Geometry/Transform.h

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// 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>
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;

} // end namespace internal

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,     
    HDim = _Dim+1,  
    Rows = int(Mode)==(AffineCompact) ? Dim : HDim
  };
  typedef _Scalar Scalar;
  typedef DenseIndex Index;
  typedef typename internal::make_proper_matrix_type<Scalar,Rows,HDim,Options>::type MatrixType;
  typedef const MatrixType ConstMatrixType;
  typedef Matrix<Scalar,Dim,Dim,Options> LinearMatrixType;
  typedef Block<MatrixType,Dim,Dim,int(Mode)==(AffineCompact)> LinearPart;
  typedef const Block<ConstMatrixType,Dim,Dim,int(Mode)==(AffineCompact)> ConstLinearPart;
  typedef typename internal::conditional<int(Mode)==int(AffineCompact),
                              MatrixType&,
                              Block<MatrixType,Dim,HDim> >::type AffinePart;
  typedef typename internal::conditional<int(Mode)==int(AffineCompact),
                              const MatrixType&,
                              const Block<const MatrixType,Dim,HDim> >::type ConstAffinePart;
  typedef Matrix<Scalar,Dim,1> VectorType;
  typedef Block<MatrixType,Dim,1,int(Mode)==(AffineCompact)> TranslationPart;
  typedef const Block<ConstMatrixType,Dim,1,int(Mode)==(AffineCompact)> ConstTranslationPart;
  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)) };
  typedef Transform<Scalar,Dim,TransformTimeDiagonalMode> TransformTimeDiagonalReturnType;

protected:

  MatrixType m_matrix;

public:

  inline Transform()
  {
    check_template_params();
    if (int(Mode)==Affine)
      makeAffine();
  }

  inline Transform(const Transform& other)
  {
    check_template_params();
    m_matrix = other.m_matrix;
  }

  inline explicit Transform(const TranslationType& t)
  {
    check_template_params();
    *this = t;
  }
  inline explicit Transform(const UniformScaling<Scalar>& s)
  {
    check_template_params();
    *this = s;
  }
  template<typename Derived>
  inline explicit Transform(const RotationBase<Derived, Dim>& r)
  {
    check_template_params();
    *this = r;
  }

  inline Transform& operator=(const Transform& other)
  { m_matrix = other.m_matrix; return *this; }

  typedef internal::transform_take_affine_part<Transform> take_affine_part;

  template<typename OtherDerived>
  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());
  }

  template<typename OtherDerived>
  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>
  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>
  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(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(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>
  Transform(const ReturnByValue<OtherDerived>& other)
  {
    check_template_params();
    other.evalTo(*this);
  }

  template<typename OtherDerived>
  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

  inline Scalar operator() (Index row, Index col) const { return m_matrix(row,col); }
  inline Scalar& operator() (Index row, Index col) { return m_matrix(row,col); }

  inline const MatrixType& matrix() const { return m_matrix; }
  inline MatrixType& matrix() { return m_matrix; }

  inline ConstLinearPart linear() const { return ConstLinearPart(m_matrix,0,0); }
  inline LinearPart linear() { return LinearPart(m_matrix,0,0); }

  inline ConstAffinePart affine() const { return take_affine_part::run(m_matrix); }
  inline AffinePart affine() { return take_affine_part::run(m_matrix); }

  inline ConstTranslationPart translation() const { return ConstTranslationPart(m_matrix,0,Dim); }
  inline TranslationPart translation() { return TranslationPart(m_matrix,0,Dim); }

  // note: this function is defined here because some compilers cannot find the respective declaration
  template<typename OtherDerived>
  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()); }

  template<typename OtherDerived> friend
  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); }

  template<typename DiagonalDerived>
  inline const TransformTimeDiagonalReturnType
    operator * (const DiagonalBase<DiagonalDerived> &b) const
  {
    TransformTimeDiagonalReturnType res(*this);
    res.linear() *= b;
    return res;
  }

  template<typename DiagonalDerived>
  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 (Mode!=int(AffineCompact))
      res.matrix().row(Dim) = b.matrix().row(Dim);
    return res;
  }

  template<typename OtherDerived>
  inline Transform& operator*=(const EigenBase<OtherDerived>& other) { return *this = *this * other; }

  inline const Transform operator * (const Transform& other) const
  {
    return internal::transform_transform_product_impl<Transform,Transform>::run(*this,other);
  }
  
  #ifdef __INTEL_COMPILER
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:
  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

  template<int OtherMode,int OtherOptions>
  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

  void setIdentity() { m_matrix.setIdentity(); }

  static const Transform Identity()
  {
    return Transform(MatrixType::Identity());
  }

  template<typename OtherDerived>
  inline Transform& scale(const MatrixBase<OtherDerived> &other);

  template<typename OtherDerived>
  inline Transform& prescale(const MatrixBase<OtherDerived> &other);

  inline Transform& scale(const Scalar& s);
  inline Transform& prescale(const Scalar& s);

  template<typename OtherDerived>
  inline Transform& translate(const MatrixBase<OtherDerived> &other);

  template<typename OtherDerived>
  inline Transform& pretranslate(const MatrixBase<OtherDerived> &other);

  template<typename RotationType>
  inline Transform& rotate(const RotationType& rotation);

  template<typename RotationType>
  inline Transform& prerotate(const RotationType& rotation);

  Transform& shear(const Scalar& sx, const Scalar& sy);
  Transform& preshear(const Scalar& sx, const Scalar& sy);

  inline Transform& operator=(const TranslationType& t);
  inline Transform& operator*=(const TranslationType& t) { return translate(t.vector()); }
  inline Transform operator*(const TranslationType& t) const;

  inline Transform& operator=(const UniformScaling<Scalar>& t);
  inline Transform& operator*=(const UniformScaling<Scalar>& s) { return scale(s.factor()); }
  inline Transform<Scalar,Dim,(int(Mode)==int(Isometry)?Affine:Isometry)> operator*(const UniformScaling<Scalar>& s) const
  {
    Transform<Scalar,Dim,(int(Mode)==int(Isometry)?Affine:Isometry),Options> res = *this;
    res.scale(s.factor());
    return res;
  }

  inline Transform& operator*=(const DiagonalMatrix<Scalar,Dim>& s) { linear() *= s; return *this; }

  template<typename Derived>
  inline Transform& operator=(const RotationBase<Derived,Dim>& r);
  template<typename Derived>
  inline Transform& operator*=(const RotationBase<Derived,Dim>& r) { return rotate(r.toRotationMatrix()); }
  template<typename Derived>
  inline Transform operator*(const RotationBase<Derived,Dim>& r) const;

  const LinearMatrixType rotation() const;
  template<typename RotationMatrixType, typename ScalingMatrixType>
  void computeRotationScaling(RotationMatrixType *rotation, ScalingMatrixType *scaling) const;
  template<typename ScalingMatrixType, typename RotationMatrixType>
  void computeScalingRotation(ScalingMatrixType *scaling, RotationMatrixType *rotation) const;

  template<typename PositionDerived, typename OrientationType, typename ScaleDerived>
  Transform& fromPositionOrientationScale(const MatrixBase<PositionDerived> &position,
    const OrientationType& orientation, const MatrixBase<ScaleDerived> &scale);

  inline Transform inverse(TransformTraits traits = (TransformTraits)Mode) const;

  const Scalar* data() const { return m_matrix.data(); }
  Scalar* data() { return m_matrix.data(); }

  template<typename NewScalarType>
  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); }

  template<typename OtherScalarType>
  inline explicit Transform(const Transform<OtherScalarType,Dim,Mode,Options>& other)
  {
    check_template_params();
    m_matrix = other.matrix().template cast<Scalar>();
  }

  bool isApprox(const Transform& other, const typename NumTraits<Scalar>::Real& prec = NumTraits<Scalar>::dummy_precision()) const
  { return m_matrix.isApprox(other.m_matrix, prec); }

  void makeAffine()
  {
    if(int(Mode)!=int(AffineCompact))
    {
      matrix().template block<1,Dim>(Dim,0).setZero();
      matrix().coeffRef(Dim,Dim) = Scalar(1);
    }
  }

  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); }
  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); }

  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); }
  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
    static EIGEN_STRONG_INLINE void check_template_params()
    {
      EIGEN_STATIC_ASSERT((Options & (DontAlign|RowMajor)) == Options, INVALID_MATRIX_TEMPLATE_PARAMETERS)
    }
  #endif

};

typedef Transform<float,2,Isometry> Isometry2f;
typedef Transform<float,3,Isometry> Isometry3f;
typedef Transform<double,2,Isometry> Isometry2d;
typedef Transform<double,3,Isometry> Isometry3d;

typedef Transform<float,2,Affine> Affine2f;
typedef Transform<float,3,Affine> Affine3f;
typedef Transform<double,2,Affine> Affine2d;
typedef Transform<double,3,Affine> Affine3d;

typedef Transform<float,2,AffineCompact> AffineCompact2f;
typedef Transform<float,3,AffineCompact> AffineCompact3f;
typedef Transform<double,2,AffineCompact> AffineCompact2d;
typedef Transform<double,3,AffineCompact> AffineCompact3d;

typedef Transform<float,2,Projective> Projective2f;
typedef Transform<float,3,Projective> Projective3f;
typedef Transform<double,2,Projective> Projective2d;
typedef Transform<double,3,Projective> Projective3d;

/**************************
*** Optional QT support ***
**************************/

#ifdef EIGEN_QT_SUPPORT

template<typename Scalar, int Dim, int Mode,int Options>
Transform<Scalar,Dim,Mode,Options>::Transform(const QMatrix& other)
{
  check_template_params();
  *this = other;
}

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)
  m_matrix << other.m11(), other.m21(), other.dx(),
              other.m12(), other.m22(), other.dy(),
              0, 0, 1;
  return *this;
}

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));
}

template<typename Scalar, int Dim, int Mode,int Options>
Transform<Scalar,Dim,Mode,Options>::Transform(const QTransform& other)
{
  check_template_params();
  *this = other;
}

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 (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;
}

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 (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 ***
*********************/

template<typename Scalar, int Dim, int Mode, int Options>
template<typename OtherDerived>
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;
}

template<typename Scalar, int Dim, int Mode, int Options>
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;
}

template<typename Scalar, int Dim, int Mode, int Options>
template<typename OtherDerived>
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)
  m_matrix.template block<Dim,HDim>(0,0).noalias() = (other.asDiagonal() * m_matrix.template block<Dim,HDim>(0,0));
  return *this;
}

template<typename Scalar, int Dim, int Mode, int Options>
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;
}

template<typename Scalar, int Dim, int Mode, int Options>
template<typename OtherDerived>
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;
}

template<typename Scalar, int Dim, int Mode, int Options>
template<typename OtherDerived>
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(int(Mode)==int(Projective))
    affine() += other * m_matrix.row(Dim);
  else
    translation() += other;
  return *this;
}

template<typename Scalar, int Dim, int Mode, int Options>
template<typename RotationType>
Transform<Scalar,Dim,Mode,Options>&
Transform<Scalar,Dim,Mode,Options>::rotate(const RotationType& rotation)
{
  linearExt() *= internal::toRotationMatrix<Scalar,Dim>(rotation);
  return *this;
}

template<typename Scalar, int Dim, int Mode, int Options>
template<typename RotationType>
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;
}

template<typename Scalar, int Dim, int Mode, int Options>
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;
}

template<typename Scalar, int Dim, int Mode, int Options>
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>
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>
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>
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>
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>
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 ***
************************/

template<typename Scalar, int Dim, int Mode, int Options>
const typename Transform<Scalar,Dim,Mode,Options>::LinearMatrixType
Transform<Scalar,Dim,Mode,Options>::rotation() const
{
  LinearMatrixType result;
  computeRotationScaling(&result, (LinearMatrixType*)0);
  return result;
}


template<typename Scalar, int Dim, int Mode, int Options>
template<typename RotationMatrixType, typename ScalingMatrixType>
void Transform<Scalar,Dim,Mode,Options>::computeRotationScaling(RotationMatrixType *rotation, ScalingMatrixType *scaling) const
{
  JacobiSVD<LinearMatrixType> svd(linear(), ComputeFullU | ComputeFullV);

  Scalar x = (svd.matrixU() * svd.matrixV().adjoint()).determinant(); // so x has absolute value 1
  VectorType sv(svd.singularValues());
  sv.coeffRef(0) *= x;
  if(scaling) scaling->lazyAssign(svd.matrixV() * sv.asDiagonal() * svd.matrixV().adjoint());
  if(rotation)
  {
    LinearMatrixType m(svd.matrixU());
    m.col(0) /= x;
    rotation->lazyAssign(m * svd.matrixV().adjoint());
  }
}

template<typename Scalar, int Dim, int Mode, int Options>
template<typename ScalingMatrixType, typename RotationMatrixType>
void Transform<Scalar,Dim,Mode,Options>::computeScalingRotation(ScalingMatrixType *scaling, RotationMatrixType *rotation) const
{
  JacobiSVD<LinearMatrixType> svd(linear(), ComputeFullU | ComputeFullV);

  Scalar x = (svd.matrixU() * svd.matrixV().adjoint()).determinant(); // so x has absolute value 1
  VectorType sv(svd.singularValues());
  sv.coeffRef(0) *= x;
  if(scaling) scaling->lazyAssign(svd.matrixU() * sv.asDiagonal() * svd.matrixU().adjoint());
  if(rotation)
  {
    LinearMatrixType m(svd.matrixU());
    m.col(0) /= x;
    rotation->lazyAssign(m * svd.matrixV().adjoint());
  }
}

template<typename Scalar, int Dim, int Mode, int Options>
template<typename PositionDerived, typename OrientationType, typename ScaleDerived>
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 {

// selector needed to avoid taking the inverse of a 3x4 matrix
template<typename TransformType, int Mode=TransformType::Mode>
struct projective_transform_inverse
{
  static inline void run(const TransformType&, TransformType&)
  {}
};

template<typename TransformType>
struct projective_transform_inverse<TransformType, Projective>
{
  static inline void run(const TransformType& m, TransformType& res)
  {
    res.matrix() = m.matrix().inverse();
  }
};

} // end namespace internal


template<typename Scalar, int Dim, int Mode, int Options>
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 >
struct transform_right_product_impl< TransformType, MatrixType, 0 >
{
  typedef typename MatrixType::PlainObject ResultType;

  static EIGEN_STRONG_INLINE ResultType run(const TransformType& T, const MatrixType& other)
  {
    return T.matrix() * other;
  }
};

template< typename TransformType, typename MatrixType >
struct transform_right_product_impl< TransformType, MatrixType, 1 >
{
  enum { 
    Dim = TransformType::Dim, 
    HDim = TransformType::HDim,
    OtherRows = MatrixType::RowsAtCompileTime,
    OtherCols = MatrixType::ColsAtCompileTime
  };

  typedef typename MatrixType::PlainObject ResultType;

  static 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 >
struct transform_right_product_impl< TransformType, MatrixType, 2 >
{
  enum { 
    Dim = TransformType::Dim, 
    HDim = TransformType::HDim,
    OtherRows = MatrixType::RowsAtCompileTime,
    OtherCols = MatrixType::ColsAtCompileTime
  };

  typedef typename MatrixType::PlainObject ResultType;

  static 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;
  }
};

/**********************************************************
***   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