Transform.h

// 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.
  * \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 sightly 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>