternion
800 template<typename Other>
801 struct quaternionbase_assign_impl<Other,4,1>
802 {
803  typedef typename Other::Scalar Scalar;
804  template<class Derived> EIGEN_DEVICE_FUNC static inline void run(QuaternionBase<Derived>& q, const Other& vec)
805  {
806  q.coeffs() = vec;
807  }
808 };
809 
810 } // end namespace internal
811 
812 } // end namespace Eigen
813 
814 #endif // EIGEN_QUATERNION_H
const Vector3 & axis() const
Definition: AngleAxis.h:96
NonConstCoeffReturnType x()
Definition: Quaternion.h:75
Definition: Constants.h:326
CoeffReturnType w() const
Definition: Quaternion.h:72
NonConstCoeffReturnType y()
Definition: Quaternion.h:77
A matrix or vector expression mapping an existing array of data.
Definition: Map.h:94
Matrix< Scalar, 3, 1 > Vector3
Definition: Quaternion.h:57
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_sqrt_op< typename Derived::Scalar >, const Derived > sqrt(const Eigen::ArrayBase< Derived > &x)
bool isApprox(const QuaternionBase< OtherDerived > &other, const RealScalar &prec=NumTraits< Scalar >::dummy_precision()) const
Definition: Quaternion.h:166
Scalar angle() const
Definition: AngleAxis.h:91
Quaternion< double > Quaterniond
Definition: Quaternion.h:310
const unsigned int LvalueBit
Definition: Constants.h:139
Scalar dot(const QuaternionBase< OtherDerived > &other) const
Definition: Quaternion.h:139
Namespace containing all symbols from the Eigen library.
Definition: Core:303
Holds information about the various numeric (i.e. scalar) types allowed by Eigen. ...
Definition: NumTraits.h:150
const PlainObject normalized() const
Definition: Dot.h:121
Scalar norm() const
Definition: Quaternion.h:125
internal::cast_return_type< Derived, Quaternion< NewScalarType > >::type cast() const
Definition: Quaternion.h:178
AngleAxis< Scalar > AngleAxisType
Definition: Quaternion.h:61
Derived & derived()
Definition: EigenBase.h:45
Quaternion()
Definition: Quaternion.h:247
NonConstCoeffReturnType w()
Definition: Quaternion.h:81
Derived & operator*=(const QuaternionBase< OtherDerived > &q)
Definition: Quaternion.h:460
static Quaternion< Scalar > Identity()
Definition: Quaternion.h:111
NonConstCoeffReturnType z()
Definition: Quaternion.h:79
Map(const Scalar *coeffs)
Definition: Quaternion.h:365
void normalize()
Definition: Quaternion.h:129
Derived & setFromTwoVectors(const MatrixBase< Derived1 > &a, const MatrixBase< Derived2 > &b)
Definition: Quaternion.h:582
Scalar & coeffRef(Index rowId, Index colId)
Definition: PlainObjectBase.h:183
CoeffReturnType z() const
Definition: Quaternion.h:70
Expression of a fixed-size or dynamic-size sub-vector.
Definition: ForwardDeclarations.h:87
Scalar squaredNorm() const
Definition: Quaternion.h:120
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_acos_op< typename Derived::Scalar >, const Derived > acos(const Eigen::ArrayBase< Derived > &x)
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_inverse_op< typename Derived::Scalar >, const Derived > inverse(const Eigen::ArrayBase< Derived > &x)
internal::traits< Derived >::Coefficients & coeffs()
Definition: Quaternion.h:93
const VectorBlock< const Coefficients, 3 > vec() const
Definition: Quaternion.h:84
static Quaternion UnitRandom()
Definition: Quaternion.h:623
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_cos_op< typename Derived::Scalar >, const Derived > cos(const Eigen::ArrayBase< Derived > &x)
const Product< MatrixDerived, PermutationDerived, AliasFreeProduct > operator*(const MatrixBase< MatrixDerived > &matrix, const PermutationBase< PermutationDerived > &permutation)
Definition: PermutationMatrix.h:543
EIGEN_DEFAULT_DENSE_INDEX_TYPE Index
The Index type as used for the API.
Definition: Meta.h:33
Map(Scalar *coeffs)
Definition: Quaternion.h:402
Definition: Constants.h:235
Quaternion(const Quaternion< OtherScalar, OtherOptions > &other)
Definition: Quaternion.h:276
Quaternion< Scalar > normalized() const
Definition: Quaternion.h:132
CoeffReturnType x() const
Definition: Quaternion.h:66
Base class for quaternion expressions.
Definition: ForwardDeclarations.h:268
Quaternion< Scalar > inverse() const
Definition: Quaternion.h:664
CoeffReturnType y() const
Definition: Quaternion.h:68
Map< Quaternion< float >, 0 > QuaternionMapf
Definition: Quaternion.h:413
Definition: Eigen_Colamd.h:50
Quaternion(const AngleAxisType &aa)
Definition: Quaternion.h:265
The quaternion class used to represent 3D orientations and rotations.
Definition: ForwardDeclarations.h:273
Quaternion(const MatrixBase< Derived > &other)
Definition: Quaternion.h:272
Map< Quaternion< double >, 0 > QuaternionMapd
Definition: Quaternion.h:416
Two-sided Jacobi SVD decomposition of a rectangular matrix.
Definition: ForwardDeclarations.h:258
Map< Quaternion< float >, Aligned > QuaternionMapAlignedf
Definition: Quaternion.h:419
QuaternionBase & setIdentity()
Definition: Quaternion.h:115
Quaternion< Scalar > conjugate() const
Definition: Quaternion.h:695
const MatrixVType & matrixV() const
Definition: SVDBase.h:99
Matrix< Scalar, 3, 3 > Matrix3
Definition: Quaternion.h:59
VectorBlock< Coefficients, 3 > vec()
Definition: Quaternion.h:87
Definition: Constants.h:387
Vector3 _transformVector(const Vector3 &v) const
Definition: Quaternion.h:475
Quaternion(const Scalar &w, const Scalar &x, const Scalar &y, const Scalar &z)
Definition: Quaternion.h:256
Base class for all dense matrices, vectors, and expressions.
Definition: MatrixBase.h:48
Represents a 3D rotation as a rotation angle around an arbitrary 3D axis.
Definition: ForwardDeclarations.h:270
Matrix3 toRotationMatrix() const
Definition: Quaternion.h:536
Quaternion(const Scalar *data)
Definition: Quaternion.h:259
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_sin_op< typename Derived::Scalar >, const Derived > sin(const Eigen::ArrayBase< Derived > &x)
const internal::traits< Derived >::Coefficients & coeffs() const
Definition: Quaternion.h:90
Quaternion< float > Quaternionf
Definition: Quaternion.h:307
Map< Quaternion< double >, Aligned > QuaternionMapAlignedd
Definition: Quaternion.h:422
Quaternion(const QuaternionBase< Derived > &other)
Definition: Quaternion.h:262