Quaternion.h

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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;
}

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;
  enum {
    Flags = Eigen::internal::traits<Derived>::Flags
  };

 // typedef typename Matrix<Scalar,4,1> Coefficients;
  typedef Matrix<Scalar,3,1> Vector3;
  typedef Matrix<Scalar,3,3> Matrix3;
  typedef AngleAxis<Scalar> AngleAxisType;



  EIGEN_DEVICE_FUNC inline Scalar x() const { return this->derived().coeffs().coeff(0); }
  EIGEN_DEVICE_FUNC inline Scalar y() const { return this->derived().coeffs().coeff(1); }
  EIGEN_DEVICE_FUNC inline Scalar z() const { return this->derived().coeffs().coeff(2); }
  EIGEN_DEVICE_FUNC inline Scalar w() const { return this->derived().coeffs().coeff(3); }

  EIGEN_DEVICE_FUNC inline Scalar& x() { return this->derived().coeffs().coeffRef(0); }
  EIGEN_DEVICE_FUNC inline Scalar& y() { return this->derived().coeffs().coeffRef(1); }
  EIGEN_DEVICE_FUNC inline Scalar& z() { return this->derived().coeffs().coeffRef(2); }
  EIGEN_DEVICE_FUNC inline Scalar& w() { return this->derived().coeffs().coeffRef(3); }

  EIGEN_DEVICE_FUNC inline const VectorBlock<const Coefficients,3> vec() const { return coeffs().template head<3>(); }

  EIGEN_DEVICE_FUNC inline VectorBlock<Coefficients,3> vec() { return coeffs().template head<3>(); }

  EIGEN_DEVICE_FUNC inline const typename internal::traits<Derived>::Coefficients& coeffs() const { return derived().coeffs(); }

  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);

  EIGEN_DEVICE_FUNC static inline Quaternion<Scalar> Identity() { return Quaternion<Scalar>(Scalar(1), Scalar(0), Scalar(0), Scalar(0)); }

  EIGEN_DEVICE_FUNC inline QuaternionBase& setIdentity() { coeffs() << Scalar(0), Scalar(0), Scalar(0), Scalar(1); return *this; }

  EIGEN_DEVICE_FUNC inline Scalar squaredNorm() const { return coeffs().squaredNorm(); }

  EIGEN_DEVICE_FUNC inline Scalar norm() const { return coeffs().norm(); }

  EIGEN_DEVICE_FUNC inline void normalize() { coeffs().normalize(); }
  EIGEN_DEVICE_FUNC inline Quaternion<Scalar> normalized() const { return Quaternion<Scalar>(coeffs().normalized()); }

  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;

  EIGEN_DEVICE_FUNC Matrix3 toRotationMatrix() const;

  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);

  EIGEN_DEVICE_FUNC Quaternion<Scalar> inverse() const;

  EIGEN_DEVICE_FUNC Quaternion<Scalar> conjugate() const;

  template<class OtherDerived> EIGEN_DEVICE_FUNC Quaternion<Scalar> slerp(const Scalar& t, const QuaternionBase<OtherDerived>& other) const;

  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); }

  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Vector3 _transformVector(const Vector3& v) const;

  template<typename NewScalarType>
  EIGEN_DEVICE_FUNC inline typename internal::cast_return_type<Derived,Quaternion<NewScalarType> >::type cast() const
  {
    return typename internal::cast_return_type<Derived,Quaternion<NewScalarType> >::type(derived());
  }

#ifdef EIGEN_QUATERNIONBASE_PLUGIN
# include EIGEN_QUATERNIONBASE_PLUGIN
#endif
};

/***************************************************************************
* Definition/implementation of Quaternion<Scalar>
***************************************************************************/

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;

  EIGEN_DEVICE_FUNC inline Quaternion() {}

  EIGEN_DEVICE_FUNC inline Quaternion(const Scalar& w, const Scalar& x, const Scalar& y, const Scalar& z) : m_coeffs(x, y, z, w){}

  EIGEN_DEVICE_FUNC explicit inline Quaternion(const Scalar* data) : m_coeffs(data) {}

  template<class Derived> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Quaternion(const QuaternionBase<Derived>& other) { this->Base::operator=(other); }

  EIGEN_DEVICE_FUNC explicit inline Quaternion(const AngleAxisType& aa) { *this = aa; }

  template<typename Derived>
  EIGEN_DEVICE_FUNC explicit inline Quaternion(const MatrixBase<Derived>& other) { *this = other; }

  template<typename OtherScalar, int OtherOptions>
  EIGEN_DEVICE_FUNC explicit inline Quaternion(const Quaternion<OtherScalar, OtherOptions>& other)
  { m_coeffs = other.coeffs().template cast<Scalar>(); }

  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
};

typedef Quaternion<float> Quaternionf;
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
    };
  };
}

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*=;

    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;
};

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*=;

    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;
};

typedef Map<Quaternion<float>, 0>         QuaternionMapf;
typedef Map<Quaternion<double>, 0>        QuaternionMapd;
typedef Map<Quaternion<float>, Aligned>   QuaternionMapAlignedf;
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, int _Options> 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()
    );
  }
};
}

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,
                         EIGEN_PLAIN_ENUM_MIN(internal::traits<Derived>::Alignment, internal::traits<OtherDerived>::Alignment)>::run(*this, other);
}

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();
}

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();
}

template<class Derived>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& QuaternionBase<Derived>::operator=(const AngleAxisType& aa)
{
  EIGEN_USING_STD_MATH(cos)
  EIGEN_USING_STD_MATH(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();
}

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();
}

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;
}

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_MATH(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();
}

template<typename Scalar, int Options>
EIGEN_DEVICE_FUNC Quaternion<Scalar,Options> Quaternion<Scalar,Options>::UnitRandom()
{
  EIGEN_USING_STD_MATH(sqrt)
  EIGEN_USING_STD_MATH(sin)
  EIGEN_USING_STD_MATH(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(1 - u1),
               b = sqrt(u1);
  return Quaternion (a * sin(u2), a * cos(u2), b * sin(u3), b * cos(u3));
}


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;
}


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, int _Options> 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());
  }
};
}
                         
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,
                         internal::traits<Derived>::Alignment>::run(*this);
                         
}

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_MATH(atan2)
  Quaternion<Scalar> d = (*this) * other.conjugate();
  return Scalar(2) * atan2( d.vec().norm(), numext::abs(d.w()) );
}

 
    
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_MATH(acos)
  EIGEN_USING_STD_MATH(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_MATH(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