Re: [eigen] Mapping array of scalars into quaternions

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

Hi,

Thanks a lot. I just pushed your changeset, followed by another
changeset to make things work here with GCC.  I replaced the existing
Quaternion class by your Quat (so there's no Quat anymore) and got it
to the point that the unit-tests succeed. I didn't look very closely
if there were more things to do.

Here's the commit msg of my changeset:

* rename new Quat class to Quaternion, remove existing Quaternion
* add Copyright line for Mathieu
* cast() was broken (compile errors) and needed anyway to be in QuaternionBase
* it's VectorBlock<T,3>, don't pass additional parameter 1, it has
different meaning!!
* make it compile with GCC (put 'typename' at the right location)

In the future, in order to test compilation yourself, you just need to do:
mkdir build
cd build
cmake ../eigen2
make test_geo_quaternion

What's needed next:
* update documentation
* extend unit tests to cover Map<Quaternion>, no need to redo all the
arithmetic there but focus on the mapping of existing memory, test
both aligned and not aligned, etc... see doc here:
http://eigen.tuxfamily.org/index.php?title=Developer%27s_Corner#Writing_unit_tests

Thanks,
Benoit

2009/10/27 Mathieu Gautier <mathieu.gautier@xxxxxx>:
>> So instead of QuaternionWrapper, what about using a specialization of Map
>> for Quaternion:
>
> This version of the patch used Map<Quaternion<_Scalar> > instead of
> QuaternionWrapper. It's work with the last revision (1750).
>
> I had to replace the ei_quaternion_product function by a struct with a
> static function to allow partial specialization to use the vectorized
> multiplication between quaternion and mapped quaternion. It's possible that
> I had used to much template arguments.
>
> --
> Mathieu Gautier
>
> # HG changeset patch
> # User Mathieu Gautier <mathieu.gautier@xxxxxx>
> # Date 1256649556 0
> # Node ID 8eae2834af38b30e009ed88ee12cbe63f4adab0a
> # Parent  b065d97337165b7a9d7282b01aed52d4c47cc39a
> Quaternion could now map an array of 4 scalars :
>
> new classes :
> * QuaternionBase
> * Map<Quaternion>
>
> diff -r b065d9733716 -r 8eae2834af38
> Eigen/src/Core/util/ForwardDeclarations.h
> --- a/Eigen/src/Core/util/ForwardDeclarations.h Sat Oct 24 14:48:34 2009
> +0200
> +++ b/Eigen/src/Core/util/ForwardDeclarations.h Tue Oct 27 13:19:16 2009
> +0000
> @@ -129,6 +129,7 @@
>  // Geometry module:
>  template<typename Derived, int _Dim> class RotationBase;
>  template<typename Lhs, typename Rhs> class Cross;
> +template<typename Derived> class QuaternionBase;
>  template<typename Scalar> class Quaternion;
>  template<typename Scalar> class Rotation2D;
>  template<typename Scalar> class AngleAxis;
> diff -r b065d9733716 -r 8eae2834af38 Eigen/src/Geometry/Quaternion.h
> --- a/Eigen/src/Geometry/Quaternion.h   Sat Oct 24 14:48:34 2009 +0200
> +++ b/Eigen/src/Geometry/Quaternion.h   Tue Oct 27 13:19:16 2009 +0000
> @@ -507,4 +507,549 @@
>   }
>  };
>
> +/*###################################################################
> +      QuaternionBase and Map<Quaternion> 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 when Quat becomes
> Quaternion
> +
> +template<typename Derived>
> +struct ei_traits<QuaternionBase<Derived> >
> +{
> +  typedef typename ei_traits<Derived>::Scalar Scalar;
> +  enum {
> +    PacketAccess = ei_traits<Derived>::PacketAccess
> +  };
> +};
> +
> +template<class Derived>
> +class QuaternionBase : public RotationBase<Derived, 3> {
> +  typedef RotationBase<Derived, 3> Base;
> +public:
> +  using Base::operator*;
> +
> +  typedef typename ei_traits<QuaternionBase<Derived> >::Scalar Scalar;
> +  typedef typename NumTraits<Scalar>::Real RealScalar;
> +
> + // 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<typename ei_traits<Derived>::Coefficients,3,1>
> vec() const { return this->derived().coeffs().template start<3>(); }
> +
> +  /** \returns a vector expression of the imaginary part (x,y,z) */
> +  inline VectorBlock<typename ei_traits<Derived>::Coefficients,3,1> vec() {
> return this->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 this->derived().coeffs(); }
> +
> +  /** \returns a vector expression of the coefficients (x,y,z,w) */
> +  inline typename ei_traits<Derived>::Coefficients& coeffs() { return
> this->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 RealScalar 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> >
> +{
> +  typedef _Scalar Scalar;
> +  typedef Matrix<_Scalar,4,1> Coefficients;
> +  enum{
> +    PacketAccess = Aligned
> +  };
> +};
> +
> +template<typename _Scalar>
> +class Quat : public QuaternionBase<Quat<_Scalar> >{
> +  typedef QuaternionBase<Quat<_Scalar> > Base;
> +public:
> +  using Base::operator=;
> +
> +  typedef _Scalar Scalar;
> +
> +  typedef typename ei_traits<Quat<Scalar> >::Coefficients Coefficients;
> +  typedef typename Base::AngleAxisType AngleAxisType;
> +
> +  /** 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 */
> +//  template<class Derived> inline Quat(const QuaternionBase<Derived>&
> other) { m_coeffs = other.coeffs(); } [XXX] redundant with 703
> +
> +  /** 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<class Derived>
> +  inline typename ei_cast_return_type<Quat, QuaternionBase<Derived> >::type
> cast() const
> +  { return typename ei_cast_return_type<Quat, QuaternionBase<Derived>
>>::type(*this); }
> +
> +  /** Copy constructor with scalar type conversion */
> +  template<class Derived>
> +  inline explicit Quat(const QuaternionBase<Derived>& 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;
> +};
> +
> +/* ########### Map<Quat> */
> +
> +/** \class Map<Quat>
> +  * \nonstableyet
> +  *
> +  * \brief Expression of a quaternion
> +  *
> +  * \param Scalar the type of the vector of diagonal coefficients
> +  *
> +  * \sa class Quaternion, class QuaternionBase
> +  */
> +template<typename _Scalar, int _PacketAccess>
> +struct ei_traits<Map<Quat<_Scalar>, _PacketAccess> >:
> +ei_traits<Quat<_Scalar> >
> +{
> +  typedef _Scalar Scalar;
> +  typedef Map<Matrix<_Scalar,4,1> > Coefficients;
> +  enum {
> +    PacketAccess = _PacketAccess
> +  };
> +};
> +
> +template<typename _Scalar, int PacketAccess>
> +class Map<Quat<_Scalar>, PacketAccess > : public
> QuaternionBase<Map<Quat<_Scalar>, PacketAccess> >, ei_no_assignment_operator
> {
> +  public:
> +
> +    typedef _Scalar Scalar;
> +
> +    typedef typename ei_traits<Map<Quat<Scalar>, PacketAccess>
>>::Coefficients Coefficients;
> +
> +    inline Map<Quat<Scalar>, PacketAccess >(const Scalar* coeffs) :
> m_coeffs(coeffs) {}
> +
> +    inline Coefficients& coeffs() { return m_coeffs;}
> +    inline const Coefficients& coeffs() const { return m_coeffs;}
> +
> +  protected:
> +    Coefficients m_coeffs;
> +};
> +
> +typedef Map<Quat<double> > QuaternionMapd;
> +typedef Map<Quat<float> > QuaternionMapf;
> +typedef Map<Quat<double>, Aligned> QuaternionMapAlignedd;
> +typedef Map<Quat<float>, Aligned> QuaternionMapAlignedf;
> +
> +// Generic Quaternion * Quaternion product
> +template<int Arch, class Derived, class OtherDerived, typename Scalar, int
> PacketAccess> struct ei_quat_product
> +{
> +  inline static Quat<Scalar> run(const QuaternionBase<Derived>& a, const
> QuaternionBase<OtherDerived>& b){
> +    return Quat<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<QuaternionBase<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<EiArch, Derived, OtherDerived,
> +                          ei_traits<Derived>::Scalar,
> +                          ei_traits<Derived>::PacketAccess &&
> ei_traits<OtherDerived>::PacketAccess>::run(*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)
> +{
> +  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<QuaternionBase<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>(ei_traits<Derived>::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<QuaternionBase<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<QuaternionBase<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<QuaternionBase<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
> diff -r b065d9733716 -r 8eae2834af38 Eigen/src/Geometry/arch/Geometry_SSE..h
> --- a/Eigen/src/Geometry/arch/Geometry_SSE.h    Sat Oct 24 14:48:34 2009
> +0200
> +++ b/Eigen/src/Geometry/arch/Geometry_SSE.h    Tue Oct 27 13:19:16 2009
> +0000
> @@ -45,6 +45,27 @@
>   return res;
>  }
>
> +template<class Derived, class OtherDerived> struct
> ei_quat_product<EiArch_SSE, Derived, OtherDerived, float, Aligned>
> +{
> +  inline static Quat<float> run(const QuaternionBase<Derived>& _a, const
> QuaternionBase<OtherDerived>& _b)
> +  {
> +    const __m128 mask = _mm_castsi128_ps(_mm_setr_epi32(0,0,0,0x80000000));
> +    Quat<float> res;
> +    __m128 a = _a.coeffs().packet<Aligned>(0);
> +    __m128 b = _b.coeffs().packet<Aligned>(0);
> +    __m128 flip1 = _mm_xor_ps(_mm_mul_ps(ei_vec4f_swizzle1(a,1,2,0,2),
> +
> ei_vec4f_swizzle1(b,2,0,1,2)),mask);
> +    __m128 flip2 = _mm_xor_ps(_mm_mul_ps(ei_vec4f_swizzle1(a,3,3,3,1),
> +
> ei_vec4f_swizzle1(b,0,1,2,1)),mask);
> +    ei_pstore(&res.x(),
> +
>  _mm_add_ps(_mm_sub_ps(_mm_mul_ps(a,ei_vec4f_swizzle1(b,3,3,3,3)),
> +
>  _mm_mul_ps(ei_vec4f_swizzle1(a,2,0,1,0),
> +
> ei_vec4f_swizzle1(b,1,2,0,0))),
> +                         _mm_add_ps(flip1,flip2)));
> +    return res;
> +  }
> +};
> +
>  template<typename VectorLhs,typename VectorRhs>
>  struct ei_cross3_impl<EiArch_SSE,VectorLhs,VectorRhs,float,true> {
>   inline static typename ei_plain_matrix_type<VectorLhs>::type
>
>