text: // This file is part of Eigen, a lightweight C++ template library
  // for linear algebra.
  //
  // Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
  // Copyright (C) 2009 Mathieu Gautier <mathieu.gautier@cea.fr>
  //
  // Eigen is free software; you can redistribute it and/or
  // modify it under the terms of the GNU Lesser General Public
  // License as published by the Free Software Foundation; either
  // version 3 of the License, or (at your option) any later version.
  //
  // Alternatively, you can redistribute it and/or
  // modify it under the terms of the GNU General Public License as
  // published by the Free Software Foundation; either version 2 of
  // the License, or (at your option) any later version.
  //
  // Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
  // WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
  // FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
  // GNU General Public License for more details.
  //
  // You should have received a copy of the GNU Lesser General Public
  // License and a copy of the GNU General Public License along with
  // Eigen. If not, see <http://www.gnu.org/licenses/>.
  
  #ifndef EIGEN_QUATERNION_H
  #define EIGEN_QUATERNION_H
  
  /***************************************************************************
  * Definition of QuaternionBase<Derived>
  * The implementation is at the end of the file
  ***************************************************************************/
  
  template<typename Other,
           int OtherRows=Other::RowsAtCompileTime,
           int OtherCols=Other::ColsAtCompileTime>
  struct ei_quaternionbase_assign_impl;
  
  template<class Derived>
  class QuaternionBase : public RotationBase<Derived, 3>
  {
    typedef RotationBase<Derived, 3> Base;
  public:
    using Base::operator*;
    using Base::derived;
  
    typedef typename ei_traits<Derived>::Scalar Scalar;
    typedef typename NumTraits<Scalar>::Real RealScalar;
    typedef typename ei_traits<Derived>::Coefficients Coefficients;
  
   // 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 */
    inline Scalar x() const { return this->derived().coeffs().coeff(0); }
    /** \returns the \c y coefficient */
    inline Scalar y() const { return this->derived().coeffs().coeff(1); }
    /** \returns the \c z coefficient */
    inline Scalar z() const { return this->derived().coeffs().coeff(2); }
    /** \returns the \c w coefficient */
    inline Scalar w() const { return this->derived().coeffs().coeff(3); }
  
    /** \returns a reference to the \c x coefficient */
    inline Scalar& x() { return this->derived().coeffs().coeffRef(0); }
    /** \returns a reference to the \c y coefficient */
    inline Scalar& y() { return this->derived().coeffs().coeffRef(1); }
    /** \returns a reference to the \c z coefficient */
    inline Scalar& z() { return this->derived().coeffs().coeffRef(2); }
    /** \returns a reference to the \c w coefficient */
    inline Scalar& w() { return this->derived().coeffs().coeffRef(3); }
  
    /** \returns a read-only vector expression of the imaginary part (x,y,z) */
    inline const VectorBlock<Coefficients,3> vec() const { return coeffs().template head<3>(); }
  
    /** \returns a vector expression of the imaginary part (x,y,z) */
    inline VectorBlock<Coefficients,3> vec() { return coeffs().template head<3>(); }
  
    /** \returns a read-only vector expression of the coefficients (x,y,z,w) */
    inline const typename ei_traits<Derived>::Coefficients& coeffs() const { return derived().coeffs(); }
  
    /** \returns a vector expression of the coefficients (x,y,z,w) */
    inline typename ei_traits<Derived>::Coefficients& coeffs() { return derived().coeffs(); }
  
    EIGEN_STRONG_INLINE QuaternionBase<Derived>& operator=(const QuaternionBase<Derived>& other);
    template<class OtherDerived> 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); }
  
    Derived& operator=(const AngleAxisType& aa);
    template<class OtherDerived> Derived& operator=(const MatrixBase<OtherDerived>& m);
  
    /** \returns a quaternion representing an identity rotation
      * \sa MatrixBase::Identity()
      */
    inline static Quaternion<Scalar> Identity() { return Quaternion<Scalar>(1, 0, 0, 0); }
  
    /** \sa QuaternionBase::Identity(), MatrixBase::setIdentity()
      */
    inline QuaternionBase& setIdentity() { coeffs() << 0, 0, 0, 1; return *this; }
  
    /** \returns the squared norm of the quaternion's coefficients
      * \sa QuaternionBase::norm(), MatrixBase::squaredNorm()
      */
    inline Scalar squaredNorm() const { return coeffs().squaredNorm(); }
  
    /** \returns the norm of the quaternion's coefficients
      * \sa QuaternionBase::squaredNorm(), MatrixBase::norm()
      */
    inline Scalar norm() const { return coeffs().norm(); }
  
    /** Normalizes the quaternion \c *this
      * \sa normalized(), MatrixBase::normalize() */
    inline void normalize() { coeffs().normalize(); }
    /** \returns a normalized copy of \c *this
      * \sa normalize(), MatrixBase::normalized() */
    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> inline Scalar dot(const QuaternionBase<OtherDerived>& other) const { return coeffs().dot(other.coeffs()); }
  
    template<class OtherDerived> Scalar angularDistance(const QuaternionBase<OtherDerived>& other) const;
  
    /** \returns an equivalent 3x3 rotation matrix */
    Matrix3 toRotationMatrix() const;
  
    /** \returns the quaternion which transform \a a into \a b through a rotation */
    template<typename Derived1, typename Derived2>
    Derived& setFromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b);
  
    template<class OtherDerived> EIGEN_STRONG_INLINE Quaternion<Scalar> operator* (const QuaternionBase<OtherDerived>& q) const;
    template<class OtherDerived> EIGEN_STRONG_INLINE Derived& operator*= (const QuaternionBase<OtherDerived>& q);
  
    /** \returns the quaternion describing the inverse rotation */
    Quaternion<Scalar> inverse() const;
  
    /** \returns the conjugated quaternion */
    Quaternion<Scalar> conjugate() const;
  
    /** \returns an interpolation for a constant motion between \a other and \c *this
      * \a t in [0;1]
      * see http://en.wikipedia.org/wiki/Slerp
      */
    template<class OtherDerived> Quaternion<Scalar> slerp(Scalar t, const QuaternionBase<OtherDerived>& other) const;
  
    /** \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>
    bool isApprox(const QuaternionBase<OtherDerived>& other, RealScalar prec = NumTraits<Scalar>::dummy_precision()) const
    { return coeffs().isApprox(other.coeffs(), prec); }
  
  	/** return the result vector of \a v through the rotation*/
    EIGEN_STRONG_INLINE Vector3 _transformVector(Vector3 v) const;
  
    /** \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>
    inline typename ei_cast_return_type<Derived,Quaternion<NewScalarType> >::type cast() const
    {
      return typename ei_cast_return_type<Derived,Quaternion<NewScalarType> >::type(
        coeffs().template cast<NewScalarType>());
    }
  };
  
  /***************************************************************************
  * Definition/implementation of Quaternion<Scalar>
  ***************************************************************************/
  
  /** \geometry_module \ingroup Geometry_Module
    *
    * \class Quaternion
    *
    * \brief The quaternion class used to represent 3D orientations and rotations
    *
    * \param _Scalar the scalar type, i.e., the type of the coefficients
    *
    * 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, quatertions 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
    *
    * \sa  class AngleAxis, class Transform
    */
  
  template<typename _Scalar>
  struct ei_traits<Quaternion<_Scalar> >
  {
    typedef Quaternion<_Scalar> PlainObject;
    typedef _Scalar Scalar;
    typedef Matrix<_Scalar,4,1> Coefficients;
    enum{
      PacketAccess = Aligned
    };
  };
  
  template<typename _Scalar>
  class Quaternion : public QuaternionBase<Quaternion<_Scalar> >{
    typedef QuaternionBase<Quaternion<_Scalar> > Base;
  public:
    typedef _Scalar Scalar;
  
    EIGEN_INHERIT_ASSIGNMENT_EQUAL_OPERATOR(Quaternion<Scalar>)
    using Base::operator*=;
  
    typedef typename ei_traits<Quaternion<Scalar> >::Coefficients Coefficients;
    typedef typename Base::AngleAxisType AngleAxisType;
  
    /** Default constructor leaving the quaternion uninitialized. */
    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]
      */
    inline Quaternion(Scalar w, Scalar x, Scalar y, Scalar z) : m_coeffs(x, y, z, w){}
  
    /** Constructs and initialize a quaternion from the array data */
    inline Quaternion(const Scalar* data) : m_coeffs(data) {}
  
    /** Copy constructor */
    template<class Derived> EIGEN_STRONG_INLINE Quaternion(const QuaternionBase<Derived>& other) { this->Base::operator=(other); }
  
    /** Constructs and initializes a quaternion from the angle-axis \a aa */
    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>
    explicit inline Quaternion(const MatrixBase<Derived>& other) { *this = other; }
  
    inline Coefficients& coeffs() { return m_coeffs;}
    inline const Coefficients& coeffs() const { return m_coeffs;}
  
  protected:
    Coefficients m_coeffs;
  };
  
  /** \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>>
  ***************************************************************************/
  
  /** \class Map<Quaternion>
    *
    *
    * \brief Expression of a quaternion from a memory buffer
    *
    * \param _Scalar the type of the Quaternion coefficients
    * \param PacketAccess 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 _PacketAccess>
  struct ei_traits<Map<Quaternion<_Scalar>, _PacketAccess> >:
  ei_traits<Quaternion<_Scalar> >
  {
    typedef _Scalar Scalar;
    typedef Map<Matrix<_Scalar,4,1>, _PacketAccess> Coefficients;
    enum {
      PacketAccess = _PacketAccess
    };
  };
  
  template<typename _Scalar, int PacketAccess>
  class Map<Quaternion<_Scalar>, PacketAccess >
    : public QuaternionBase<Map<Quaternion<_Scalar>, PacketAccess> >
  {
      typedef QuaternionBase<Map<Quaternion<_Scalar>, PacketAccess> > Base;
  
    public:
      typedef _Scalar Scalar;
      typedef typename ei_traits<Map>::Coefficients Coefficients;
      EIGEN_INHERIT_ASSIGNMENT_EQUAL_OPERATOR(Map)
      using Base::operator*=;
  
      /** Constructs a Mapped Quaternion object from the pointer \a coeffs
        *
        * The pointer \a coeffs must reference the four coeffecients of Quaternion in the following order:
        * \code *coeffs == {x, y, z, w} \endcode
        *
        * If the template parameter PacketAccess is set to Aligned, then the pointer coeffs must be aligned. */
      EIGEN_STRONG_INLINE Map(const Scalar* coeffs) : m_coeffs(coeffs) {}
  
      inline Coefficients& coeffs() { return m_coeffs;}
      inline const Coefficients& coeffs() const { return m_coeffs;}
  
    protected:
      Coefficients m_coeffs;
  };
  
  /** \ingroup Geometry_Module
    * Map an unaligned array of single precision scalar as a quaternion */
  typedef Map<Quaternion<float>, 0>         QuaternionMapf;
  /** \ingroup Geometry_Module
    * Map an unaligned array of double precision scalar as a quaternion */
  typedef Map<Quaternion<double>, 0>        QuaternionMapd;
  /** \ingroup Geometry_Module
    * Map a 16-bits aligned array of double precision scalars as a quaternion */
  typedef Map<Quaternion<float>, Aligned>   QuaternionMapAlignedf;
  /** \ingroup Geometry_Module
    * Map a 16-bits 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.
  template<int Arch, class Derived1, class Derived2, typename Scalar, int PacketAccess> struct ei_quat_product
  {
    EIGEN_STRONG_INLINE static 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_STRONG_INLINE Quaternion<typename ei_traits<Derived>::Scalar>
  QuaternionBase<Derived>::operator* (const QuaternionBase<OtherDerived>& other) const
  {
    EIGEN_STATIC_ASSERT((ei_is_same_type<typename Derived::Scalar, typename OtherDerived::Scalar>::ret),
     YOU_MIXED_DIFFERENT_NUMERIC_TYPES__YOU_NEED_TO_USE_THE_CAST_METHOD_OF_MATRIXBASE_TO_CAST_NUMERIC_TYPES_EXPLICITLY)
    return ei_quat_product<Architecture::Target, Derived, OtherDerived,
                           typename ei_traits<Derived>::Scalar,
                           ei_traits<Derived>::PacketAccess && ei_traits<OtherDerived>::PacketAccess>::run(*this, other);
  }
  
  /** \sa operator*(Quaternion) */
  template <class Derived>
  template <class OtherDerived>
  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_STRONG_INLINE typename QuaternionBase<Derived>::Vector3
  QuaternionBase<Derived>::_transformVector(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 litterature (30 versus 39 flops). It also requires two
      // Vector3 as temporaries.
      Vector3 uv = Scalar(2) * this->vec().cross(v);
      return v + this->w() * uv + this->vec().cross(uv);
  }
  
  template<class Derived>
  EIGEN_STRONG_INLINE QuaternionBase<Derived>& QuaternionBase<Derived>::operator=(const QuaternionBase<Derived>& other)
  {
    coeffs() = other.coeffs();
    return derived();
  }
  
  template<class Derived>
  template<class OtherDerived>
  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_STRONG_INLINE Derived& QuaternionBase<Derived>::operator=(const AngleAxisType& aa)
  {
    Scalar ha = Scalar(0.5)*aa.angle(); // Scalar(0.5) to suppress precision loss warnings
    this->w() = ei_cos(ha);
    this->vec() = ei_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>
  inline Derived& QuaternionBase<Derived>::operator=(const MatrixBase<MatrixDerived>& xpr)
  {
    EIGEN_STATIC_ASSERT((ei_is_same_type<typename Derived::Scalar, typename MatrixDerived::Scalar>::ret),
     YOU_MIXED_DIFFERENT_NUMERIC_TYPES__YOU_NEED_TO_USE_THE_CAST_METHOD_OF_MATRIXBASE_TO_CAST_NUMERIC_TYPES_EXPLICITLY)
    ei_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>
  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  = 2*this->x();
    const Scalar ty  = 2*this->y();
    const Scalar tz  = 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) = 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) = 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) = 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>
  inline Derived& QuaternionBase<Derived>::setFromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b)
  {
    Vector3 v0 = a.normalized();
    Vector3 v1 = b.normalized();
    Scalar c = v1.dot(v0);
  
    // if dot == -1, vectors are nearly opposites
    // => accuraletly 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 = std::max<Scalar>(c,-1);
      Matrix<Scalar,2,3> m; m << v0.transpose(), v1.transpose();
      JacobiSVD<Matrix<Scalar,2,3> > svd(m);
      Vector3 axis = svd.matrixV().col(2);
  
      Scalar w2 = (Scalar(1)+c)*Scalar(0.5);
      this->w() = ei_sqrt(w2);
      this->vec() = axis * ei_sqrt(Scalar(1) - w2);
      return derived();
    }
    Vector3 axis = v0.cross(v1);
    Scalar s = ei_sqrt((Scalar(1)+c)*Scalar(2));
    Scalar invs = Scalar(1)/s;
    this->vec() = axis * invs;
    this->w() = s * Scalar(0.5);
  
    return derived();
  }
  
  /** \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>
  inline Quaternion<typename ei_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 > 0)
      return Quaternion<Scalar>(conjugate().coeffs() / n2);
    else
    {
      // return an invalid result to flag the error
      return Quaternion<Scalar>(Coefficients::Zero());
    }
  }
  
  /** \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>
  inline Quaternion<typename ei_traits<Derived>::Scalar>
  QuaternionBase<Derived>::conjugate() const
  {
    return Quaternion<Scalar>(this->w(),-this->x(),-this->y(),-this->z());
  }
  
  /** \returns the angle (in radian) between two rotations
    * \sa dot()
    */
  template <class Derived>
  template <class OtherDerived>
  inline typename ei_traits<Derived>::Scalar
  QuaternionBase<Derived>::angularDistance(const QuaternionBase<OtherDerived>& other) const
  {
    double d = ei_abs(this->dot(other));
    if (d>=1.0)
      return Scalar(0);
    return static_cast<Scalar>(2 * std::acos(d));
  }
  
  /** \returns the spherical linear interpolation between the two quaternions
    * \c *this and \a other at the parameter \a t
    */
  template <class Derived>
  template <class OtherDerived>
  Quaternion<typename ei_traits<Derived>::Scalar>
  QuaternionBase<Derived>::slerp(Scalar t, const QuaternionBase<OtherDerived>& other) const
  {
    static const Scalar one = Scalar(1) - NumTraits<Scalar>::epsilon();
    Scalar d = this->dot(other);
    Scalar absD = ei_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 = std::acos(absD);
      Scalar sinTheta = ei_sin(theta);
  
      scale0 = ei_sin( ( Scalar(1) - t ) * theta) / sinTheta;
      scale1 = ei_sin( ( t * theta) ) / sinTheta;
      if (d<0)
        scale1 = -scale1;
    }
  
    return Quaternion<Scalar>(scale0 * coeffs() + scale1 * other.coeffs());
  }
  
  // set from a rotation matrix
  template<typename Other>
  struct ei_quaternionbase_assign_impl<Other,3,3>
  {
    typedef typename Other::Scalar Scalar;
    typedef DenseIndex Index;
    template<class Derived> inline static void run(QuaternionBase<Derived>& q, const Other& mat)
    {
      // This algorithm comes from  "Quaternion Calculus and Fast Animation",
      // Ken Shoemake, 1987 SIGGRAPH course notes
      Scalar t = mat.trace();
      if (t > Scalar(0))
      {
        t = ei_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
      {
        DenseIndex i = 0;
        if (mat.coeff(1,1) > mat.coeff(0,0))
          i = 1;
        if (mat.coeff(2,2) > mat.coeff(i,i))
          i = 2;
        DenseIndex j = (i+1)%3;
        DenseIndex k = (j+1)%3;
  
        t = ei_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 ei_quaternionbase_assign_impl<Other,4,1>
  {
    typedef typename Other::Scalar Scalar;
    template<class Derived> inline static void run(QuaternionBase<Derived>& q, const Other& vec)
    {
      q.coeffs() = vec;
    }
  };
  
  
  #endif // EIGEN_QUATERNION_H
  