// 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/Fixed_border_parameterizer_3.h>
#include <CGAL/surface_mesh_parameterization_assertions.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.
///
/// One-to-one mapping is guaranteed if 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()`.
/// `Discrete_authalic_parameterizer_3`:
/// - It provides default `BorderParameterizer_3` and `SparseLinearAlgebraTraits_d` template
///   parameters that make sense.
/// - It implements `compute_w_ij()` to compute w_ij = (i, j) coefficient of matrix A
///   for j neighbor vertex of i based on Discrete Authalic Parameterization algorithm.
///
/// \cgalModels `ParameterizerTraits_3`
///
/// \sa `CGAL::Parameterizer_traits_3<ParameterizationMesh_3>`
/// \sa `CGAL::Fixed_border_parameterizer_3<ParameterizationMesh_3, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
/// \sa `CGAL::Barycentric_mapping_parameterizer_3<ParameterizationMesh_3, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
/// \sa `CGAL::Discrete_conformal_map_parameterizer_3<ParameterizationMesh_3, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
/// \sa `CGAL::LSCM_parameterizer_3<ParameterizationMesh_3, BorderParameterizer_3>`
/// \sa `CGAL::Mean_value_coordinates_parameterizer_3<ParameterizationMesh_3, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`

template
<
    class ParameterizationMesh_3,     ///< 3D surface mesh
    class BorderParameterizer_3       ///< Strategy to parameterize the surface border
                = Circular_border_arc_length_parameterizer_3<ParameterizationMesh_3>,
    class SparseLinearAlgebraTraits_d ///< Traits class to solve a sparse linear system
                =  Eigen_solver_traits<Eigen::BiCGSTAB<Eigen_sparse_matrix<double>::EigenType, Eigen::IncompleteLUT< double > > >
>
class Discrete_authalic_parameterizer_3
    : public Fixed_border_parameterizer_3<ParameterizationMesh_3,
                                          BorderParameterizer_3,
                                          SparseLinearAlgebraTraits_d>
{
// Private types
private:
    // Superclass
    typedef Fixed_border_parameterizer_3<ParameterizationMesh_3,
                                         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 ParameterizationMesh_3          Adaptor;
    typedef BorderParameterizer_3           Border_param;
    typedef SparseLinearAlgebraTraits_d     Sparse_LA;
    /// @endcond

// Private types
private:
    // Mesh_Adaptor_3 subtypes:
    typedef typename Adaptor::NT            NT;
    typedef typename Adaptor::Point_2       Point_2;
    typedef typename Adaptor::Point_3       Point_3;
    typedef typename Adaptor::Vector_2      Vector_2;
    typedef typename Adaptor::Vector_3      Vector_3;
    typedef typename Adaptor::Facet         Facet;
    typedef typename Adaptor::Facet_handle  Facet_handle;
    typedef typename Adaptor::Facet_const_handle
                                            Facet_const_handle;
    typedef typename Adaptor::Facet_iterator Facet_iterator;
    typedef typename Adaptor::Facet_const_iterator
                                            Facet_const_iterator;
    typedef typename Adaptor::Vertex        Vertex;
    typedef typename Adaptor::Vertex_handle Vertex_handle;
    typedef typename Adaptor::Vertex_const_handle
                                            Vertex_const_handle;
    typedef typename Adaptor::Vertex_iterator Vertex_iterator;
    typedef typename Adaptor::Vertex_const_iterator
                                            Vertex_const_iterator;
    typedef typename Adaptor::Border_vertex_iterator
                                            Border_vertex_iterator;
    typedef typename Adaptor::Border_vertex_const_iterator
                                            Border_vertex_const_iterator;
    typedef typename Adaptor::Vertex_around_facet_circulator
                                            Vertex_around_facet_circulator;
    typedef typename Adaptor::Vertex_around_facet_const_circulator
                                            Vertex_around_facet_const_circulator;
    typedef typename Adaptor::Vertex_around_vertex_circulator
                                            Vertex_around_vertex_circulator;
    typedef typename Adaptor::Vertex_around_vertex_const_circulator
                                            Vertex_around_vertex_const_circulator;

    // 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<Adaptor,
                                     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.
    virtual NT compute_w_ij(const Adaptor& mesh,
                            Vertex_const_handle main_vertex_v_i,
                            Vertex_around_vertex_const_circulator neighbor_vertex_v_j)
    {
        Point_3 position_v_i = mesh.get_vertex_position(main_vertex_v_i);
        Point_3 position_v_j = mesh.get_vertex_position(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_vertex_const_circulator previous_vertex_v_k = neighbor_vertex_v_j;
        previous_vertex_v_k --;
        Point_3 position_v_k = mesh.get_vertex_position(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_vertex_const_circulator next_vertex_v_l = neighbor_vertex_v_j;
        next_vertex_v_l ++;
        Point_3 position_v_l = mesh.get_vertex_position(next_vertex_v_l);
        double cotg_theta_ij = cotangent(position_v_i, position_v_j, position_v_l);

        double weight = 0.0;
        CGAL_surface_mesh_parameterization_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