text: // 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
  