Re: [eigen] Mapping array of scalars into quaternions

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

ah and one more thing. For vectors, we have VectorBlock instead of Block.

Benoit

2009/10/25 Benoit Jacob <jacob.benoit.1@xxxxxxxxx>:
> ok, sorry it got me so long. i've been on a honeymoon with my new
> computer, getting the new hardware working with linux...
>
> First of all I want to say that your patch is good work already and
> your questions are good too. It just needs a bit more work before it
> can go in (most importantly, it currently doesnt compile). comments
> below.
>
> 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.
>
> ok so here are my comments.
>
> The first thing that i did after applying your patch was:
>
>    make test_geo_quaternion
>
> which is just a basic syntax check as your new classes aren't touched
> by the existing test. I got many errors, starting with:
>
> eigen2/Eigen/src/Geometry/Quaternion.h:529: error: ‘Scalar’ was not
> declared in this scope
>
> Indeed, 'Scalar' isn't known by the QuaternionBase class. Yet this
> class uses 'Scalar', for example in the prototype of the x() method.
> So i'm puzzled, how could the code work for you?
>
> The solution to this problem might interest you, since below you write
> that you aren't sure what ei_traits is for.
>
> The naive solution, to define Scalar in QuaternionBase, would be to
> add this typedef in QuaternionBase:
>
> typedef typename Derived::Scalar Scalar;
>
> however, this can't work, because Derived can't be instantiated before
> QuaternionBase<Derived> is. The problem is that these 2 classes want
> to use each other! The solution that we found to this problem was to
> add an external struct ei_traits<Derived> that's used by both Derived
> and Quaternion<Derived>, but doesn't use them. This struct would
> provide the Scalar typedef. Chech for example what we do in
> Eigen/src/Core/Matrix.h: we define ei_traits<Matrix> containing the
> Scalar typedef. Then check in MatrixBase.h, we use ei_traits<Derived>
> to get the Scalar typedef.
>
> Since you created a ei_traits<Derived> struct already, you just have
> to add the Scalar typedef there:
>
>  typedef _Scalar Scalar;   // at line 635
>
> then in QuaternionBase you do:
>
>  typedef typename ei_traits<Derived>::Scalar Scalar;
>
> Notice that in ei_traits<QuaternionWrapper>, line 714 you do:
>
>  typedef typename Coefficients::Scalar Scalar;
>
> that can be simplified as
>
>  typedef _Scalar Scalar;
>
>
> The second error that I got was that the derived() methods that you
> call in QuaternionBase, aren't defined. It should be just a matter of
> static_cast'ing the 'this' pointer, like in MatrixBase.h.
>
> Then i decided to let you fix your patch before continuing ;)
>
> Here is another nitpick. In QuaternionWrapper you define Coefficients
> and CoeffType. CoeffType:
>  * is unused
>  * has a name too similar to Coefficients
>  * is a bit dangerous because it has an alignment requirement that
> isn't guaranteed at all.
> So please remove it :)
>
> Finally let me mention what the supreme refinement would be. As you
> can see from the other thread on this list, Gael just improved Map so
> it is now possible to specify that the pointer is 16-byte aligned.
> This allows vectorization. You could add a templated parameter to
> QuaternionWrapper to allow specifying that, and define typedefs
> QuaternionMapAlignedf, etc.
>
> Now if you do this, one should let QuaternionBase know that the
> pointer is aligned. in the ei_traits, you'd add an enum that says
> whether the pointer is aligned. When it's aligned,
> QuaternionBase::operator* could use the vectorized path!
>
>> 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.
>
> That would be awesome. Start from test/geo_quaternion.cpp, to cover
> QuaternionWrapper, or perhaps copy it (hg cp ...) into a new test
> geo_quaternionwrapper.cpp. Then you add it in test/CMakeLists.txt.
>
> Thanks for your effort.
> Benoit
>
>>
>> --
>> 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
>>
>>
>