// Copyright (c) 2005  INRIA (France).
// All rights reserved.
//
// This file is part of CGAL (www.cgal.org).
// You can redistribute it and/or modify it under the terms of the GNU
// General Public License as published by the Free Software Foundation,
// either version 3 of the License, or (at your option) any later version.
//
// Licensees holding a valid commercial license may use this file in
// accordance with the commercial license agreement provided with the software.
//
// This file is provided AS IS with NO WARRANTY OF ANY KIND, INCLUDING THE
// WARRANTY OF DESIGN, MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
//
// $URL$
// $Id$
//
//
// Author(s)     : Laurent Saboret, Pierre Alliez, Bruno Levy

#ifndef CGAL_DISCRETE_AUTHALIC_PARAMETERIZER_3_H
#define CGAL_DISCRETE_AUTHALIC_PARAMETERIZER_3_H

#include <CGAL/license/Surface_mesh_parameterization.h>


#include <CGAL/Fixed_border_parameterizer_3.h>
#include <CGAL/Circular_border_parameterizer_3.h>
#include <CGAL/Eigen_solver_traits.h>

/// \file Discrete_authalic_parameterizer_3.h

namespace CGAL {

/// \ingroup  PkgSurfaceParameterizationMethods
///
/// The class `Discrete_authalic_parameterizer_3`
/// implements the *Discrete Authalic Parameterization* algorithm \cgalCite{cgal:dma-ipsm-02}. This method
/// is sometimes called <i>DAP</i> or just <i>Authalic parameterization</i>.
///
/// DAP is a weak area-preserving parameterization. It is a compromise between
/// area-preserving and angle-preserving.
///
/// A one-to-one mapping is guaranteed if the surface's border is mapped onto a convex polygon.
///
/// This class is a Strategy \cgalCite{cgal:ghjv-dpero-95} called by the main
/// parameterization algorithm `Fixed_border_parameterizer_3::parameterize()` and it:
/// - provides the template parameters `BorderParameterizer_3` and `SparseLinearAlgebraTraits_d`.
/// - implements `compute_w_ij()` to compute w_ij = (i, j), coefficient of the matrix A
///   for j neighbor vertex of i, based on Discrete Authalic Parameterization algorithm.
///
/// \cgalModels `ParameterizerTraits_3`
///
/// \tparam TriangleMesh must be a model of `FaceGraph`
/// \tparam BorderParameterizer_3 is a Strategy to parameterize the surface border.
/// \tparam SparseLinearAlgebraTraits_d is a Traits class to solve a sparse linear system. <br>
///         Note: the system is *not* symmetric because `Fixed_border_parameterizer_3`
///         does not remove (yet) border vertices from the system.
///
/// \sa `CGAL::Parameterizer_traits_3<TriangleMesh>`.
/// \sa `CGAL::Fixed_border_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
/// \sa `CGAL::ARAP_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
/// \sa `CGAL::Barycentric_mapping_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
/// \sa `CGAL::Discrete_conformal_map_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
/// \sa `CGAL::LSCM_parameterizer_3<TriangleMesh, BorderParameterizer_3>`
/// \sa `CGAL::Mean_value_coordinates_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
///
template
<
  class TriangleMesh,
  class BorderParameterizer_3
    = Circular_border_arc_length_parameterizer_3<TriangleMesh>,
  class SparseLinearAlgebraTraits_d
    =  Eigen_solver_traits<Eigen::BiCGSTAB<Eigen_sparse_matrix<double>::EigenType,
                                           Eigen::IncompleteLUT<double> > >
>
class Discrete_authalic_parameterizer_3
  : public Fixed_border_parameterizer_3<TriangleMesh,
                                        BorderParameterizer_3,
                                        SparseLinearAlgebraTraits_d>
{
// Private types
private:
  // Superclass
  typedef Fixed_border_parameterizer_3<TriangleMesh,
                                       BorderParameterizer_3,
                                       SparseLinearAlgebraTraits_d>    Base;

// Public types
public:
  // We have to repeat the types exported by superclass
  /// @cond SKIP_IN_MANUAL
  typedef typename Base::Error_code       Error_code;
  typedef BorderParameterizer_3           Border_param;
  typedef SparseLinearAlgebraTraits_d     Sparse_LA;
  /// @endcond

// Private types
private:
  typedef typename boost::graph_traits<TriangleMesh>::vertex_descriptor vertex_descriptor;
  typedef CGAL::Vertex_around_target_circulator<TriangleMesh> vertex_around_target_circulator;

  // Traits subtypes:
  typedef typename Parameterizer_traits_3<TriangleMesh>::NT            NT;
  typedef typename Parameterizer_traits_3<TriangleMesh>::Point_3       Point_3;
  typedef typename Parameterizer_traits_3<TriangleMesh>::Vector_3      Vector_3;

  // SparseLinearAlgebraTraits_d subtypes:
  typedef typename Sparse_LA::Vector      Vector;
  typedef typename Sparse_LA::Matrix      Matrix;

  using Base::cotangent;

// Public operations
public:
  /// Constructor
  Discrete_authalic_parameterizer_3(Border_param border_param = Border_param(),
                                    ///< %Object that maps the surface's border to 2D space.
                                    Sparse_LA sparse_la = Sparse_LA())
                                    ///< Traits object to access a sparse linear system.
    : Fixed_border_parameterizer_3<TriangleMesh,
                                   Border_param,
                                   Sparse_LA>(border_param, sparse_la)
  { }

  // Default copy constructor and operator =() are fine

// Protected operations
protected:
  /// Compute w_ij = (i, j), coefficient of matrix A for j neighbor vertex of i.
  ///
  /// \param mesh a triangulated surface.
  /// \param main_vertex_v_i the vertex of `mesh` with index `i`
  /// \param neighbor_vertex_v_j the vertex of `mesh` with index `j`
  virtual NT compute_w_ij(const TriangleMesh& mesh,
                          vertex_descriptor main_vertex_v_i,
                          vertex_around_target_circulator neighbor_vertex_v_j)
  {
    typedef typename Parameterizer_traits_3<TriangleMesh>::VPM PPmap;
    PPmap ppmap = get(vertex_point, mesh);

    Point_3 position_v_i = get(ppmap, main_vertex_v_i);
    Point_3 position_v_j = get(ppmap, *neighbor_vertex_v_j);

    // Compute the square norm of v_j -> v_i vector
    Vector_3 edge = position_v_i - position_v_j;
    double square_len = edge*edge;

    // Compute cotangent of (v_k,v_j,v_i) corner (i.e. cotan of v_j corner)
    // if v_k is the vertex before v_j when circulating around v_i
    vertex_around_target_circulator previous_vertex_v_k = neighbor_vertex_v_j;
    previous_vertex_v_k--;
    Point_3 position_v_k = get(ppmap, *previous_vertex_v_k);
    double cotg_psi_ij = cotangent(position_v_k, position_v_j, position_v_i);

    // Compute cotangent of (v_i,v_j,v_l) corner (i.e. cotan of v_j corner)
    // if v_l is the vertex after v_j when circulating around v_i
    vertex_around_target_circulator next_vertex_v_l = neighbor_vertex_v_j;
    next_vertex_v_l++;
    Point_3 position_v_l = get(ppmap,*next_vertex_v_l);
    double cotg_theta_ij = cotangent(position_v_i, position_v_j, position_v_l);

    double weight = 0.0;
    CGAL_assertion(square_len != 0.0); // two points are identical!
    if(square_len != 0.0)
      weight = (cotg_psi_ij+cotg_theta_ij)/square_len;

    return weight;
  }
};

} // namespace CGAL

#endif // CGAL_DISCRETE_AUTHALIC_PARAMETERIZER_3_H