cart-elc

Transform.h (61930B)

---

 // This file is part of Eigen, a lightweight C++ template library
 // for linear algebra.
 //
 // Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
 // Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
 // Copyright (C) 2010 Hauke Heibel <hauke.heibel@gmail.com>
 //
 // 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_TRANSFORM_H
 #define EIGEN_TRANSFORM_H
 
 namespace Eigen {
 
 namespace internal {
 
 template<typename Transform>
 struct transform_traits
 {
   enum
   {
     Dim = Transform::Dim,
     HDim = Transform::HDim,
     Mode = Transform::Mode,
     IsProjective = (int(Mode)==int(Projective))
   };
 };
 
 template< typename TransformType,
           typename MatrixType,
           int Case = transform_traits<TransformType>::IsProjective ? 0
                    : int(MatrixType::RowsAtCompileTime) == int(transform_traits<TransformType>::HDim) ? 1
                    : 2,
           int RhsCols = MatrixType::ColsAtCompileTime>
 struct transform_right_product_impl;
 
 template< typename Other,
           int Mode,
           int Options,
           int Dim,
           int HDim,
           int OtherRows=Other::RowsAtCompileTime,
           int OtherCols=Other::ColsAtCompileTime>
 struct transform_left_product_impl;
 
 template< typename Lhs,
           typename Rhs,
           bool AnyProjective =
             transform_traits<Lhs>::IsProjective ||
             transform_traits<Rhs>::IsProjective>
 struct transform_transform_product_impl;
 
 template< typename Other,
           int Mode,
           int Options,
           int Dim,
           int HDim,
           int OtherRows=Other::RowsAtCompileTime,
           int OtherCols=Other::ColsAtCompileTime>
 struct transform_construct_from_matrix;
 
 template<typename TransformType> struct transform_take_affine_part;
 
 template<typename _Scalar, int _Dim, int _Mode, int _Options>
 struct traits<Transform<_Scalar,_Dim,_Mode,_Options> >
 {
   typedef _Scalar Scalar;
   typedef Eigen::Index StorageIndex;
   typedef Dense StorageKind;
   enum {
     Dim1 = _Dim==Dynamic ? _Dim : _Dim + 1,
     RowsAtCompileTime = _Mode==Projective ? Dim1 : _Dim,
     ColsAtCompileTime = Dim1,
     MaxRowsAtCompileTime = RowsAtCompileTime,
     MaxColsAtCompileTime = ColsAtCompileTime,
     Flags = 0
   };
 };
 
 template<int Mode> struct transform_make_affine;
 
 } // end namespace internal
 
 /** \geometry_module \ingroup Geometry_Module
   *
   * \class Transform
   *
   * \brief Represents an homogeneous transformation in a N dimensional space
   *
   * \tparam _Scalar the scalar type, i.e., the type of the coefficients
   * \tparam _Dim the dimension of the space
   * \tparam _Mode the type of the transformation. Can be:
   *              - #Affine: the transformation is stored as a (Dim+1)^2 matrix,
   *                         where the last row is assumed to be [0 ... 0 1].
   *              - #AffineCompact: the transformation is stored as a (Dim)x(Dim+1) matrix.
   *              - #Projective: the transformation is stored as a (Dim+1)^2 matrix
   *                             without any assumption.
   *              - #Isometry: same as #Affine with the additional assumption that
   *                           the linear part represents a rotation. This assumption is exploited
   *                           to speed up some functions such as inverse() and rotation().
   * \tparam _Options has the same meaning as in class Matrix. It allows to specify DontAlign and/or RowMajor.
   *                  These Options are passed directly to the underlying matrix type.
   *
   * The homography is internally represented and stored by a matrix which
   * is available through the matrix() method. To understand the behavior of
   * this class you have to think a Transform object as its internal
   * matrix representation. The chosen convention is right multiply:
   *
   * \code v' = T * v \endcode
   *
   * Therefore, an affine transformation matrix M is shaped like this:
   *
   * \f$ \left( \begin{array}{cc}
   * linear & translation\\
   * 0 ... 0 & 1
   * \end{array} \right) \f$
   *
   * Note that for a projective transformation the last row can be anything,
   * and then the interpretation of different parts might be slightly different.
   *
   * However, unlike a plain matrix, the Transform class provides many features
   * simplifying both its assembly and usage. In particular, it can be composed
   * with any other transformations (Transform,Translation,RotationBase,DiagonalMatrix)
   * and can be directly used to transform implicit homogeneous vectors. All these
   * operations are handled via the operator*. For the composition of transformations,
   * its principle consists to first convert the right/left hand sides of the product
   * to a compatible (Dim+1)^2 matrix and then perform a pure matrix product.
   * Of course, internally, operator* tries to perform the minimal number of operations
   * according to the nature of each terms. Likewise, when applying the transform
   * to points, the latters are automatically promoted to homogeneous vectors
   * before doing the matrix product. The conventions to homogeneous representations
   * are performed as follow:
   *
   * \b Translation t (Dim)x(1):
   * \f$ \left( \begin{array}{cc}
   * I & t \\
   * 0\,...\,0 & 1
   * \end{array} \right) \f$
   *
   * \b Rotation R (Dim)x(Dim):
   * \f$ \left( \begin{array}{cc}
   * R & 0\\
   * 0\,...\,0 & 1
   * \end{array} \right) \f$
   *<!--
   * \b Linear \b Matrix L (Dim)x(Dim):
   * \f$ \left( \begin{array}{cc}
   * L & 0\\
   * 0\,...\,0 & 1
   * \end{array} \right) \f$
   *
   * \b Affine \b Matrix A (Dim)x(Dim+1):
   * \f$ \left( \begin{array}{c}
   * A\\
   * 0\,...\,0\,1
   * \end{array} \right) \f$
   *-->
   * \b Scaling \b DiagonalMatrix S (Dim)x(Dim):
   * \f$ \left( \begin{array}{cc}
   * S & 0\\
   * 0\,...\,0 & 1
   * \end{array} \right) \f$
   *
   * \b Column \b point v (Dim)x(1):
   * \f$ \left( \begin{array}{c}
   * v\\
   * 1
   * \end{array} \right) \f$
   *
   * \b Set \b of \b column \b points V1...Vn (Dim)x(n):
   * \f$ \left( \begin{array}{ccc}
   * v_1 & ... & v_n\\
   * 1 & ... & 1
   * \end{array} \right) \f$
   *
   * The concatenation of a Transform object with any kind of other transformation
   * always returns a Transform object.
   *
   * A little exception to the "as pure matrix product" rule is the case of the
   * transformation of non homogeneous vectors by an affine transformation. In
   * that case the last matrix row can be ignored, and the product returns non
   * homogeneous vectors.
   *
   * Since, for instance, a Dim x Dim matrix is interpreted as a linear transformation,
   * it is not possible to directly transform Dim vectors stored in a Dim x Dim matrix.
   * The solution is either to use a Dim x Dynamic matrix or explicitly request a
   * vector transformation by making the vector homogeneous:
   * \code
   * m' = T * m.colwise().homogeneous();
   * \endcode
   * Note that there is zero overhead.
   *
   * Conversion methods from/to Qt's QMatrix and QTransform are available if the
   * preprocessor token EIGEN_QT_SUPPORT is defined.
   *
   * This class can be extended with the help of the plugin mechanism described on the page
   * \ref TopicCustomizing_Plugins by defining the preprocessor symbol \c EIGEN_TRANSFORM_PLUGIN.
   *
   * \sa class Matrix, class Quaternion
   */
 template<typename _Scalar, int _Dim, int _Mode, int _Options>
 class Transform
 {
 public:
   EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(_Scalar,_Dim==Dynamic ? Dynamic : (_Dim+1)*(_Dim+1))
   enum {
     Mode = _Mode,
     Options = _Options,
     Dim = _Dim,     ///< space dimension in which the transformation holds
     HDim = _Dim+1,  ///< size of a respective homogeneous vector
     Rows = int(Mode)==(AffineCompact) ? Dim : HDim
   };
   /** the scalar type of the coefficients */
   typedef _Scalar Scalar;
   typedef Eigen::Index StorageIndex;
   typedef Eigen::Index Index; ///< \deprecated since Eigen 3.3
   /** type of the matrix used to represent the transformation */
   typedef typename internal::make_proper_matrix_type<Scalar,Rows,HDim,Options>::type MatrixType;
   /** constified MatrixType */
   typedef const MatrixType ConstMatrixType;
   /** type of the matrix used to represent the linear part of the transformation */
   typedef Matrix<Scalar,Dim,Dim,Options> LinearMatrixType;
   /** type of read/write reference to the linear part of the transformation */
   typedef Block<MatrixType,Dim,Dim,int(Mode)==(AffineCompact) && (int(Options)&RowMajor)==0> LinearPart;
   /** type of read reference to the linear part of the transformation */
   typedef const Block<ConstMatrixType,Dim,Dim,int(Mode)==(AffineCompact) && (int(Options)&RowMajor)==0> ConstLinearPart;
   /** type of read/write reference to the affine part of the transformation */
   typedef typename internal::conditional<int(Mode)==int(AffineCompact),
                               MatrixType&,
                               Block<MatrixType,Dim,HDim> >::type AffinePart;
   /** type of read reference to the affine part of the transformation */
   typedef typename internal::conditional<int(Mode)==int(AffineCompact),
                               const MatrixType&,
                               const Block<const MatrixType,Dim,HDim> >::type ConstAffinePart;
   /** type of a vector */
   typedef Matrix<Scalar,Dim,1> VectorType;
   /** type of a read/write reference to the translation part of the rotation */
   typedef Block<MatrixType,Dim,1,!(internal::traits<MatrixType>::Flags & RowMajorBit)> TranslationPart;
   /** type of a read reference to the translation part of the rotation */
   typedef const Block<ConstMatrixType,Dim,1,!(internal::traits<MatrixType>::Flags & RowMajorBit)> ConstTranslationPart;
   /** corresponding translation type */
   typedef Translation<Scalar,Dim> TranslationType;
 
   // this intermediate enum is needed to avoid an ICE with gcc 3.4 and 4.0
   enum { TransformTimeDiagonalMode = ((Mode==int(Isometry))?Affine:int(Mode)) };
   /** The return type of the product between a diagonal matrix and a transform */
   typedef Transform<Scalar,Dim,TransformTimeDiagonalMode> TransformTimeDiagonalReturnType;
 
 protected:
 
   MatrixType m_matrix;
 
 public:
 
   /** Default constructor without initialization of the meaningful coefficients.
     * If Mode==Affine or Mode==Isometry, then the last row is set to [0 ... 0 1] */
   EIGEN_DEVICE_FUNC inline Transform()
   {
     check_template_params();
     internal::transform_make_affine<(int(Mode)==Affine || int(Mode)==Isometry) ? Affine : AffineCompact>::run(m_matrix);
   }
 
   EIGEN_DEVICE_FUNC inline explicit Transform(const TranslationType& t)
   {
     check_template_params();
     *this = t;
   }
   EIGEN_DEVICE_FUNC inline explicit Transform(const UniformScaling<Scalar>& s)
   {
     check_template_params();
     *this = s;
   }
   template<typename Derived>
   EIGEN_DEVICE_FUNC inline explicit Transform(const RotationBase<Derived, Dim>& r)
   {
     check_template_params();
     *this = r;
   }
 
   typedef internal::transform_take_affine_part<Transform> take_affine_part;
 
   /** Constructs and initializes a transformation from a Dim^2 or a (Dim+1)^2 matrix. */
   template<typename OtherDerived>
   EIGEN_DEVICE_FUNC inline explicit Transform(const EigenBase<OtherDerived>& other)
   {
     EIGEN_STATIC_ASSERT((internal::is_same<Scalar,typename OtherDerived::Scalar>::value),
       YOU_MIXED_DIFFERENT_NUMERIC_TYPES__YOU_NEED_TO_USE_THE_CAST_METHOD_OF_MATRIXBASE_TO_CAST_NUMERIC_TYPES_EXPLICITLY);
 
     check_template_params();
     internal::transform_construct_from_matrix<OtherDerived,Mode,Options,Dim,HDim>::run(this, other.derived());
   }
 
   /** Set \c *this from a Dim^2 or (Dim+1)^2 matrix. */
   template<typename OtherDerived>
   EIGEN_DEVICE_FUNC inline Transform& operator=(const EigenBase<OtherDerived>& other)
   {
     EIGEN_STATIC_ASSERT((internal::is_same<Scalar,typename OtherDerived::Scalar>::value),
       YOU_MIXED_DIFFERENT_NUMERIC_TYPES__YOU_NEED_TO_USE_THE_CAST_METHOD_OF_MATRIXBASE_TO_CAST_NUMERIC_TYPES_EXPLICITLY);
 
     internal::transform_construct_from_matrix<OtherDerived,Mode,Options,Dim,HDim>::run(this, other.derived());
     return *this;
   }
 
   template<int OtherOptions>
   EIGEN_DEVICE_FUNC inline Transform(const Transform<Scalar,Dim,Mode,OtherOptions>& other)
   {
     check_template_params();
     // only the options change, we can directly copy the matrices
     m_matrix = other.matrix();
   }
 
   template<int OtherMode,int OtherOptions>
   EIGEN_DEVICE_FUNC inline Transform(const Transform<Scalar,Dim,OtherMode,OtherOptions>& other)
   {
     check_template_params();
     // prevent conversions as:
     // Affine | AffineCompact | Isometry = Projective
     EIGEN_STATIC_ASSERT(EIGEN_IMPLIES(OtherMode==int(Projective), Mode==int(Projective)),
                         YOU_PERFORMED_AN_INVALID_TRANSFORMATION_CONVERSION)
 
     // prevent conversions as:
     // Isometry = Affine | AffineCompact
     EIGEN_STATIC_ASSERT(EIGEN_IMPLIES(OtherMode==int(Affine)||OtherMode==int(AffineCompact), Mode!=int(Isometry)),
                         YOU_PERFORMED_AN_INVALID_TRANSFORMATION_CONVERSION)
 
     enum { ModeIsAffineCompact = Mode == int(AffineCompact),
            OtherModeIsAffineCompact = OtherMode == int(AffineCompact)
     };
 
     if(EIGEN_CONST_CONDITIONAL(ModeIsAffineCompact == OtherModeIsAffineCompact))
     {
       // We need the block expression because the code is compiled for all
       // combinations of transformations and will trigger a compile time error
       // if one tries to assign the matrices directly
       m_matrix.template block<Dim,Dim+1>(0,0) = other.matrix().template block<Dim,Dim+1>(0,0);
       makeAffine();
     }
     else if(EIGEN_CONST_CONDITIONAL(OtherModeIsAffineCompact))
     {
       typedef typename Transform<Scalar,Dim,OtherMode,OtherOptions>::MatrixType OtherMatrixType;
       internal::transform_construct_from_matrix<OtherMatrixType,Mode,Options,Dim,HDim>::run(this, other.matrix());
     }
     else
     {
       // here we know that Mode == AffineCompact and OtherMode != AffineCompact.
       // if OtherMode were Projective, the static assert above would already have caught it.
       // So the only possibility is that OtherMode == Affine
       linear() = other.linear();
       translation() = other.translation();
     }
   }
 
   template<typename OtherDerived>
   EIGEN_DEVICE_FUNC Transform(const ReturnByValue<OtherDerived>& other)
   {
     check_template_params();
     other.evalTo(*this);
   }
 
   template<typename OtherDerived>
   EIGEN_DEVICE_FUNC Transform& operator=(const ReturnByValue<OtherDerived>& other)
   {
     other.evalTo(*this);
     return *this;
   }
 
   #ifdef EIGEN_QT_SUPPORT
   inline Transform(const QMatrix& other);
   inline Transform& operator=(const QMatrix& other);
   inline QMatrix toQMatrix(void) const;
   inline Transform(const QTransform& other);
   inline Transform& operator=(const QTransform& other);
   inline QTransform toQTransform(void) const;
   #endif
 
   EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index rows() const EIGEN_NOEXCEPT { return int(Mode)==int(Projective) ? m_matrix.cols() : (m_matrix.cols()-1); }
   EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index cols() const EIGEN_NOEXCEPT { return m_matrix.cols(); }
 
   /** shortcut for m_matrix(row,col);
     * \sa MatrixBase::operator(Index,Index) const */
   EIGEN_DEVICE_FUNC inline Scalar operator() (Index row, Index col) const { return m_matrix(row,col); }
   /** shortcut for m_matrix(row,col);
     * \sa MatrixBase::operator(Index,Index) */
   EIGEN_DEVICE_FUNC inline Scalar& operator() (Index row, Index col) { return m_matrix(row,col); }
 
   /** \returns a read-only expression of the transformation matrix */
   EIGEN_DEVICE_FUNC inline const MatrixType& matrix() const { return m_matrix; }
   /** \returns a writable expression of the transformation matrix */
   EIGEN_DEVICE_FUNC inline MatrixType& matrix() { return m_matrix; }
 
   /** \returns a read-only expression of the linear part of the transformation */
   EIGEN_DEVICE_FUNC inline ConstLinearPart linear() const { return ConstLinearPart(m_matrix,0,0); }
   /** \returns a writable expression of the linear part of the transformation */
   EIGEN_DEVICE_FUNC inline LinearPart linear() { return LinearPart(m_matrix,0,0); }
 
   /** \returns a read-only expression of the Dim x HDim affine part of the transformation */
   EIGEN_DEVICE_FUNC inline ConstAffinePart affine() const { return take_affine_part::run(m_matrix); }
   /** \returns a writable expression of the Dim x HDim affine part of the transformation */
   EIGEN_DEVICE_FUNC inline AffinePart affine() { return take_affine_part::run(m_matrix); }
 
   /** \returns a read-only expression of the translation vector of the transformation */
   EIGEN_DEVICE_FUNC inline ConstTranslationPart translation() const { return ConstTranslationPart(m_matrix,0,Dim); }
   /** \returns a writable expression of the translation vector of the transformation */
   EIGEN_DEVICE_FUNC inline TranslationPart translation() { return TranslationPart(m_matrix,0,Dim); }
 
   /** \returns an expression of the product between the transform \c *this and a matrix expression \a other.
     *
     * The right-hand-side \a other can be either:
     * \li an homogeneous vector of size Dim+1,
     * \li a set of homogeneous vectors of size Dim+1 x N,
     * \li a transformation matrix of size Dim+1 x Dim+1.
     *
     * Moreover, if \c *this represents an affine transformation (i.e., Mode!=Projective), then \a other can also be:
     * \li a point of size Dim (computes: \code this->linear() * other + this->translation()\endcode),
     * \li a set of N points as a Dim x N matrix (computes: \code (this->linear() * other).colwise() + this->translation()\endcode),
     *
     * In all cases, the return type is a matrix or vector of same sizes as the right-hand-side \a other.
     *
     * If you want to interpret \a other as a linear or affine transformation, then first convert it to a Transform<> type,
     * or do your own cooking.
     *
     * Finally, if you want to apply Affine transformations to vectors, then explicitly apply the linear part only:
     * \code
     * Affine3f A;
     * Vector3f v1, v2;
     * v2 = A.linear() * v1;
     * \endcode
     *
     */
   // note: this function is defined here because some compilers cannot find the respective declaration
   template<typename OtherDerived>
   EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const typename internal::transform_right_product_impl<Transform, OtherDerived>::ResultType
   operator * (const EigenBase<OtherDerived> &other) const
   { return internal::transform_right_product_impl<Transform, OtherDerived>::run(*this,other.derived()); }
 
   /** \returns the product expression of a transformation matrix \a a times a transform \a b
     *
     * The left hand side \a other can be either:
     * \li a linear transformation matrix of size Dim x Dim,
     * \li an affine transformation matrix of size Dim x Dim+1,
     * \li a general transformation matrix of size Dim+1 x Dim+1.
     */
   template<typename OtherDerived> friend
   EIGEN_DEVICE_FUNC inline const typename internal::transform_left_product_impl<OtherDerived,Mode,Options,_Dim,_Dim+1>::ResultType
     operator * (const EigenBase<OtherDerived> &a, const Transform &b)
   { return internal::transform_left_product_impl<OtherDerived,Mode,Options,Dim,HDim>::run(a.derived(),b); }
 
   /** \returns The product expression of a transform \a a times a diagonal matrix \a b
     *
     * The rhs diagonal matrix is interpreted as an affine scaling transformation. The
     * product results in a Transform of the same type (mode) as the lhs only if the lhs
     * mode is no isometry. In that case, the returned transform is an affinity.
     */
   template<typename DiagonalDerived>
   EIGEN_DEVICE_FUNC inline const TransformTimeDiagonalReturnType
     operator * (const DiagonalBase<DiagonalDerived> &b) const
   {
     TransformTimeDiagonalReturnType res(*this);
     res.linearExt() *= b;
     return res;
   }
 
   /** \returns The product expression of a diagonal matrix \a a times a transform \a b
     *
     * The lhs diagonal matrix is interpreted as an affine scaling transformation. The
     * product results in a Transform of the same type (mode) as the lhs only if the lhs
     * mode is no isometry. In that case, the returned transform is an affinity.
     */
   template<typename DiagonalDerived>
   EIGEN_DEVICE_FUNC friend inline TransformTimeDiagonalReturnType
     operator * (const DiagonalBase<DiagonalDerived> &a, const Transform &b)
   {
     TransformTimeDiagonalReturnType res;
     res.linear().noalias() = a*b.linear();
     res.translation().noalias() = a*b.translation();
     if (EIGEN_CONST_CONDITIONAL(Mode!=int(AffineCompact)))
       res.matrix().row(Dim) = b.matrix().row(Dim);
     return res;
   }
 
   template<typename OtherDerived>
   EIGEN_DEVICE_FUNC inline Transform& operator*=(const EigenBase<OtherDerived>& other) { return *this = *this * other; }
 
   /** Concatenates two transformations */
   EIGEN_DEVICE_FUNC inline const Transform operator * (const Transform& other) const
   {
     return internal::transform_transform_product_impl<Transform,Transform>::run(*this,other);
   }
 
   #if EIGEN_COMP_ICC
 private:
   // this intermediate structure permits to workaround a bug in ICC 11:
   //   error: template instantiation resulted in unexpected function type of "Eigen::Transform<double, 3, 32, 0>
   //             (const Eigen::Transform<double, 3, 2, 0> &) const"
   //  (the meaning of a name may have changed since the template declaration -- the type of the template is:
   // "Eigen::internal::transform_transform_product_impl<Eigen::Transform<double, 3, 32, 0>,
   //     Eigen::Transform<double, 3, Mode, Options>, <expression>>::ResultType (const Eigen::Transform<double, 3, Mode, Options> &) const")
   //
   template<int OtherMode,int OtherOptions> struct icc_11_workaround
   {
     typedef internal::transform_transform_product_impl<Transform,Transform<Scalar,Dim,OtherMode,OtherOptions> > ProductType;
     typedef typename ProductType::ResultType ResultType;
   };
 
 public:
   /** Concatenates two different transformations */
   template<int OtherMode,int OtherOptions>
   inline typename icc_11_workaround<OtherMode,OtherOptions>::ResultType
     operator * (const Transform<Scalar,Dim,OtherMode,OtherOptions>& other) const
   {
     typedef typename icc_11_workaround<OtherMode,OtherOptions>::ProductType ProductType;
     return ProductType::run(*this,other);
   }
   #else
   /** Concatenates two different transformations */
   template<int OtherMode,int OtherOptions>
   EIGEN_DEVICE_FUNC inline typename internal::transform_transform_product_impl<Transform,Transform<Scalar,Dim,OtherMode,OtherOptions> >::ResultType
     operator * (const Transform<Scalar,Dim,OtherMode,OtherOptions>& other) const
   {
     return internal::transform_transform_product_impl<Transform,Transform<Scalar,Dim,OtherMode,OtherOptions> >::run(*this,other);
   }
   #endif
 
   /** \sa MatrixBase::setIdentity() */
   EIGEN_DEVICE_FUNC void setIdentity() { m_matrix.setIdentity(); }
 
   /**
    * \brief Returns an identity transformation.
    * \todo In the future this function should be returning a Transform expression.
    */
   EIGEN_DEVICE_FUNC static const Transform Identity()
   {
     return Transform(MatrixType::Identity());
   }
 
   template<typename OtherDerived>
   EIGEN_DEVICE_FUNC
   inline Transform& scale(const MatrixBase<OtherDerived> &other);
 
   template<typename OtherDerived>
   EIGEN_DEVICE_FUNC
   inline Transform& prescale(const MatrixBase<OtherDerived> &other);
 
   EIGEN_DEVICE_FUNC inline Transform& scale(const Scalar& s);
   EIGEN_DEVICE_FUNC inline Transform& prescale(const Scalar& s);
 
   template<typename OtherDerived>
   EIGEN_DEVICE_FUNC
   inline Transform& translate(const MatrixBase<OtherDerived> &other);
 
   template<typename OtherDerived>
   EIGEN_DEVICE_FUNC
   inline Transform& pretranslate(const MatrixBase<OtherDerived> &other);
 
   template<typename RotationType>
   EIGEN_DEVICE_FUNC
   inline Transform& rotate(const RotationType& rotation);
 
   template<typename RotationType>
   EIGEN_DEVICE_FUNC
   inline Transform& prerotate(const RotationType& rotation);
 
   EIGEN_DEVICE_FUNC Transform& shear(const Scalar& sx, const Scalar& sy);
   EIGEN_DEVICE_FUNC Transform& preshear(const Scalar& sx, const Scalar& sy);
 
   EIGEN_DEVICE_FUNC inline Transform& operator=(const TranslationType& t);
 
   EIGEN_DEVICE_FUNC
   inline Transform& operator*=(const TranslationType& t) { return translate(t.vector()); }
 
   EIGEN_DEVICE_FUNC inline Transform operator*(const TranslationType& t) const;
 
   EIGEN_DEVICE_FUNC
   inline Transform& operator=(const UniformScaling<Scalar>& t);
 
   EIGEN_DEVICE_FUNC
   inline Transform& operator*=(const UniformScaling<Scalar>& s) { return scale(s.factor()); }
 
   EIGEN_DEVICE_FUNC
   inline TransformTimeDiagonalReturnType operator*(const UniformScaling<Scalar>& s) const
   {
     TransformTimeDiagonalReturnType res = *this;
     res.scale(s.factor());
     return res;
   }
 
   EIGEN_DEVICE_FUNC
   inline Transform& operator*=(const DiagonalMatrix<Scalar,Dim>& s) { linearExt() *= s; return *this; }
 
   template<typename Derived>
   EIGEN_DEVICE_FUNC inline Transform& operator=(const RotationBase<Derived,Dim>& r);
   template<typename Derived>
   EIGEN_DEVICE_FUNC inline Transform& operator*=(const RotationBase<Derived,Dim>& r) { return rotate(r.toRotationMatrix()); }
   template<typename Derived>
   EIGEN_DEVICE_FUNC inline Transform operator*(const RotationBase<Derived,Dim>& r) const;
 
   typedef typename internal::conditional<int(Mode)==Isometry,ConstLinearPart,const LinearMatrixType>::type RotationReturnType;
   EIGEN_DEVICE_FUNC RotationReturnType rotation() const;
 
   template<typename RotationMatrixType, typename ScalingMatrixType>
   EIGEN_DEVICE_FUNC
   void computeRotationScaling(RotationMatrixType *rotation, ScalingMatrixType *scaling) const;
   template<typename ScalingMatrixType, typename RotationMatrixType>
   EIGEN_DEVICE_FUNC
   void computeScalingRotation(ScalingMatrixType *scaling, RotationMatrixType *rotation) const;
 
   template<typename PositionDerived, typename OrientationType, typename ScaleDerived>
   EIGEN_DEVICE_FUNC
   Transform& fromPositionOrientationScale(const MatrixBase<PositionDerived> &position,
     const OrientationType& orientation, const MatrixBase<ScaleDerived> &scale);
 
   EIGEN_DEVICE_FUNC
   inline Transform inverse(TransformTraits traits = (TransformTraits)Mode) const;
 
   /** \returns a const pointer to the column major internal matrix */
   EIGEN_DEVICE_FUNC const Scalar* data() const { return m_matrix.data(); }
   /** \returns a non-const pointer to the column major internal matrix */
   EIGEN_DEVICE_FUNC Scalar* data() { return m_matrix.data(); }
 
   /** \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>
   EIGEN_DEVICE_FUNC inline typename internal::cast_return_type<Transform,Transform<NewScalarType,Dim,Mode,Options> >::type cast() const
   { return typename internal::cast_return_type<Transform,Transform<NewScalarType,Dim,Mode,Options> >::type(*this); }
 
   /** Copy constructor with scalar type conversion */
   template<typename OtherScalarType>
   EIGEN_DEVICE_FUNC inline explicit Transform(const Transform<OtherScalarType,Dim,Mode,Options>& other)
   {
     check_template_params();
     m_matrix = other.matrix().template cast<Scalar>();
   }
 
   /** \returns \c true if \c *this is approximately equal to \a other, within the precision
     * determined by \a prec.
     *
     * \sa MatrixBase::isApprox() */
   EIGEN_DEVICE_FUNC bool isApprox(const Transform& other, const typename NumTraits<Scalar>::Real& prec = NumTraits<Scalar>::dummy_precision()) const
   { return m_matrix.isApprox(other.m_matrix, prec); }
 
   /** Sets the last row to [0 ... 0 1]
     */
   EIGEN_DEVICE_FUNC void makeAffine()
   {
     internal::transform_make_affine<int(Mode)>::run(m_matrix);
   }
 
   /** \internal
     * \returns the Dim x Dim linear part if the transformation is affine,
     *          and the HDim x Dim part for projective transformations.
     */
   EIGEN_DEVICE_FUNC inline Block<MatrixType,int(Mode)==int(Projective)?HDim:Dim,Dim> linearExt()
   { return m_matrix.template block<int(Mode)==int(Projective)?HDim:Dim,Dim>(0,0); }
   /** \internal
     * \returns the Dim x Dim linear part if the transformation is affine,
     *          and the HDim x Dim part for projective transformations.
     */
   EIGEN_DEVICE_FUNC inline const Block<MatrixType,int(Mode)==int(Projective)?HDim:Dim,Dim> linearExt() const
   { return m_matrix.template block<int(Mode)==int(Projective)?HDim:Dim,Dim>(0,0); }
 
   /** \internal
     * \returns the translation part if the transformation is affine,
     *          and the last column for projective transformations.
     */
   EIGEN_DEVICE_FUNC inline Block<MatrixType,int(Mode)==int(Projective)?HDim:Dim,1> translationExt()
   { return m_matrix.template block<int(Mode)==int(Projective)?HDim:Dim,1>(0,Dim); }
   /** \internal
     * \returns the translation part if the transformation is affine,
     *          and the last column for projective transformations.
     */
   EIGEN_DEVICE_FUNC inline const Block<MatrixType,int(Mode)==int(Projective)?HDim:Dim,1> translationExt() const
   { return m_matrix.template block<int(Mode)==int(Projective)?HDim:Dim,1>(0,Dim); }
 
 
   #ifdef EIGEN_TRANSFORM_PLUGIN
   #include EIGEN_TRANSFORM_PLUGIN
   #endif
 
 protected:
   #ifndef EIGEN_PARSED_BY_DOXYGEN
     EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE void check_template_params()
     {
       EIGEN_STATIC_ASSERT((Options & (DontAlign|RowMajor)) == Options, INVALID_MATRIX_TEMPLATE_PARAMETERS)
     }
   #endif
 
 };
 
 /** \ingroup Geometry_Module */
 typedef Transform<float,2,Isometry> Isometry2f;
 /** \ingroup Geometry_Module */
 typedef Transform<float,3,Isometry> Isometry3f;
 /** \ingroup Geometry_Module */
 typedef Transform<double,2,Isometry> Isometry2d;
 /** \ingroup Geometry_Module */
 typedef Transform<double,3,Isometry> Isometry3d;
 
 /** \ingroup Geometry_Module */
 typedef Transform<float,2,Affine> Affine2f;
 /** \ingroup Geometry_Module */
 typedef Transform<float,3,Affine> Affine3f;
 /** \ingroup Geometry_Module */
 typedef Transform<double,2,Affine> Affine2d;
 /** \ingroup Geometry_Module */
 typedef Transform<double,3,Affine> Affine3d;
 
 /** \ingroup Geometry_Module */
 typedef Transform<float,2,AffineCompact> AffineCompact2f;
 /** \ingroup Geometry_Module */
 typedef Transform<float,3,AffineCompact> AffineCompact3f;
 /** \ingroup Geometry_Module */
 typedef Transform<double,2,AffineCompact> AffineCompact2d;
 /** \ingroup Geometry_Module */
 typedef Transform<double,3,AffineCompact> AffineCompact3d;
 
 /** \ingroup Geometry_Module */
 typedef Transform<float,2,Projective> Projective2f;
 /** \ingroup Geometry_Module */
 typedef Transform<float,3,Projective> Projective3f;
 /** \ingroup Geometry_Module */
 typedef Transform<double,2,Projective> Projective2d;
 /** \ingroup Geometry_Module */
 typedef Transform<double,3,Projective> Projective3d;
 
 /**************************
 *** Optional QT support ***
 **************************/
 
 #ifdef EIGEN_QT_SUPPORT
 /** Initializes \c *this from a QMatrix assuming the dimension is 2.
   *
   * This function is available only if the token EIGEN_QT_SUPPORT is defined.
   */
 template<typename Scalar, int Dim, int Mode,int Options>
 Transform<Scalar,Dim,Mode,Options>::Transform(const QMatrix& other)
 {
   check_template_params();
   *this = other;
 }
 
 /** Set \c *this from a QMatrix assuming the dimension is 2.
   *
   * This function is available only if the token EIGEN_QT_SUPPORT is defined.
   */
 template<typename Scalar, int Dim, int Mode,int Options>
 Transform<Scalar,Dim,Mode,Options>& Transform<Scalar,Dim,Mode,Options>::operator=(const QMatrix& other)
 {
   EIGEN_STATIC_ASSERT(Dim==2, YOU_MADE_A_PROGRAMMING_MISTAKE)
   if (EIGEN_CONST_CONDITIONAL(Mode == int(AffineCompact)))
     m_matrix << other.m11(), other.m21(), other.dx(),
                 other.m12(), other.m22(), other.dy();
   else
     m_matrix << other.m11(), other.m21(), other.dx(),
                 other.m12(), other.m22(), other.dy(),
                 0, 0, 1;
   return *this;
 }
 
 /** \returns a QMatrix from \c *this assuming the dimension is 2.
   *
   * \warning this conversion might loss data if \c *this is not affine
   *
   * This function is available only if the token EIGEN_QT_SUPPORT is defined.
   */
 template<typename Scalar, int Dim, int Mode, int Options>
 QMatrix Transform<Scalar,Dim,Mode,Options>::toQMatrix(void) const
 {
   check_template_params();
   EIGEN_STATIC_ASSERT(Dim==2, YOU_MADE_A_PROGRAMMING_MISTAKE)
   return QMatrix(m_matrix.coeff(0,0), m_matrix.coeff(1,0),
                  m_matrix.coeff(0,1), m_matrix.coeff(1,1),
                  m_matrix.coeff(0,2), m_matrix.coeff(1,2));
 }
 
 /** Initializes \c *this from a QTransform assuming the dimension is 2.
   *
   * This function is available only if the token EIGEN_QT_SUPPORT is defined.
   */
 template<typename Scalar, int Dim, int Mode,int Options>
 Transform<Scalar,Dim,Mode,Options>::Transform(const QTransform& other)
 {
   check_template_params();
   *this = other;
 }
 
 /** Set \c *this from a QTransform assuming the dimension is 2.
   *
   * This function is available only if the token EIGEN_QT_SUPPORT is defined.
   */
 template<typename Scalar, int Dim, int Mode, int Options>
 Transform<Scalar,Dim,Mode,Options>& Transform<Scalar,Dim,Mode,Options>::operator=(const QTransform& other)
 {
   check_template_params();
   EIGEN_STATIC_ASSERT(Dim==2, YOU_MADE_A_PROGRAMMING_MISTAKE)
   if (EIGEN_CONST_CONDITIONAL(Mode == int(AffineCompact)))
     m_matrix << other.m11(), other.m21(), other.dx(),
                 other.m12(), other.m22(), other.dy();
   else
     m_matrix << other.m11(), other.m21(), other.dx(),
                 other.m12(), other.m22(), other.dy(),
                 other.m13(), other.m23(), other.m33();
   return *this;
 }
 
 /** \returns a QTransform from \c *this assuming the dimension is 2.
   *
   * This function is available only if the token EIGEN_QT_SUPPORT is defined.
   */
 template<typename Scalar, int Dim, int Mode, int Options>
 QTransform Transform<Scalar,Dim,Mode,Options>::toQTransform(void) const
 {
   EIGEN_STATIC_ASSERT(Dim==2, YOU_MADE_A_PROGRAMMING_MISTAKE)
   if (EIGEN_CONST_CONDITIONAL(Mode == int(AffineCompact)))
     return QTransform(m_matrix.coeff(0,0), m_matrix.coeff(1,0),
                       m_matrix.coeff(0,1), m_matrix.coeff(1,1),
                       m_matrix.coeff(0,2), m_matrix.coeff(1,2));
   else
     return QTransform(m_matrix.coeff(0,0), m_matrix.coeff(1,0), m_matrix.coeff(2,0),
                       m_matrix.coeff(0,1), m_matrix.coeff(1,1), m_matrix.coeff(2,1),
                       m_matrix.coeff(0,2), m_matrix.coeff(1,2), m_matrix.coeff(2,2));
 }
 #endif
 
 /*********************
 *** Procedural API ***
 *********************/
 
 /** Applies on the right the non uniform scale transformation represented
   * by the vector \a other to \c *this and returns a reference to \c *this.
   * \sa prescale()
   */
 template<typename Scalar, int Dim, int Mode, int Options>
 template<typename OtherDerived>
 EIGEN_DEVICE_FUNC Transform<Scalar,Dim,Mode,Options>&
 Transform<Scalar,Dim,Mode,Options>::scale(const MatrixBase<OtherDerived> &other)
 {
   EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(OtherDerived,int(Dim))
   EIGEN_STATIC_ASSERT(Mode!=int(Isometry), THIS_METHOD_IS_ONLY_FOR_SPECIFIC_TRANSFORMATIONS)
   linearExt().noalias() = (linearExt() * other.asDiagonal());
   return *this;
 }
 
 /** Applies on the right a uniform scale of a factor \a c to \c *this
   * and returns a reference to \c *this.
   * \sa prescale(Scalar)
   */
 template<typename Scalar, int Dim, int Mode, int Options>
 EIGEN_DEVICE_FUNC inline Transform<Scalar,Dim,Mode,Options>& Transform<Scalar,Dim,Mode,Options>::scale(const Scalar& s)
 {
   EIGEN_STATIC_ASSERT(Mode!=int(Isometry), THIS_METHOD_IS_ONLY_FOR_SPECIFIC_TRANSFORMATIONS)
   linearExt() *= s;
   return *this;
 }
 
 /** Applies on the left the non uniform scale transformation represented
   * by the vector \a other to \c *this and returns a reference to \c *this.
   * \sa scale()
   */
 template<typename Scalar, int Dim, int Mode, int Options>
 template<typename OtherDerived>
 EIGEN_DEVICE_FUNC Transform<Scalar,Dim,Mode,Options>&
 Transform<Scalar,Dim,Mode,Options>::prescale(const MatrixBase<OtherDerived> &other)
 {
   EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(OtherDerived,int(Dim))
   EIGEN_STATIC_ASSERT(Mode!=int(Isometry), THIS_METHOD_IS_ONLY_FOR_SPECIFIC_TRANSFORMATIONS)
   affine().noalias() = (other.asDiagonal() * affine());
   return *this;
 }
 
 /** Applies on the left a uniform scale of a factor \a c to \c *this
   * and returns a reference to \c *this.
   * \sa scale(Scalar)
   */
 template<typename Scalar, int Dim, int Mode, int Options>
 EIGEN_DEVICE_FUNC inline Transform<Scalar,Dim,Mode,Options>& Transform<Scalar,Dim,Mode,Options>::prescale(const Scalar& s)
 {
   EIGEN_STATIC_ASSERT(Mode!=int(Isometry), THIS_METHOD_IS_ONLY_FOR_SPECIFIC_TRANSFORMATIONS)
   m_matrix.template topRows<Dim>() *= s;
   return *this;
 }
 
 /** Applies on the right the translation matrix represented by the vector \a other
   * to \c *this and returns a reference to \c *this.
   * \sa pretranslate()
   */
 template<typename Scalar, int Dim, int Mode, int Options>
 template<typename OtherDerived>
 EIGEN_DEVICE_FUNC Transform<Scalar,Dim,Mode,Options>&
 Transform<Scalar,Dim,Mode,Options>::translate(const MatrixBase<OtherDerived> &other)
 {
   EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(OtherDerived,int(Dim))
   translationExt() += linearExt() * other;
   return *this;
 }
 
 /** Applies on the left the translation matrix represented by the vector \a other
   * to \c *this and returns a reference to \c *this.
   * \sa translate()
   */
 template<typename Scalar, int Dim, int Mode, int Options>
 template<typename OtherDerived>
 EIGEN_DEVICE_FUNC Transform<Scalar,Dim,Mode,Options>&
 Transform<Scalar,Dim,Mode,Options>::pretranslate(const MatrixBase<OtherDerived> &other)
 {
   EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(OtherDerived,int(Dim))
   if(EIGEN_CONST_CONDITIONAL(int(Mode)==int(Projective)))
     affine() += other * m_matrix.row(Dim);
   else
     translation() += other;
   return *this;
 }
 
 /** Applies on the right the rotation represented by the rotation \a rotation
   * to \c *this and returns a reference to \c *this.
   *
   * The template parameter \a RotationType is the type of the rotation which
   * must be known by internal::toRotationMatrix<>.
   *
   * Natively supported types includes:
   *   - any scalar (2D),
   *   - a Dim x Dim matrix expression,
   *   - a Quaternion (3D),
   *   - a AngleAxis (3D)
   *
   * This mechanism is easily extendable to support user types such as Euler angles,
   * or a pair of Quaternion for 4D rotations.
   *
   * \sa rotate(Scalar), class Quaternion, class AngleAxis, prerotate(RotationType)
   */
 template<typename Scalar, int Dim, int Mode, int Options>
 template<typename RotationType>
 EIGEN_DEVICE_FUNC Transform<Scalar,Dim,Mode,Options>&
 Transform<Scalar,Dim,Mode,Options>::rotate(const RotationType& rotation)
 {
   linearExt() *= internal::toRotationMatrix<Scalar,Dim>(rotation);
   return *this;
 }
 
 /** Applies on the left the rotation represented by the rotation \a rotation
   * to \c *this and returns a reference to \c *this.
   *
   * See rotate() for further details.
   *
   * \sa rotate()
   */
 template<typename Scalar, int Dim, int Mode, int Options>
 template<typename RotationType>
 EIGEN_DEVICE_FUNC Transform<Scalar,Dim,Mode,Options>&
 Transform<Scalar,Dim,Mode,Options>::prerotate(const RotationType& rotation)
 {
   m_matrix.template block<Dim,HDim>(0,0) = internal::toRotationMatrix<Scalar,Dim>(rotation)
                                          * m_matrix.template block<Dim,HDim>(0,0);
   return *this;
 }
 
 /** Applies on the right the shear transformation represented
   * by the vector \a other to \c *this and returns a reference to \c *this.
   * \warning 2D only.
   * \sa preshear()
   */
 template<typename Scalar, int Dim, int Mode, int Options>
 EIGEN_DEVICE_FUNC Transform<Scalar,Dim,Mode,Options>&
 Transform<Scalar,Dim,Mode,Options>::shear(const Scalar& sx, const Scalar& sy)
 {
   EIGEN_STATIC_ASSERT(int(Dim)==2, YOU_MADE_A_PROGRAMMING_MISTAKE)
   EIGEN_STATIC_ASSERT(Mode!=int(Isometry), THIS_METHOD_IS_ONLY_FOR_SPECIFIC_TRANSFORMATIONS)
   VectorType tmp = linear().col(0)*sy + linear().col(1);
   linear() << linear().col(0) + linear().col(1)*sx, tmp;
   return *this;
 }
 
 /** Applies on the left the shear transformation represented
   * by the vector \a other to \c *this and returns a reference to \c *this.
   * \warning 2D only.
   * \sa shear()
   */
 template<typename Scalar, int Dim, int Mode, int Options>
 EIGEN_DEVICE_FUNC Transform<Scalar,Dim,Mode,Options>&
 Transform<Scalar,Dim,Mode,Options>::preshear(const Scalar& sx, const Scalar& sy)
 {
   EIGEN_STATIC_ASSERT(int(Dim)==2, YOU_MADE_A_PROGRAMMING_MISTAKE)
   EIGEN_STATIC_ASSERT(Mode!=int(Isometry), THIS_METHOD_IS_ONLY_FOR_SPECIFIC_TRANSFORMATIONS)
   m_matrix.template block<Dim,HDim>(0,0) = LinearMatrixType(1, sx, sy, 1) * m_matrix.template block<Dim,HDim>(0,0);
   return *this;
 }
 
 /******************************************************
 *** Scaling, Translation and Rotation compatibility ***
 ******************************************************/
 
 template<typename Scalar, int Dim, int Mode, int Options>
 EIGEN_DEVICE_FUNC inline Transform<Scalar,Dim,Mode,Options>& Transform<Scalar,Dim,Mode,Options>::operator=(const TranslationType& t)
 {
   linear().setIdentity();
   translation() = t.vector();
   makeAffine();
   return *this;
 }
 
 template<typename Scalar, int Dim, int Mode, int Options>
 EIGEN_DEVICE_FUNC inline Transform<Scalar,Dim,Mode,Options> Transform<Scalar,Dim,Mode,Options>::operator*(const TranslationType& t) const
 {
   Transform res = *this;
   res.translate(t.vector());
   return res;
 }
 
 template<typename Scalar, int Dim, int Mode, int Options>
 EIGEN_DEVICE_FUNC inline Transform<Scalar,Dim,Mode,Options>& Transform<Scalar,Dim,Mode,Options>::operator=(const UniformScaling<Scalar>& s)
 {
   m_matrix.setZero();
   linear().diagonal().fill(s.factor());
   makeAffine();
   return *this;
 }
 
 template<typename Scalar, int Dim, int Mode, int Options>
 template<typename Derived>
 EIGEN_DEVICE_FUNC inline Transform<Scalar,Dim,Mode,Options>& Transform<Scalar,Dim,Mode,Options>::operator=(const RotationBase<Derived,Dim>& r)
 {
   linear() = internal::toRotationMatrix<Scalar,Dim>(r);
   translation().setZero();
   makeAffine();
   return *this;
 }
 
 template<typename Scalar, int Dim, int Mode, int Options>
 template<typename Derived>
 EIGEN_DEVICE_FUNC inline Transform<Scalar,Dim,Mode,Options> Transform<Scalar,Dim,Mode,Options>::operator*(const RotationBase<Derived,Dim>& r) const
 {
   Transform res = *this;
   res.rotate(r.derived());
   return res;
 }
 
 /************************
 *** Special functions ***
 ************************/
 
 namespace internal {
 template<int Mode> struct transform_rotation_impl {
   template<typename TransformType>
   EIGEN_DEVICE_FUNC static inline
   const typename TransformType::LinearMatrixType run(const TransformType& t)
   {
     typedef typename TransformType::LinearMatrixType LinearMatrixType;
     LinearMatrixType result;
     t.computeRotationScaling(&result, (LinearMatrixType*)0);
     return result;
   }
 };
 template<> struct transform_rotation_impl<Isometry> {
   template<typename TransformType>
   EIGEN_DEVICE_FUNC static inline
   typename TransformType::ConstLinearPart run(const TransformType& t)
   {
     return t.linear();
   }
 };
 }
 /** \returns the rotation part of the transformation
   *
   * If Mode==Isometry, then this method is an alias for linear(),
   * otherwise it calls computeRotationScaling() to extract the rotation
   * through a SVD decomposition.
   *
   * \svd_module
   *
   * \sa computeRotationScaling(), computeScalingRotation(), class SVD
   */
 template<typename Scalar, int Dim, int Mode, int Options>
 EIGEN_DEVICE_FUNC
 typename Transform<Scalar,Dim,Mode,Options>::RotationReturnType
 Transform<Scalar,Dim,Mode,Options>::rotation() const
 {
   return internal::transform_rotation_impl<Mode>::run(*this);
 }
 
 
 /** decomposes the linear part of the transformation as a product rotation x scaling, the scaling being
   * not necessarily positive.
   *
   * If either pointer is zero, the corresponding computation is skipped.
   *
   *
   *
   * \svd_module
   *
   * \sa computeScalingRotation(), rotation(), class SVD
   */
 template<typename Scalar, int Dim, int Mode, int Options>
 template<typename RotationMatrixType, typename ScalingMatrixType>
 EIGEN_DEVICE_FUNC void Transform<Scalar,Dim,Mode,Options>::computeRotationScaling(RotationMatrixType *rotation, ScalingMatrixType *scaling) const
 {
   // Note that JacobiSVD is faster than BDCSVD for small matrices.
   JacobiSVD<LinearMatrixType> svd(linear(), ComputeFullU | ComputeFullV);
 
   Scalar x = (svd.matrixU() * svd.matrixV().adjoint()).determinant() < Scalar(0) ? Scalar(-1) : Scalar(1); // so x has absolute value 1
   VectorType sv(svd.singularValues());
   sv.coeffRef(Dim-1) *= x;
   if(scaling) *scaling = svd.matrixV() * sv.asDiagonal() * svd.matrixV().adjoint();
   if(rotation)
   {
     LinearMatrixType m(svd.matrixU());
     m.col(Dim-1) *= x;
     *rotation = m * svd.matrixV().adjoint();
   }
 }
 
 /** decomposes the linear part of the transformation as a product scaling x rotation, the scaling being
   * not necessarily positive.
   *
   * If either pointer is zero, the corresponding computation is skipped.
   *
   *
   *
   * \svd_module
   *
   * \sa computeRotationScaling(), rotation(), class SVD
   */
 template<typename Scalar, int Dim, int Mode, int Options>
 template<typename ScalingMatrixType, typename RotationMatrixType>
 EIGEN_DEVICE_FUNC void Transform<Scalar,Dim,Mode,Options>::computeScalingRotation(ScalingMatrixType *scaling, RotationMatrixType *rotation) const
 {
   // Note that JacobiSVD is faster than BDCSVD for small matrices.
   JacobiSVD<LinearMatrixType> svd(linear(), ComputeFullU | ComputeFullV);
 
   Scalar x = (svd.matrixU() * svd.matrixV().adjoint()).determinant() < Scalar(0) ? Scalar(-1) : Scalar(1); // so x has absolute value 1
   VectorType sv(svd.singularValues());
   sv.coeffRef(Dim-1) *= x;
   if(scaling) *scaling = svd.matrixU() * sv.asDiagonal() * svd.matrixU().adjoint();
   if(rotation)
   {
     LinearMatrixType m(svd.matrixU());
     m.col(Dim-1) *= x;
     *rotation = m * svd.matrixV().adjoint();
   }
 }
 
 /** Convenient method to set \c *this from a position, orientation and scale
   * of a 3D object.
   */
 template<typename Scalar, int Dim, int Mode, int Options>
 template<typename PositionDerived, typename OrientationType, typename ScaleDerived>
 EIGEN_DEVICE_FUNC Transform<Scalar,Dim,Mode,Options>&
 Transform<Scalar,Dim,Mode,Options>::fromPositionOrientationScale(const MatrixBase<PositionDerived> &position,
   const OrientationType& orientation, const MatrixBase<ScaleDerived> &scale)
 {
   linear() = internal::toRotationMatrix<Scalar,Dim>(orientation);
   linear() *= scale.asDiagonal();
   translation() = position;
   makeAffine();
   return *this;
 }
 
 namespace internal {
 
 template<int Mode>
 struct transform_make_affine
 {
   template<typename MatrixType>
   EIGEN_DEVICE_FUNC static void run(MatrixType &mat)
   {
     static const int Dim = MatrixType::ColsAtCompileTime-1;
     mat.template block<1,Dim>(Dim,0).setZero();
     mat.coeffRef(Dim,Dim) = typename MatrixType::Scalar(1);
   }
 };
 
 template<>
 struct transform_make_affine<AffineCompact>
 {
   template<typename MatrixType> EIGEN_DEVICE_FUNC static void run(MatrixType &) { }
 };
 
 // selector needed to avoid taking the inverse of a 3x4 matrix
 template<typename TransformType, int Mode=TransformType::Mode>
 struct projective_transform_inverse
 {
   EIGEN_DEVICE_FUNC static inline void run(const TransformType&, TransformType&)
   {}
 };
 
 template<typename TransformType>
 struct projective_transform_inverse<TransformType, Projective>
 {
   EIGEN_DEVICE_FUNC static inline void run(const TransformType& m, TransformType& res)
   {
     res.matrix() = m.matrix().inverse();
   }
 };
 
 } // end namespace internal
 
 
 /**
   *
   * \returns the inverse transformation according to some given knowledge
   * on \c *this.
   *
   * \param hint allows to optimize the inversion process when the transformation
   * is known to be not a general transformation (optional). The possible values are:
   *  - #Projective if the transformation is not necessarily affine, i.e., if the
   *    last row is not guaranteed to be [0 ... 0 1]
   *  - #Affine if the last row can be assumed to be [0 ... 0 1]
   *  - #Isometry if the transformation is only a concatenations of translations
   *    and rotations.
   *  The default is the template class parameter \c Mode.
   *
   * \warning unless \a traits is always set to NoShear or NoScaling, this function
   * requires the generic inverse method of MatrixBase defined in the LU module. If
   * you forget to include this module, then you will get hard to debug linking errors.
   *
   * \sa MatrixBase::inverse()
   */
 template<typename Scalar, int Dim, int Mode, int Options>
 EIGEN_DEVICE_FUNC Transform<Scalar,Dim,Mode,Options>
 Transform<Scalar,Dim,Mode,Options>::inverse(TransformTraits hint) const
 {
   Transform res;
   if (hint == Projective)
   {
     internal::projective_transform_inverse<Transform>::run(*this, res);
   }
   else
   {
     if (hint == Isometry)
     {
       res.matrix().template topLeftCorner<Dim,Dim>() = linear().transpose();
     }
     else if(hint&Affine)
     {
       res.matrix().template topLeftCorner<Dim,Dim>() = linear().inverse();
     }
     else
     {
       eigen_assert(false && "Invalid transform traits in Transform::Inverse");
     }
     // translation and remaining parts
     res.matrix().template topRightCorner<Dim,1>()
       = - res.matrix().template topLeftCorner<Dim,Dim>() * translation();
     res.makeAffine(); // we do need this, because in the beginning res is uninitialized
   }
   return res;
 }
 
 namespace internal {
 
 /*****************************************************
 *** Specializations of take affine part            ***
 *****************************************************/
 
 template<typename TransformType> struct transform_take_affine_part {
   typedef typename TransformType::MatrixType MatrixType;
   typedef typename TransformType::AffinePart AffinePart;
   typedef typename TransformType::ConstAffinePart ConstAffinePart;
   static inline AffinePart run(MatrixType& m)
   { return m.template block<TransformType::Dim,TransformType::HDim>(0,0); }
   static inline ConstAffinePart run(const MatrixType& m)
   { return m.template block<TransformType::Dim,TransformType::HDim>(0,0); }
 };
 
 template<typename Scalar, int Dim, int Options>
 struct transform_take_affine_part<Transform<Scalar,Dim,AffineCompact, Options> > {
   typedef typename Transform<Scalar,Dim,AffineCompact,Options>::MatrixType MatrixType;
   static inline MatrixType& run(MatrixType& m) { return m; }
   static inline const MatrixType& run(const MatrixType& m) { return m; }
 };
 
 /*****************************************************
 *** Specializations of construct from matrix       ***
 *****************************************************/
 
 template<typename Other, int Mode, int Options, int Dim, int HDim>
 struct transform_construct_from_matrix<Other, Mode,Options,Dim,HDim, Dim,Dim>
 {
   static inline void run(Transform<typename Other::Scalar,Dim,Mode,Options> *transform, const Other& other)
   {
     transform->linear() = other;
     transform->translation().setZero();
     transform->makeAffine();
   }
 };
 
 template<typename Other, int Mode, int Options, int Dim, int HDim>
 struct transform_construct_from_matrix<Other, Mode,Options,Dim,HDim, Dim,HDim>
 {
   static inline void run(Transform<typename Other::Scalar,Dim,Mode,Options> *transform, const Other& other)
   {
     transform->affine() = other;
     transform->makeAffine();
   }
 };
 
 template<typename Other, int Mode, int Options, int Dim, int HDim>
 struct transform_construct_from_matrix<Other, Mode,Options,Dim,HDim, HDim,HDim>
 {
   static inline void run(Transform<typename Other::Scalar,Dim,Mode,Options> *transform, const Other& other)
   { transform->matrix() = other; }
 };
 
 template<typename Other, int Options, int Dim, int HDim>
 struct transform_construct_from_matrix<Other, AffineCompact,Options,Dim,HDim, HDim,HDim>
 {
   static inline void run(Transform<typename Other::Scalar,Dim,AffineCompact,Options> *transform, const Other& other)
   { transform->matrix() = other.template block<Dim,HDim>(0,0); }
 };
 
 /**********************************************************
 ***   Specializations of operator* with rhs EigenBase   ***
 **********************************************************/
 
 template<int LhsMode,int RhsMode>
 struct transform_product_result
 {
   enum
   {
     Mode =
       (LhsMode == (int)Projective    || RhsMode == (int)Projective    ) ? Projective :
       (LhsMode == (int)Affine        || RhsMode == (int)Affine        ) ? Affine :
       (LhsMode == (int)AffineCompact || RhsMode == (int)AffineCompact ) ? AffineCompact :
       (LhsMode == (int)Isometry      || RhsMode == (int)Isometry      ) ? Isometry : Projective
   };
 };
 
 template< typename TransformType, typename MatrixType, int RhsCols>
 struct transform_right_product_impl< TransformType, MatrixType, 0, RhsCols>
 {
   typedef typename MatrixType::PlainObject ResultType;
 
   static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE ResultType run(const TransformType& T, const MatrixType& other)
   {
     return T.matrix() * other;
   }
 };
 
 template< typename TransformType, typename MatrixType, int RhsCols>
 struct transform_right_product_impl< TransformType, MatrixType, 1, RhsCols>
 {
   enum {
     Dim = TransformType::Dim,
     HDim = TransformType::HDim,
     OtherRows = MatrixType::RowsAtCompileTime,
     OtherCols = MatrixType::ColsAtCompileTime
   };
 
   typedef typename MatrixType::PlainObject ResultType;
 
   static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE ResultType run(const TransformType& T, const MatrixType& other)
   {
     EIGEN_STATIC_ASSERT(OtherRows==HDim, YOU_MIXED_MATRICES_OF_DIFFERENT_SIZES);
 
     typedef Block<ResultType, Dim, OtherCols, int(MatrixType::RowsAtCompileTime)==Dim> TopLeftLhs;
 
     ResultType res(other.rows(),other.cols());
     TopLeftLhs(res, 0, 0, Dim, other.cols()).noalias() = T.affine() * other;
     res.row(OtherRows-1) = other.row(OtherRows-1);
 
     return res;
   }
 };
 
 template< typename TransformType, typename MatrixType, int RhsCols>
 struct transform_right_product_impl< TransformType, MatrixType, 2, RhsCols>
 {
   enum {
     Dim = TransformType::Dim,
     HDim = TransformType::HDim,
     OtherRows = MatrixType::RowsAtCompileTime,
     OtherCols = MatrixType::ColsAtCompileTime
   };
 
   typedef typename MatrixType::PlainObject ResultType;
 
   static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE ResultType run(const TransformType& T, const MatrixType& other)
   {
     EIGEN_STATIC_ASSERT(OtherRows==Dim, YOU_MIXED_MATRICES_OF_DIFFERENT_SIZES);
 
     typedef Block<ResultType, Dim, OtherCols, true> TopLeftLhs;
     ResultType res(Replicate<typename TransformType::ConstTranslationPart, 1, OtherCols>(T.translation(),1,other.cols()));
     TopLeftLhs(res, 0, 0, Dim, other.cols()).noalias() += T.linear() * other;
 
     return res;
   }
 };
 
 template< typename TransformType, typename MatrixType >
 struct transform_right_product_impl< TransformType, MatrixType, 2, 1> // rhs is a vector of size Dim
 {
   typedef typename TransformType::MatrixType TransformMatrix;
   enum {
     Dim = TransformType::Dim,
     HDim = TransformType::HDim,
     OtherRows = MatrixType::RowsAtCompileTime,
     WorkingRows = EIGEN_PLAIN_ENUM_MIN(TransformMatrix::RowsAtCompileTime,HDim)
   };
 
   typedef typename MatrixType::PlainObject ResultType;
 
   static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE ResultType run(const TransformType& T, const MatrixType& other)
   {
     EIGEN_STATIC_ASSERT(OtherRows==Dim, YOU_MIXED_MATRICES_OF_DIFFERENT_SIZES);
 
     Matrix<typename ResultType::Scalar, Dim+1, 1> rhs;
     rhs.template head<Dim>() = other; rhs[Dim] = typename ResultType::Scalar(1);
     Matrix<typename ResultType::Scalar, WorkingRows, 1> res(T.matrix() * rhs);
     return res.template head<Dim>();
   }
 };
 
 /**********************************************************
 ***   Specializations of operator* with lhs EigenBase   ***
 **********************************************************/
 
 // generic HDim x HDim matrix * T => Projective
 template<typename Other,int Mode, int Options, int Dim, int HDim>
 struct transform_left_product_impl<Other,Mode,Options,Dim,HDim, HDim,HDim>
 {
   typedef Transform<typename Other::Scalar,Dim,Mode,Options> TransformType;
   typedef typename TransformType::MatrixType MatrixType;
   typedef Transform<typename Other::Scalar,Dim,Projective,Options> ResultType;
   static ResultType run(const Other& other,const TransformType& tr)
   { return ResultType(other * tr.matrix()); }
 };
 
 // generic HDim x HDim matrix * AffineCompact => Projective
 template<typename Other, int Options, int Dim, int HDim>
 struct transform_left_product_impl<Other,AffineCompact,Options,Dim,HDim, HDim,HDim>
 {
   typedef Transform<typename Other::Scalar,Dim,AffineCompact,Options> TransformType;
   typedef typename TransformType::MatrixType MatrixType;
   typedef Transform<typename Other::Scalar,Dim,Projective,Options> ResultType;
   static ResultType run(const Other& other,const TransformType& tr)
   {
     ResultType res;
     res.matrix().noalias() = other.template block<HDim,Dim>(0,0) * tr.matrix();
     res.matrix().col(Dim) += other.col(Dim);
     return res;
   }
 };
 
 // affine matrix * T
 template<typename Other,int Mode, int Options, int Dim, int HDim>
 struct transform_left_product_impl<Other,Mode,Options,Dim,HDim, Dim,HDim>
 {
   typedef Transform<typename Other::Scalar,Dim,Mode,Options> TransformType;
   typedef typename TransformType::MatrixType MatrixType;
   typedef TransformType ResultType;
   static ResultType run(const Other& other,const TransformType& tr)
   {
     ResultType res;
     res.affine().noalias() = other * tr.matrix();
     res.matrix().row(Dim) = tr.matrix().row(Dim);
     return res;
   }
 };
 
 // affine matrix * AffineCompact
 template<typename Other, int Options, int Dim, int HDim>
 struct transform_left_product_impl<Other,AffineCompact,Options,Dim,HDim, Dim,HDim>
 {
   typedef Transform<typename Other::Scalar,Dim,AffineCompact,Options> TransformType;
   typedef typename TransformType::MatrixType MatrixType;
   typedef TransformType ResultType;
   static ResultType run(const Other& other,const TransformType& tr)
   {
     ResultType res;
     res.matrix().noalias() = other.template block<Dim,Dim>(0,0) * tr.matrix();
     res.translation() += other.col(Dim);
     return res;
   }
 };
 
 // linear matrix * T
 template<typename Other,int Mode, int Options, int Dim, int HDim>
 struct transform_left_product_impl<Other,Mode,Options,Dim,HDim, Dim,Dim>
 {
   typedef Transform<typename Other::Scalar,Dim,Mode,Options> TransformType;
   typedef typename TransformType::MatrixType MatrixType;
   typedef TransformType ResultType;
   static ResultType run(const Other& other, const TransformType& tr)
   {
     TransformType res;
     if(Mode!=int(AffineCompact))
       res.matrix().row(Dim) = tr.matrix().row(Dim);
     res.matrix().template topRows<Dim>().noalias()
       = other * tr.matrix().template topRows<Dim>();
     return res;
   }
 };
 
 /**********************************************************
 *** Specializations of operator* with another Transform ***
 **********************************************************/
 
 template<typename Scalar, int Dim, int LhsMode, int LhsOptions, int RhsMode, int RhsOptions>
 struct transform_transform_product_impl<Transform<Scalar,Dim,LhsMode,LhsOptions>,Transform<Scalar,Dim,RhsMode,RhsOptions>,false >
 {
   enum { ResultMode = transform_product_result<LhsMode,RhsMode>::Mode };
   typedef Transform<Scalar,Dim,LhsMode,LhsOptions> Lhs;
   typedef Transform<Scalar,Dim,RhsMode,RhsOptions> Rhs;
   typedef Transform<Scalar,Dim,ResultMode,LhsOptions> ResultType;
   static ResultType run(const Lhs& lhs, const Rhs& rhs)
   {
     ResultType res;
     res.linear() = lhs.linear() * rhs.linear();
     res.translation() = lhs.linear() * rhs.translation() + lhs.translation();
     res.makeAffine();
     return res;
   }
 };
 
 template<typename Scalar, int Dim, int LhsMode, int LhsOptions, int RhsMode, int RhsOptions>
 struct transform_transform_product_impl<Transform<Scalar,Dim,LhsMode,LhsOptions>,Transform<Scalar,Dim,RhsMode,RhsOptions>,true >
 {
   typedef Transform<Scalar,Dim,LhsMode,LhsOptions> Lhs;
   typedef Transform<Scalar,Dim,RhsMode,RhsOptions> Rhs;
   typedef Transform<Scalar,Dim,Projective> ResultType;
   static ResultType run(const Lhs& lhs, const Rhs& rhs)
   {
     return ResultType( lhs.matrix() * rhs.matrix() );
   }
 };
 
 template<typename Scalar, int Dim, int LhsOptions, int RhsOptions>
 struct transform_transform_product_impl<Transform<Scalar,Dim,AffineCompact,LhsOptions>,Transform<Scalar,Dim,Projective,RhsOptions>,true >
 {
   typedef Transform<Scalar,Dim,AffineCompact,LhsOptions> Lhs;
   typedef Transform<Scalar,Dim,Projective,RhsOptions> Rhs;
   typedef Transform<Scalar,Dim,Projective> ResultType;
   static ResultType run(const Lhs& lhs, const Rhs& rhs)
   {
     ResultType res;
     res.matrix().template topRows<Dim>() = lhs.matrix() * rhs.matrix();
     res.matrix().row(Dim) = rhs.matrix().row(Dim);
     return res;
   }
 };
 
 template<typename Scalar, int Dim, int LhsOptions, int RhsOptions>
 struct transform_transform_product_impl<Transform<Scalar,Dim,Projective,LhsOptions>,Transform<Scalar,Dim,AffineCompact,RhsOptions>,true >
 {
   typedef Transform<Scalar,Dim,Projective,LhsOptions> Lhs;
   typedef Transform<Scalar,Dim,AffineCompact,RhsOptions> Rhs;
   typedef Transform<Scalar,Dim,Projective> ResultType;
   static ResultType run(const Lhs& lhs, const Rhs& rhs)
   {
     ResultType res(lhs.matrix().template leftCols<Dim>() * rhs.matrix());
     res.matrix().col(Dim) += lhs.matrix().col(Dim);
     return res;
   }
 };
 
 } // end namespace internal
 
 } // end namespace Eigen
 
 #endif // EIGEN_TRANSFORM_H