Re: [eigen] Mapping array of scalars into quaternions

[Benoit Jacob <jacob.benoit.1@xxxxxxxxx>]

just wanted to say that i've not forgotten about your patch. i'll
study it and reply tomorrow (if nobody does before). i can already say
about asserts that indeed we have a EIGEN_STATIC_ASSERT mechanism that
works at compile time.
Benoit

2009/10/23 Mathieu Gautier <mathieu.gautier@xxxxxx>:
>> That would be great -- i could take care of doing Quaternion then,
>> that would be a piece of cake once you've done the rest.
>
> It's done. I put the classes at the end of the Quaternion.h file.
>
> I have implemented :
>
> * QuaternionBase class which hold much of the former Quaternion class method
> * QuaternionWrapper which map a array of scalars
> * Quat to test the operation between a QuaternionWrapper and a Quaternion
> and some operation (such as Identity) return a Quaternion.
>
>> In your patch, just let the Quaternion class untouched.
>
> You just have to replace the Quat class by the class Quaternion that you
> will modify and replace the occurence of Quat by Quaternion.
>
> I have some remarks.
>
> I used asserts to check the scalars type on some operation (assignation and
> *), but I don't know if it's the best choice or if I can achieve with a
> smarter use of templates.
>
> I am not sure about my uses of traits and typedef, I try to stay close to
> DiagonalMatrix, but I'm not comfortable with that.
>
> If the modification are included, I can extend the units tests to fit these
> changes and try to complete the class documentation.
>
> --
> Mathieu
>
>
> # HG changeset patch
> # User Mathieu Gautier <mathieu.gautier@xxxxxx>
> # Date 1256310829 -7200
> # Node ID a2c5adefbf0204bddd1ea6922dca2b1c6d0782ed
> # Parent  754dfef12c186fb259e088dc5a6a58e06445d3bd
> * QuaternionBase and QuaternionWrapper (Quat is needed in order to build)
>
> diff -r 754dfef12c18 -r a2c5adefbf02
> Eigen/src/Core/util/ForwardDeclarations.h
> --- a/Eigen/src/Core/util/ForwardDeclarations.h Tue Oct 20 23:25:49 2009
> -0400
> +++ b/Eigen/src/Core/util/ForwardDeclarations.h Fri Oct 23 17:13:49 2009
> +0200
> @@ -130,6 +130,7 @@
>  template<typename Derived, int _Dim> class RotationBase;
>  template<typename Lhs, typename Rhs> class Cross;
>  template<typename Scalar> class Quaternion;
> +template<typename Scalar> class QuaternionWrapper;
>  template<typename Scalar> class Rotation2D;
>  template<typename Scalar> class AngleAxis;
>  template<typename Scalar,int Dim,int Mode=Affine> class Transform;
> diff -r 754dfef12c18 -r a2c5adefbf02 Eigen/src/Geometry/Quaternion.h
> --- a/Eigen/src/Geometry/Quaternion.h   Tue Oct 20 23:25:49 2009 -0400
> +++ b/Eigen/src/Geometry/Quaternion.h   Fri Oct 23 17:13:49 2009 +0200
> @@ -507,4 +507,529 @@
>   }
>  };
>
> +/*###################################################################
> +      QuaternionBase and QuaternionWrapper and Quat
> +  ###################################################################*/
> +
> +template<typename Other,
> +         int OtherRows=Other::RowsAtCompileTime,
> +         int OtherCols=Other::ColsAtCompileTime>
> +struct ei_quaternionbase_assign_impl;
> +
> +template<typename Scalar> class Quat; // [XXX] remove
> +
> +template<class Derived>
> +class QuaternionBase : public RotationBase<Derived, 3> {
> +  typedef RotationBase<Derived, 3> Base;
> +public:
> +  using Base::operator*;
> +
> + // 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 derived().coeffs().coeff(0); }
> +  /** \returns the \c y coefficient */
> +  inline Scalar y() const { return derived().coeffs().coeff(1); }
> +  /** \returns the \c z coefficient */
> +  inline Scalar z() const { return derived().coeffs().coeff(2); }
> +  /** \returns the \c w coefficient */
> +  inline Scalar w() const { return derived().coeffs().coeff(3); }
> +
> +  /** \returns a reference to the \c x coefficient */
> +  inline Scalar& x() { return derived().coeffs().coeffRef(0); }
> +  /** \returns a reference to the \c y coefficient */
> +  inline Scalar& y() { return derived().coeffs().coeffRef(1); }
> +  /** \returns a reference to the \c z coefficient */
> +  inline Scalar& z() { return derived().coeffs().coeffRef(2); }
> +  /** \returns a reference to the \c w coefficient */
> +  inline Scalar& w() { return derived().coeffs().coeffRef(3); }
> +
> +  /** \returns a read-only vector expression of the imaginary part (x,y,z)
> */
> +  inline const Block<typename ei_traits<Derived>::Coefficients,3,1> vec()
> const { return derived().coeffs().template start<3>(); }
> +
> +  /** \returns a vector expression of the imaginary part (x,y,z) */
> +  inline Block<typename ei_traits<Derived>::Coefficients,3,1> vec() {
> return derived().coeffs().template start<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(); }
> +
> +  template<class OtherDerived> QuaternionBase& operator=(const
> QuaternionBase<OtherDerived>& other);
> +  QuaternionBase& operator=(const AngleAxisType& aa);
> +  template<class OtherDerived>
> +  QuaternionBase& operator=(const MatrixBase<OtherDerived>& m);
> +
> +  /** \returns a quaternion representing an identity rotation
> +    * \sa MatrixBase::Identity()
> +    */
> +  inline static Quat<Scalar> Identity() { return Quat<Scalar>(1, 0, 0, 0);
> }
> +
> +  /** \sa Quaternion2::Identity(), MatrixBase::setIdentity()
> +    */
> +  inline QuaternionBase& setIdentity() { coeffs() << 0, 0, 0, 1; return
> *this; }
> +
> +  /** \returns the squared norm of the quaternion's coefficients
> +    * \sa Quaternion2::norm(), MatrixBase::squaredNorm()
> +    */
> +  inline Scalar squaredNorm() const { return coeffs().squaredNorm(); }
> +
> +  /** \returns the norm of the quaternion's coefficients
> +    * \sa Quaternion2::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 version of \c *this
> +    * \sa normalize(), MatrixBase::normalized() */
> +  inline Quat<Scalar> normalized() const { return
> Quat<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> inline Scalar angularDistance(const
> QuaternionBase<OtherDerived>& other) const;
> +
> +  Matrix3 toRotationMatrix(void) const;
> +
> +  template<typename Derived1, typename Derived2>
> +  QuaternionBase& setFromTwoVectors(const MatrixBase<Derived1>& a, const
> MatrixBase<Derived2>& b);
> +
> +  template<class OtherDerived> inline Quat<Scalar> operator* (const
> QuaternionBase<OtherDerived>& q) const;
> +  template<class OtherDerived> inline QuaternionBase& operator*= (const
> QuaternionBase<OtherDerived>& q);
> +
> +  Quat<Scalar> inverse(void) const;
> +  Quat<Scalar> conjugate(void) const;
> +
> +  template<class OtherDerived> Quat<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() */
> +  bool isApprox(const QuaternionBase& other, typename
> NumTraits<Scalar>::Real prec = precision<Scalar>()) const
> +  { return coeffs().isApprox(other.coeffs(), prec); }
> +
> +  Vector3 _transformVector(Vector3 v) const;
> +
> +};
> +
> +/* ########### Quat -> Quaternion */
> +
> +template<typename _Scalar>
> +struct ei_traits<Quat<_Scalar> >
> + : ei_traits<Matrix<_Scalar,4,1> >
> +{
> +  typedef typename Matrix<_Scalar,4,1> Coefficients;
> +};
> +
> +template<typename _Scalar>
> +class Quat : public QuaternionBase<Quat<_Scalar> >{
> +  typedef QuaternionBase<Quat<Scalar> > Base;
> +public:
> +
> +  typename typedef ei_traits<Quat<_Scalar> >::Coefficients Coefficients;
> +
> +  using Base::operator=;
> +
> +  /** Default constructor leaving the quaternion uninitialized. */
> +  inline Quat() {}
> +
> +  /** 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 Quat(Scalar w, Scalar x, Scalar y, Scalar z)
> +  { coeffs() << x, y, z, w; }
> +
> +  /** Constructs and initialize a quaternion from the array data
> +    * This constructor is also used to map an array */
> +  inline Quat(const Scalar* data) : m_coeffs(data) {}
> +
> +  /** Copy constructor */
> +  inline Quat(const QuaternionBase& other) { m_coeffs = other.coeffs(); }
> +
> +  /** Constructs and initializes a quaternion from the angle-axis \a aa */
> +  explicit inline Quat(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 Quat(const MatrixBase<Derived>& other) { *this = other; }
> +
> +  /** \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<QuaternionBase,
> QuaternionBase<NewScalarType> >::type cast() const
> +  { return typename ei_cast_return_type<QuaternionBase,
> QuaternionBase<NewScalarType> >::type(*this); }
> +
> +  /** Copy constructor with scalar type conversion */
> +  template<typename OtherScalarType>
> +  inline explicit Quat(const QuaternionBase<OtherScalarType>& other)
> +  { m_coeffs = other.coeffs().template cast<Scalar>(); }
> +
> +  inline Coefficients& coeffs() { return m_coeffs;}
> +  inline const Coefficients& coeffs() const { return m_coeffs;}
> +
> +protected:
> +  Coefficients m_coeffs;
> +};
> +
> +/* ########### QuaternionWrapper */
> +
> +/** \class QuaternionWrapper
> +  * \nonstableyet
> +  *
> +  * \brief Expression of a quaternion
> +  *
> +  * \param Scalar the type of the vector of diagonal coefficients
> +  *
> +  * \sa class Quaternion, class QuaternionBase
> +  */
> +template<typename _Scalar>
> +struct ei_traits<QuaternionWrapper<_Scalar> >
> +{
> +  typedef Map<Matrix<_Scalar, 4, 1> > Coefficients;
> +  typedef Matrix<_Scalar, 4, 1> CoeffType;
> +  typedef typename Coefficients::Scalar Scalar;
> + /* enum {
> +    RowsAtCompileTime = Coefficients::SizeAtCompileTime,
> +    ColsAtCompileTime = Coefficients::SizeAtCompileTime,
> +    MaxRowsAtCompileTime = Coefficients::SizeAtCompileTime,
> +    MaxColsAtCompileTime = Coefficients::SizeAtCompileTime,
> +    Flags = 0
> +  }; */ //[XXX] ??
> +};
> +
> +template<typename _Scalar>
> +class QuaternionWrapper : public QuaternionBase<QuaternionWrapper<_Scalar>
>>, ei_no_assignment_operator {
> +  public:
> +    typename typedef ei_traits<QuaternionWrapper<_Scalar> >::Coefficients
> Coefficients;
> +    typename typedef ei_traits<QuaternionWrapper<_Scalar> >::CoeffType
> CoeffType;
> +
> +    inline QuaternionWrapper(const Scalar* coeffs) : m_coeffs(coeffs) {}
> +
> +    inline Coefficients& coeffs() { return m_coeffs;}
> +    inline const Coefficients& coeffs() const { return m_coeffs;}
> +
> +  protected:
> +    typename Coefficients m_coeffs;
> +};
> +
> +typedef QuaternionWrapper<double> QuaternionMapd;
> +typedef QuaternionWrapper<float> QuaternionMapf;
> +
> +// Generic Quaternion * Quaternion product
> +template<class Derived, class OtherDerived> inline Quat<typename
> ei_traits<Derived>::Scalar>
> +ei_quat_product(const QuaternionBase<Derived>& a, const
> QuaternionBase<OtherDerived>& b)
> +{
> +  return Quat<ei_traits<Derived>::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>
> +inline Quat<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(*this, other);
> +}
> +
> +/** \sa operator*(Quaternion) */
> +template <class Derived>
> +template <class OtherDerived>
> +inline QuaternionBase<Derived>& QuaternionBase<Derived>::operator*= (const
> QuaternionBase<OtherDerived>& other)
> +{
> +  return (*this = *this * other);
> +}
> +
> +/** 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>
> +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>
> +template<class OtherDerived>
> +inline QuaternionBase<Derived>& QuaternionBase<Derived>::operator=(const
> QuaternionBase<OtherDerived>& other)
> +{
> +  m_coeffs = other.coeffs();
> +  return *this;
> +}
> +
> +/** Set \c *this from an angle-axis \a aa and returns a reference to \c
> *this
> +  */
> +template<class Derived>
> +inline QuaternionBase<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 *this;
> +}
> +
> +/** 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 QuaternionBase<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 *this;
> +}
> +
> +/** 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 QuaternionBase<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)+precision<Scalar>())
> +  {
> +    c = std::max<Scalar>(c,-1);
> +    Matrix<Scalar,2,3> m; m << v0.transpose(), v1.transpose();
> +    SVD<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 *this;
> +  }
> +  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 *this;
> +}
> +
> +/** \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 Quaternion2::conjugate()
> +  */
> +template <class Derived>
> +inline Quat<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 Quat<Scalar>(conjugate().coeffs() / n2);
> +  else
> +  {
> +    // return an invalid result to flag the error
> +    return Quat<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 Quat<typename ei_traits<Derived>::Scalar>
> QuaternionBase<Derived>::conjugate() const
> +{
> +  return Quat<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 0;
> +  return 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>
> +Quat<typename ei_traits<Derived>::Scalar>
> QuaternionBase<Derived>::slerp(Scalar t, const QuaternionBase<OtherDerived>&
> other) const
> +{
> +  static const Scalar one = Scalar(1) - precision<Scalar>();
> +  Scalar d = this->dot(other);
> +  Scalar absD = ei_abs(d);
> +  if (absD>=one)
> +    return Quat<Scalar>(*this);
> +
> +  // theta is the angle between the 2 quaternions
> +  Scalar theta = std::acos(absD);
> +  Scalar sinTheta = ei_sin(theta);
> +
> +  Scalar scale0 = ei_sin( ( Scalar(1) - t ) * theta) / sinTheta;
> +  Scalar scale1 = ei_sin( ( t * theta) ) / sinTheta;
> +  if (d<0)
> +    scale1 = -scale1;
> +
> +  return Quat<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;
> +  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 > 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
> +    {
> +      int i = 0;
> +      if (mat.coeff(1,1) > mat.coeff(0,0))
> +        i = 1;
> +      if (mat.coeff(2,2) > mat.coeff(i,i))
> +        i = 2;
> +      int j = (i+1)%3;
> +      int 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
>
>