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 // This file is part of Eigen, a lightweight C++ template library
 // for linear algebra.
 //
 // Copyright (C) 2008-2010 Gael Guennebaud <gael.guennebaud@inria.fr>
 // Copyright (C) 2009 Mathieu Gautier <mathieu.gautier@cea.fr>
 //
 // This Source Code Form is subject to the terms of the Mozilla
 // Public License v. 2.0. If a copy of the MPL was not distributed
 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
 
 #ifndef EIGEN_QUATERNION_H
 #define EIGEN_QUATERNION_H
 namespace Eigen { 
 
 
 /***************************************************************************
 * Definition of QuaternionBase<Derived>
 * The implementation is at the end of the file
 ***************************************************************************/
 
 namespace internal {
 template<typename Other,
          int OtherRows=Other::RowsAtCompileTime,
          int OtherCols=Other::ColsAtCompileTime>
 struct quaternionbase_assign_impl;
 }
 
 /** \geometry_module \ingroup Geometry_Module
   * \class QuaternionBase
   * \brief Base class for quaternion expressions
   * \tparam Derived derived type (CRTP)
   * \sa class Quaternion
   */
 template<class Derived>
 class QuaternionBase : public RotationBase<Derived, 3>
 {
  public:
   typedef RotationBase<Derived, 3> Base;
 
   using Base::operator*;
   using Base::derived;
 
   typedef typename internal::traits<Derived>::Scalar Scalar;
   typedef typename NumTraits<Scalar>::Real RealScalar;
   typedef typename internal::traits<Derived>::Coefficients Coefficients;
   typedef typename Coefficients::CoeffReturnType CoeffReturnType;
   typedef typename internal::conditional<bool(internal::traits<Derived>::Flags&LvalueBit),
                                         Scalar&, CoeffReturnType>::type NonConstCoeffReturnType;
 
 
   enum {
     Flags = Eigen::internal::traits<Derived>::Flags
   };
 
  // typedef typename Matrix<Scalar,4,1> Coefficients;
   /** the type of a 3D vector */
   typedef Matrix<Scalar,3,1> Vector3;
   /** the equivalent rotation matrix type */
   typedef Matrix<Scalar,3,3> Matrix3;
   /** the equivalent angle-axis type */
   typedef AngleAxis<Scalar> AngleAxisType;
 
 
 
   /** \returns the \c x coefficient */
   EIGEN_DEVICE_FUNC inline CoeffReturnType x() const { return this->derived().coeffs().coeff(0); }
   /** \returns the \c y coefficient */
   EIGEN_DEVICE_FUNC inline CoeffReturnType y() const { return this->derived().coeffs().coeff(1); }
   /** \returns the \c z coefficient */
   EIGEN_DEVICE_FUNC inline CoeffReturnType z() const { return this->derived().coeffs().coeff(2); }
   /** \returns the \c w coefficient */
   EIGEN_DEVICE_FUNC inline CoeffReturnType w() const { return this->derived().coeffs().coeff(3); }
 
   /** \returns a reference to the \c x coefficient (if Derived is a non-const lvalue) */
   EIGEN_DEVICE_FUNC inline NonConstCoeffReturnType x() { return this->derived().coeffs().x(); }
   /** \returns a reference to the \c y coefficient (if Derived is a non-const lvalue) */
   EIGEN_DEVICE_FUNC inline NonConstCoeffReturnType y() { return this->derived().coeffs().y(); }
   /** \returns a reference to the \c z coefficient (if Derived is a non-const lvalue) */
   EIGEN_DEVICE_FUNC inline NonConstCoeffReturnType z() { return this->derived().coeffs().z(); }
   /** \returns a reference to the \c w coefficient (if Derived is a non-const lvalue) */
   EIGEN_DEVICE_FUNC inline NonConstCoeffReturnType w() { return this->derived().coeffs().w(); }
 
   /** \returns a read-only vector expression of the imaginary part (x,y,z) */
   EIGEN_DEVICE_FUNC inline const VectorBlock<const Coefficients,3> vec() const { return coeffs().template head<3>(); }
 
   /** \returns a vector expression of the imaginary part (x,y,z) */
   EIGEN_DEVICE_FUNC inline VectorBlock<Coefficients,3> vec() { return coeffs().template head<3>(); }
 
   /** \returns a read-only vector expression of the coefficients (x,y,z,w) */
   EIGEN_DEVICE_FUNC inline const typename internal::traits<Derived>::Coefficients& coeffs() const { return derived().coeffs(); }
 
   /** \returns a vector expression of the coefficients (x,y,z,w) */
   EIGEN_DEVICE_FUNC inline typename internal::traits<Derived>::Coefficients& coeffs() { return derived().coeffs(); }
 
   EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE QuaternionBase<Derived>& operator=(const QuaternionBase<Derived>& other);
   template<class OtherDerived> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& operator=(const QuaternionBase<OtherDerived>& other);
 
 // disabled this copy operator as it is giving very strange compilation errors when compiling
 // test_stdvector with GCC 4.4.2. This looks like a GCC bug though, so feel free to re-enable it if it's
 // useful; however notice that we already have the templated operator= above and e.g. in MatrixBase
 // we didn't have to add, in addition to templated operator=, such a non-templated copy operator.
 //  Derived& operator=(const QuaternionBase& other)
 //  { return operator=<Derived>(other); }
 
   EIGEN_DEVICE_FUNC Derived& operator=(const AngleAxisType& aa);
   template<class OtherDerived> EIGEN_DEVICE_FUNC Derived& operator=(const MatrixBase<OtherDerived>& m);
 
   /** \returns a quaternion representing an identity rotation
     * \sa MatrixBase::Identity()
     */
   EIGEN_DEVICE_FUNC static inline Quaternion<Scalar> Identity() { return Quaternion<Scalar>(Scalar(1), Scalar(0), Scalar(0), Scalar(0)); }
 
   /** \sa QuaternionBase::Identity(), MatrixBase::setIdentity()
     */
   EIGEN_DEVICE_FUNC inline QuaternionBase& setIdentity() { coeffs() << Scalar(0), Scalar(0), Scalar(0), Scalar(1); return *this; }
 
   /** \returns the squared norm of the quaternion's coefficients
     * \sa QuaternionBase::norm(), MatrixBase::squaredNorm()
     */
   EIGEN_DEVICE_FUNC inline Scalar squaredNorm() const { return coeffs().squaredNorm(); }
 
   /** \returns the norm of the quaternion's coefficients
     * \sa QuaternionBase::squaredNorm(), MatrixBase::norm()
     */
   EIGEN_DEVICE_FUNC inline Scalar norm() const { return coeffs().norm(); }
 
   /** Normalizes the quaternion \c *this
     * \sa normalized(), MatrixBase::normalize() */
   EIGEN_DEVICE_FUNC inline void normalize() { coeffs().normalize(); }
   /** \returns a normalized copy of \c *this
     * \sa normalize(), MatrixBase::normalized() */
   EIGEN_DEVICE_FUNC inline Quaternion<Scalar> normalized() const { return Quaternion<Scalar>(coeffs().normalized()); }
 
     /** \returns the dot product of \c *this and \a other
     * Geometrically speaking, the dot product of two unit quaternions
     * corresponds to the cosine of half the angle between the two rotations.
     * \sa angularDistance()
     */
   template<class OtherDerived> EIGEN_DEVICE_FUNC inline Scalar dot(const QuaternionBase<OtherDerived>& other) const { return coeffs().dot(other.coeffs()); }
 
   template<class OtherDerived> EIGEN_DEVICE_FUNC Scalar angularDistance(const QuaternionBase<OtherDerived>& other) const;
 
   /** \returns an equivalent 3x3 rotation matrix */
   EIGEN_DEVICE_FUNC inline Matrix3 toRotationMatrix() const;
 
   /** \returns the quaternion which transform \a a into \a b through a rotation */
   template<typename Derived1, typename Derived2>
   EIGEN_DEVICE_FUNC Derived& setFromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b);
 
   template<class OtherDerived> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Quaternion<Scalar> operator* (const QuaternionBase<OtherDerived>& q) const;
   template<class OtherDerived> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& operator*= (const QuaternionBase<OtherDerived>& q);
 
   /** \returns the quaternion describing the inverse rotation */
   EIGEN_DEVICE_FUNC Quaternion<Scalar> inverse() const;
 
   /** \returns the conjugated quaternion */
   EIGEN_DEVICE_FUNC Quaternion<Scalar> conjugate() const;
 
   template<class OtherDerived> EIGEN_DEVICE_FUNC Quaternion<Scalar> slerp(const Scalar& t, const QuaternionBase<OtherDerived>& other) const;
 
   /** \returns true if each coefficients of \c *this and \a other are all exactly equal.
     * \warning When using floating point scalar values you probably should rather use a
     *          fuzzy comparison such as isApprox()
     * \sa isApprox(), operator!= */
   template<class OtherDerived>
   EIGEN_DEVICE_FUNC inline bool operator==(const QuaternionBase<OtherDerived>& other) const
   { return coeffs() == other.coeffs(); }
 
   /** \returns true if at least one pair of coefficients of \c *this and \a other are not exactly equal to each other.
     * \warning When using floating point scalar values you probably should rather use a
     *          fuzzy comparison such as isApprox()
     * \sa isApprox(), operator== */
   template<class OtherDerived>
   EIGEN_DEVICE_FUNC inline bool operator!=(const QuaternionBase<OtherDerived>& other) const
   { return coeffs() != other.coeffs(); }
 
   /** \returns \c true if \c *this is approximately equal to \a other, within the precision
     * determined by \a prec.
     *
     * \sa MatrixBase::isApprox() */
   template<class OtherDerived>
   EIGEN_DEVICE_FUNC bool isApprox(const QuaternionBase<OtherDerived>& other, const RealScalar& prec = NumTraits<Scalar>::dummy_precision()) const
   { return coeffs().isApprox(other.coeffs(), prec); }
 
   /** return the result vector of \a v through the rotation*/
   EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Vector3 _transformVector(const Vector3& v) const;
 
   #ifdef EIGEN_PARSED_BY_DOXYGEN
   /** \returns \c *this with scalar type casted to \a NewScalarType
     *
     * Note that if \a NewScalarType is equal to the current scalar type of \c *this
     * then this function smartly returns a const reference to \c *this.
     */
   template<typename NewScalarType>
   EIGEN_DEVICE_FUNC inline typename internal::cast_return_type<Derived,Quaternion<NewScalarType> >::type cast() const;
 
   #else
 
   template<typename NewScalarType>
   EIGEN_DEVICE_FUNC inline
   typename internal::enable_if<internal::is_same<Scalar,NewScalarType>::value,const Derived&>::type cast() const
   {
     return derived();
   }
 
   template<typename NewScalarType>
   EIGEN_DEVICE_FUNC inline
   typename internal::enable_if<!internal::is_same<Scalar,NewScalarType>::value,Quaternion<NewScalarType> >::type cast() const
   {
     return Quaternion<NewScalarType>(coeffs().template cast<NewScalarType>());
   }
   #endif
 
 #ifndef EIGEN_NO_IO
   friend std::ostream& operator<<(std::ostream& s, const QuaternionBase<Derived>& q) {
     s << q.x() << "i + " << q.y() << "j + " << q.z() << "k" << " + " << q.w();
     return s;
   }
 #endif
 
 #ifdef EIGEN_QUATERNIONBASE_PLUGIN
 # include EIGEN_QUATERNIONBASE_PLUGIN
 #endif
 protected:
   EIGEN_DEFAULT_COPY_CONSTRUCTOR(QuaternionBase)
   EIGEN_DEFAULT_EMPTY_CONSTRUCTOR_AND_DESTRUCTOR(QuaternionBase)
 };
 
 /***************************************************************************
 * Definition/implementation of Quaternion<Scalar>
 ***************************************************************************/
 
 /** \geometry_module \ingroup Geometry_Module
   *
   * \class Quaternion
   *
   * \brief The quaternion class used to represent 3D orientations and rotations
   *
   * \tparam _Scalar the scalar type, i.e., the type of the coefficients
   * \tparam _Options controls the memory alignment of the coefficients. Can be \# AutoAlign or \# DontAlign. Default is AutoAlign.
   *
   * This class represents a quaternion \f$ w+xi+yj+zk \f$ that is a convenient representation of
   * orientations and rotations of objects in three dimensions. Compared to other representations
   * like Euler angles or 3x3 matrices, quaternions offer the following advantages:
   * \li \b compact storage (4 scalars)
   * \li \b efficient to compose (28 flops),
   * \li \b stable spherical interpolation
   *
   * The following two typedefs are provided for convenience:
   * \li \c Quaternionf for \c float
   * \li \c Quaterniond for \c double
   *
   * \warning Operations interpreting the quaternion as rotation have undefined behavior if the quaternion is not normalized.
   *
   * \sa  class AngleAxis, class Transform
   */
 
 namespace internal {
 template<typename _Scalar,int _Options>
 struct traits<Quaternion<_Scalar,_Options> >
 {
   typedef Quaternion<_Scalar,_Options> PlainObject;
   typedef _Scalar Scalar;
   typedef Matrix<_Scalar,4,1,_Options> Coefficients;
   enum{
     Alignment = internal::traits<Coefficients>::Alignment,
     Flags = LvalueBit
   };
 };
 }
 
 template<typename _Scalar, int _Options>
 class Quaternion : public QuaternionBase<Quaternion<_Scalar,_Options> >
 {
 public:
   typedef QuaternionBase<Quaternion<_Scalar,_Options> > Base;
   enum { NeedsAlignment = internal::traits<Quaternion>::Alignment>0 };
 
   typedef _Scalar Scalar;
 
   EIGEN_INHERIT_ASSIGNMENT_OPERATORS(Quaternion)
   using Base::operator*=;
 
   typedef typename internal::traits<Quaternion>::Coefficients Coefficients;
   typedef typename Base::AngleAxisType AngleAxisType;
 
   /** Default constructor leaving the quaternion uninitialized. */
   EIGEN_DEVICE_FUNC inline Quaternion() {}
 
   /** Constructs and initializes the quaternion \f$ w+xi+yj+zk \f$ from
     * its four coefficients \a w, \a x, \a y and \a z.
     *
     * \warning Note the order of the arguments: the real \a w coefficient first,
     * while internally the coefficients are stored in the following order:
     * [\c x, \c y, \c z, \c w]
     */
   EIGEN_DEVICE_FUNC inline Quaternion(const Scalar& w, const Scalar& x, const Scalar& y, const Scalar& z) : m_coeffs(x, y, z, w){}
 
   /** Constructs and initialize a quaternion from the array data */
   EIGEN_DEVICE_FUNC explicit inline Quaternion(const Scalar* data) : m_coeffs(data) {}
 
   /** Copy constructor */
   template<class Derived> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Quaternion(const QuaternionBase<Derived>& other) { this->Base::operator=(other); }
 
   /** Constructs and initializes a quaternion from the angle-axis \a aa */
   EIGEN_DEVICE_FUNC explicit inline Quaternion(const AngleAxisType& aa) { *this = aa; }
 
   /** Constructs and initializes a quaternion from either:
     *  - a rotation matrix expression,
     *  - a 4D vector expression representing quaternion coefficients.
     */
   template<typename Derived>
   EIGEN_DEVICE_FUNC explicit inline Quaternion(const MatrixBase<Derived>& other) { *this = other; }
 
   /** Explicit copy constructor with scalar conversion */
   template<typename OtherScalar, int OtherOptions>
   EIGEN_DEVICE_FUNC explicit inline Quaternion(const Quaternion<OtherScalar, OtherOptions>& other)
   { m_coeffs = other.coeffs().template cast<Scalar>(); }
 
 #if EIGEN_HAS_RVALUE_REFERENCES
   // We define a copy constructor, which means we don't get an implicit move constructor or assignment operator.
   /** Default move constructor */
   EIGEN_DEVICE_FUNC inline Quaternion(Quaternion&& other) EIGEN_NOEXCEPT_IF(std::is_nothrow_move_constructible<Scalar>::value)
     : m_coeffs(std::move(other.coeffs()))
   {}
 
   /** Default move assignment operator */
   EIGEN_DEVICE_FUNC Quaternion& operator=(Quaternion&& other) EIGEN_NOEXCEPT_IF(std::is_nothrow_move_assignable<Scalar>::value)
   {
     m_coeffs = std::move(other.coeffs());
     return *this;
   }
 #endif
 
   EIGEN_DEVICE_FUNC static Quaternion UnitRandom();
 
   template<typename Derived1, typename Derived2>
   EIGEN_DEVICE_FUNC static Quaternion FromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b);
 
   EIGEN_DEVICE_FUNC inline Coefficients& coeffs() { return m_coeffs;}
   EIGEN_DEVICE_FUNC inline const Coefficients& coeffs() const { return m_coeffs;}
 
   EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF(bool(NeedsAlignment))
   
 #ifdef EIGEN_QUATERNION_PLUGIN
 # include EIGEN_QUATERNION_PLUGIN
 #endif
 
 protected:
   Coefficients m_coeffs;
   
 #ifndef EIGEN_PARSED_BY_DOXYGEN
     static EIGEN_STRONG_INLINE void _check_template_params()
     {
       EIGEN_STATIC_ASSERT( (_Options & DontAlign) == _Options,
         INVALID_MATRIX_TEMPLATE_PARAMETERS)
     }
 #endif
 };
 
 /** \ingroup Geometry_Module
   * single precision quaternion type */
 typedef Quaternion<float> Quaternionf;
 /** \ingroup Geometry_Module
   * double precision quaternion type */
 typedef Quaternion<double> Quaterniond;
 
 /***************************************************************************
 * Specialization of Map<Quaternion<Scalar>>
 ***************************************************************************/
 
 namespace internal {
   template<typename _Scalar, int _Options>
   struct traits<Map<Quaternion<_Scalar>, _Options> > : traits<Quaternion<_Scalar, (int(_Options)&Aligned)==Aligned ? AutoAlign : DontAlign> >
   {
     typedef Map<Matrix<_Scalar,4,1>, _Options> Coefficients;
   };
 }
 
 namespace internal {
   template<typename _Scalar, int _Options>
   struct traits<Map<const Quaternion<_Scalar>, _Options> > : traits<Quaternion<_Scalar, (int(_Options)&Aligned)==Aligned ? AutoAlign : DontAlign> >
   {
     typedef Map<const Matrix<_Scalar,4,1>, _Options> Coefficients;
     typedef traits<Quaternion<_Scalar, (int(_Options)&Aligned)==Aligned ? AutoAlign : DontAlign> > TraitsBase;
     enum {
       Flags = TraitsBase::Flags & ~LvalueBit
     };
   };
 }
 
 /** \ingroup Geometry_Module
   * \brief Quaternion expression mapping a constant memory buffer
   *
   * \tparam _Scalar the type of the Quaternion coefficients
   * \tparam _Options see class Map
   *
   * This is a specialization of class Map for Quaternion. This class allows to view
   * a 4 scalar memory buffer as an Eigen's Quaternion object.
   *
   * \sa class Map, class Quaternion, class QuaternionBase
   */
 template<typename _Scalar, int _Options>
 class Map<const Quaternion<_Scalar>, _Options >
   : public QuaternionBase<Map<const Quaternion<_Scalar>, _Options> >
 {
   public:
     typedef QuaternionBase<Map<const Quaternion<_Scalar>, _Options> > Base;
 
     typedef _Scalar Scalar;
     typedef typename internal::traits<Map>::Coefficients Coefficients;
     EIGEN_INHERIT_ASSIGNMENT_OPERATORS(Map)
     using Base::operator*=;
 
     /** Constructs a Mapped Quaternion object from the pointer \a coeffs
       *
       * The pointer \a coeffs must reference the four coefficients of Quaternion in the following order:
       * \code *coeffs == {x, y, z, w} \endcode
       *
       * If the template parameter _Options is set to #Aligned, then the pointer coeffs must be aligned. */
     EIGEN_DEVICE_FUNC explicit EIGEN_STRONG_INLINE Map(const Scalar* coeffs) : m_coeffs(coeffs) {}
 
     EIGEN_DEVICE_FUNC inline const Coefficients& coeffs() const { return m_coeffs;}
 
   protected:
     const Coefficients m_coeffs;
 };
 
 /** \ingroup Geometry_Module
   * \brief Expression of a quaternion from a memory buffer
   *
   * \tparam _Scalar the type of the Quaternion coefficients
   * \tparam _Options see class Map
   *
   * This is a specialization of class Map for Quaternion. This class allows to view
   * a 4 scalar memory buffer as an Eigen's  Quaternion object.
   *
   * \sa class Map, class Quaternion, class QuaternionBase
   */
 template<typename _Scalar, int _Options>
 class Map<Quaternion<_Scalar>, _Options >
   : public QuaternionBase<Map<Quaternion<_Scalar>, _Options> >
 {
   public:
     typedef QuaternionBase<Map<Quaternion<_Scalar>, _Options> > Base;
 
     typedef _Scalar Scalar;
     typedef typename internal::traits<Map>::Coefficients Coefficients;
     EIGEN_INHERIT_ASSIGNMENT_OPERATORS(Map)
     using Base::operator*=;
 
     /** Constructs a Mapped Quaternion object from the pointer \a coeffs
       *
       * The pointer \a coeffs must reference the four coefficients of Quaternion in the following order:
       * \code *coeffs == {x, y, z, w} \endcode
       *
       * If the template parameter _Options is set to #Aligned, then the pointer coeffs must be aligned. */
     EIGEN_DEVICE_FUNC explicit EIGEN_STRONG_INLINE Map(Scalar* coeffs) : m_coeffs(coeffs) {}
 
     EIGEN_DEVICE_FUNC inline Coefficients& coeffs() { return m_coeffs; }
     EIGEN_DEVICE_FUNC inline const Coefficients& coeffs() const { return m_coeffs; }
 
   protected:
     Coefficients m_coeffs;
 };
 
 /** \ingroup Geometry_Module
   * Map an unaligned array of single precision scalars as a quaternion */
 typedef Map<Quaternion<float>, 0>         QuaternionMapf;
 /** \ingroup Geometry_Module
   * Map an unaligned array of double precision scalars as a quaternion */
 typedef Map<Quaternion<double>, 0>        QuaternionMapd;
 /** \ingroup Geometry_Module
   * Map a 16-byte aligned array of single precision scalars as a quaternion */
 typedef Map<Quaternion<float>, Aligned>   QuaternionMapAlignedf;
 /** \ingroup Geometry_Module
   * Map a 16-byte aligned array of double precision scalars as a quaternion */
 typedef Map<Quaternion<double>, Aligned>  QuaternionMapAlignedd;
 
 /***************************************************************************
 * Implementation of QuaternionBase methods
 ***************************************************************************/
 
 // Generic Quaternion * Quaternion product
 // This product can be specialized for a given architecture via the Arch template argument.
 namespace internal {
 template<int Arch, class Derived1, class Derived2, typename Scalar> struct quat_product
 {
   EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE Quaternion<Scalar> run(const QuaternionBase<Derived1>& a, const QuaternionBase<Derived2>& b){
     return Quaternion<Scalar>
     (
       a.w() * b.w() - a.x() * b.x() - a.y() * b.y() - a.z() * b.z(),
       a.w() * b.x() + a.x() * b.w() + a.y() * b.z() - a.z() * b.y(),
       a.w() * b.y() + a.y() * b.w() + a.z() * b.x() - a.x() * b.z(),
       a.w() * b.z() + a.z() * b.w() + a.x() * b.y() - a.y() * b.x()
     );
   }
 };
 }
 
 /** \returns the concatenation of two rotations as a quaternion-quaternion product */
 template <class Derived>
 template <class OtherDerived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Quaternion<typename internal::traits<Derived>::Scalar>
 QuaternionBase<Derived>::operator* (const QuaternionBase<OtherDerived>& other) const
 {
   EIGEN_STATIC_ASSERT((internal::is_same<typename Derived::Scalar, typename OtherDerived::Scalar>::value),
    YOU_MIXED_DIFFERENT_NUMERIC_TYPES__YOU_NEED_TO_USE_THE_CAST_METHOD_OF_MATRIXBASE_TO_CAST_NUMERIC_TYPES_EXPLICITLY)
   return internal::quat_product<Architecture::Target, Derived, OtherDerived,
                          typename internal::traits<Derived>::Scalar>::run(*this, other);
 }
 
 /** \sa operator*(Quaternion) */
 template <class Derived>
 template <class OtherDerived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& QuaternionBase<Derived>::operator*= (const QuaternionBase<OtherDerived>& other)
 {
   derived() = derived() * other.derived();
   return derived();
 }
 
 /** Rotation of a vector by a quaternion.
   * \remarks If the quaternion is used to rotate several points (>1)
   * then it is much more efficient to first convert it to a 3x3 Matrix.
   * Comparison of the operation cost for n transformations:
   *   - Quaternion2:    30n
   *   - Via a Matrix3: 24 + 15n
   */
 template <class Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE typename QuaternionBase<Derived>::Vector3
 QuaternionBase<Derived>::_transformVector(const Vector3& v) const
 {
     // Note that this algorithm comes from the optimization by hand
     // of the conversion to a Matrix followed by a Matrix/Vector product.
     // It appears to be much faster than the common algorithm found
     // in the literature (30 versus 39 flops). It also requires two
     // Vector3 as temporaries.
     Vector3 uv = this->vec().cross(v);
     uv += uv;
     return v + this->w() * uv + this->vec().cross(uv);
 }
 
 template<class Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE QuaternionBase<Derived>& QuaternionBase<Derived>::operator=(const QuaternionBase<Derived>& other)
 {
   coeffs() = other.coeffs();
   return derived();
 }
 
 template<class Derived>
 template<class OtherDerived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& QuaternionBase<Derived>::operator=(const QuaternionBase<OtherDerived>& other)
 {
   coeffs() = other.coeffs();
   return derived();
 }
 
 /** Set \c *this from an angle-axis \a aa and returns a reference to \c *this
   */
 template<class Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& QuaternionBase<Derived>::operator=(const AngleAxisType& aa)
 {
   EIGEN_USING_STD(cos)
   EIGEN_USING_STD(sin)
   Scalar ha = Scalar(0.5)*aa.angle(); // Scalar(0.5) to suppress precision loss warnings
   this->w() = cos(ha);
   this->vec() = sin(ha) * aa.axis();
   return derived();
 }
 
 /** Set \c *this from the expression \a xpr:
   *   - if \a xpr is a 4x1 vector, then \a xpr is assumed to be a quaternion
   *   - if \a xpr is a 3x3 matrix, then \a xpr is assumed to be rotation matrix
   *     and \a xpr is converted to a quaternion
   */
 
 template<class Derived>
 template<class MatrixDerived>
 EIGEN_DEVICE_FUNC inline Derived& QuaternionBase<Derived>::operator=(const MatrixBase<MatrixDerived>& xpr)
 {
   EIGEN_STATIC_ASSERT((internal::is_same<typename Derived::Scalar, typename MatrixDerived::Scalar>::value),
    YOU_MIXED_DIFFERENT_NUMERIC_TYPES__YOU_NEED_TO_USE_THE_CAST_METHOD_OF_MATRIXBASE_TO_CAST_NUMERIC_TYPES_EXPLICITLY)
   internal::quaternionbase_assign_impl<MatrixDerived>::run(*this, xpr.derived());
   return derived();
 }
 
 /** Convert the quaternion to a 3x3 rotation matrix. The quaternion is required to
   * be normalized, otherwise the result is undefined.
   */
 template<class Derived>
 EIGEN_DEVICE_FUNC inline typename QuaternionBase<Derived>::Matrix3
 QuaternionBase<Derived>::toRotationMatrix(void) const
 {
   // NOTE if inlined, then gcc 4.2 and 4.4 get rid of the temporary (not gcc 4.3 !!)
   // if not inlined then the cost of the return by value is huge ~ +35%,
   // however, not inlining this function is an order of magnitude slower, so
   // it has to be inlined, and so the return by value is not an issue
   Matrix3 res;
 
   const Scalar tx  = Scalar(2)*this->x();
   const Scalar ty  = Scalar(2)*this->y();
   const Scalar tz  = Scalar(2)*this->z();
   const Scalar twx = tx*this->w();
   const Scalar twy = ty*this->w();
   const Scalar twz = tz*this->w();
   const Scalar txx = tx*this->x();
   const Scalar txy = ty*this->x();
   const Scalar txz = tz*this->x();
   const Scalar tyy = ty*this->y();
   const Scalar tyz = tz*this->y();
   const Scalar tzz = tz*this->z();
 
   res.coeffRef(0,0) = Scalar(1)-(tyy+tzz);
   res.coeffRef(0,1) = txy-twz;
   res.coeffRef(0,2) = txz+twy;
   res.coeffRef(1,0) = txy+twz;
   res.coeffRef(1,1) = Scalar(1)-(txx+tzz);
   res.coeffRef(1,2) = tyz-twx;
   res.coeffRef(2,0) = txz-twy;
   res.coeffRef(2,1) = tyz+twx;
   res.coeffRef(2,2) = Scalar(1)-(txx+tyy);
 
   return res;
 }
 
 /** Sets \c *this to be a quaternion representing a rotation between
   * the two arbitrary vectors \a a and \a b. In other words, the built
   * rotation represent a rotation sending the line of direction \a a
   * to the line of direction \a b, both lines passing through the origin.
   *
   * \returns a reference to \c *this.
   *
   * Note that the two input vectors do \b not have to be normalized, and
   * do not need to have the same norm.
   */
 template<class Derived>
 template<typename Derived1, typename Derived2>
 EIGEN_DEVICE_FUNC inline Derived& QuaternionBase<Derived>::setFromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b)
 {
   EIGEN_USING_STD(sqrt)
   Vector3 v0 = a.normalized();
   Vector3 v1 = b.normalized();
   Scalar c = v1.dot(v0);
 
   // if dot == -1, vectors are nearly opposites
   // => accurately compute the rotation axis by computing the
   //    intersection of the two planes. This is done by solving:
   //       x^T v0 = 0
   //       x^T v1 = 0
   //    under the constraint:
   //       ||x|| = 1
   //    which yields a singular value problem
   if (c < Scalar(-1)+NumTraits<Scalar>::dummy_precision())
   {
     c = numext::maxi(c,Scalar(-1));
     Matrix<Scalar,2,3> m; m << v0.transpose(), v1.transpose();
     JacobiSVD<Matrix<Scalar,2,3> > svd(m, ComputeFullV);
     Vector3 axis = svd.matrixV().col(2);
 
     Scalar w2 = (Scalar(1)+c)*Scalar(0.5);
     this->w() = sqrt(w2);
     this->vec() = axis * sqrt(Scalar(1) - w2);
     return derived();
   }
   Vector3 axis = v0.cross(v1);
   Scalar s = sqrt((Scalar(1)+c)*Scalar(2));
   Scalar invs = Scalar(1)/s;
   this->vec() = axis * invs;
   this->w() = s * Scalar(0.5);
 
   return derived();
 }
 
 /** \returns a random unit quaternion following a uniform distribution law on SO(3)
   *
   * \note The implementation is based on http://planning.cs.uiuc.edu/node198.html
   */
 template<typename Scalar, int Options>
 EIGEN_DEVICE_FUNC Quaternion<Scalar,Options> Quaternion<Scalar,Options>::UnitRandom()
 {
   EIGEN_USING_STD(sqrt)
   EIGEN_USING_STD(sin)
   EIGEN_USING_STD(cos)
   const Scalar u1 = internal::random<Scalar>(0, 1),
                u2 = internal::random<Scalar>(0, 2*EIGEN_PI),
                u3 = internal::random<Scalar>(0, 2*EIGEN_PI);
   const Scalar a = sqrt(Scalar(1) - u1),
                b = sqrt(u1);
   return Quaternion (a * sin(u2), a * cos(u2), b * sin(u3), b * cos(u3));
 }
 
 
 /** Returns a quaternion representing a rotation between
   * the two arbitrary vectors \a a and \a b. In other words, the built
   * rotation represent a rotation sending the line of direction \a a
   * to the line of direction \a b, both lines passing through the origin.
   *
   * \returns resulting quaternion
   *
   * Note that the two input vectors do \b not have to be normalized, and
   * do not need to have the same norm.
   */
 template<typename Scalar, int Options>
 template<typename Derived1, typename Derived2>
 EIGEN_DEVICE_FUNC Quaternion<Scalar,Options> Quaternion<Scalar,Options>::FromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b)
 {
     Quaternion quat;
     quat.setFromTwoVectors(a, b);
     return quat;
 }
 
 
 /** \returns the multiplicative inverse of \c *this
   * Note that in most cases, i.e., if you simply want the opposite rotation,
   * and/or the quaternion is normalized, then it is enough to use the conjugate.
   *
   * \sa QuaternionBase::conjugate()
   */
 template <class Derived>
 EIGEN_DEVICE_FUNC inline Quaternion<typename internal::traits<Derived>::Scalar> QuaternionBase<Derived>::inverse() const
 {
   // FIXME should this function be called multiplicativeInverse and conjugate() be called inverse() or opposite()  ??
   Scalar n2 = this->squaredNorm();
   if (n2 > Scalar(0))
     return Quaternion<Scalar>(conjugate().coeffs() / n2);
   else
   {
     // return an invalid result to flag the error
     return Quaternion<Scalar>(Coefficients::Zero());
   }
 }
 
 // Generic conjugate of a Quaternion
 namespace internal {
 template<int Arch, class Derived, typename Scalar> struct quat_conj
 {
   EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE Quaternion<Scalar> run(const QuaternionBase<Derived>& q){
     return Quaternion<Scalar>(q.w(),-q.x(),-q.y(),-q.z());
   }
 };
 }
                          
 /** \returns the conjugate of the \c *this which is equal to the multiplicative inverse
   * if the quaternion is normalized.
   * The conjugate of a quaternion represents the opposite rotation.
   *
   * \sa Quaternion2::inverse()
   */
 template <class Derived>
 EIGEN_DEVICE_FUNC inline Quaternion<typename internal::traits<Derived>::Scalar>
 QuaternionBase<Derived>::conjugate() const
 {
   return internal::quat_conj<Architecture::Target, Derived,
                          typename internal::traits<Derived>::Scalar>::run(*this);
                          
 }
 
 /** \returns the angle (in radian) between two rotations
   * \sa dot()
   */
 template <class Derived>
 template <class OtherDerived>
 EIGEN_DEVICE_FUNC inline typename internal::traits<Derived>::Scalar
 QuaternionBase<Derived>::angularDistance(const QuaternionBase<OtherDerived>& other) const
 {
   EIGEN_USING_STD(atan2)
   Quaternion<Scalar> d = (*this) * other.conjugate();
   return Scalar(2) * atan2( d.vec().norm(), numext::abs(d.w()) );
 }
 
  
     
 /** \returns the spherical linear interpolation between the two quaternions
   * \c *this and \a other at the parameter \a t in [0;1].
   * 
   * This represents an interpolation for a constant motion between \c *this and \a other,
   * see also http://en.wikipedia.org/wiki/Slerp.
   */
 template <class Derived>
 template <class OtherDerived>
 EIGEN_DEVICE_FUNC Quaternion<typename internal::traits<Derived>::Scalar>
 QuaternionBase<Derived>::slerp(const Scalar& t, const QuaternionBase<OtherDerived>& other) const
 {
   EIGEN_USING_STD(acos)
   EIGEN_USING_STD(sin)
   const Scalar one = Scalar(1) - NumTraits<Scalar>::epsilon();
   Scalar d = this->dot(other);
   Scalar absD = numext::abs(d);
 
   Scalar scale0;
   Scalar scale1;
 
   if(absD>=one)
   {
     scale0 = Scalar(1) - t;
     scale1 = t;
   }
   else
   {
     // theta is the angle between the 2 quaternions
     Scalar theta = acos(absD);
     Scalar sinTheta = sin(theta);
 
     scale0 = sin( ( Scalar(1) - t ) * theta) / sinTheta;
     scale1 = sin( ( t * theta) ) / sinTheta;
   }
   if(d<Scalar(0)) scale1 = -scale1;
 
   return Quaternion<Scalar>(scale0 * coeffs() + scale1 * other.coeffs());
 }
 
 namespace internal {
 
 // set from a rotation matrix
 template<typename Other>
 struct quaternionbase_assign_impl<Other,3,3>
 {
   typedef typename Other::Scalar Scalar;
   template<class Derived> EIGEN_DEVICE_FUNC static inline void run(QuaternionBase<Derived>& q, const Other& a_mat)
   {
     const typename internal::nested_eval<Other,2>::type mat(a_mat);
     EIGEN_USING_STD(sqrt)
     // This algorithm comes from  "Quaternion Calculus and Fast Animation",
     // Ken Shoemake, 1987 SIGGRAPH course notes
     Scalar t = mat.trace();
     if (t > Scalar(0))
     {
       t = sqrt(t + Scalar(1.0));
       q.w() = Scalar(0.5)*t;
       t = Scalar(0.5)/t;
       q.x() = (mat.coeff(2,1) - mat.coeff(1,2)) * t;
       q.y() = (mat.coeff(0,2) - mat.coeff(2,0)) * t;
       q.z() = (mat.coeff(1,0) - mat.coeff(0,1)) * t;
     }
     else
     {
       Index i = 0;
       if (mat.coeff(1,1) > mat.coeff(0,0))
         i = 1;
       if (mat.coeff(2,2) > mat.coeff(i,i))
         i = 2;
       Index j = (i+1)%3;
       Index k = (j+1)%3;
 
       t = sqrt(mat.coeff(i,i)-mat.coeff(j,j)-mat.coeff(k,k) + Scalar(1.0));
       q.coeffs().coeffRef(i) = Scalar(0.5) * t;
       t = Scalar(0.5)/t;
       q.w() = (mat.coeff(k,j)-mat.coeff(j,k))*t;
       q.coeffs().coeffRef(j) = (mat.coeff(j,i)+mat.coeff(i,j))*t;
       q.coeffs().coeffRef(k) = (mat.coeff(k,i)+mat.coeff(i,k))*t;
     }
   }
 };
 
 // set from a vector of coefficients assumed to be a quaternion
 template<typename Other>
 struct quaternionbase_assign_impl<Other,4,1>
 {
   typedef typename Other::Scalar Scalar;
   template<class Derived> EIGEN_DEVICE_FUNC static inline void run(QuaternionBase<Derived>& q, const Other& vec)
   {
     q.coeffs() = vec;
   }
 };
 
 } // end namespace internal
 
 } // end namespace Eigen
 
 #endif // EIGEN_QUATERNION_H

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