// Copyright (c) 2016  GeometryFactory (France).
// All rights reserved.
//
// This file is part of CGAL (www.cgal.org).
//
// $URL: https://github.com/CGAL/cgal/blob/v5.5.1/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/Orbifold_Tutte_parameterizer_3.h $
// $Id: Orbifold_Tutte_parameterizer_3.h 4ffc949 2022-02-03T17:11:20+01:00 Sébastien Loriot
// SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
//
// Author(s)     : Mael Rouxel-Labbé

#ifndef CGAL_SURFACE_MESH_PARAMETERIZATION_ORBIFOLD_TUTTE_PARAMETERIZER_3_H
#define CGAL_SURFACE_MESH_PARAMETERIZATION_ORBIFOLD_TUTTE_PARAMETERIZER_3_H

#include <CGAL/license/Surface_mesh_parameterization.h>

#include <CGAL/disable_warnings.h>

#include <CGAL/Surface_mesh_parameterization/internal/kernel_traits.h>
#include <CGAL/Surface_mesh_parameterization/internal/orbifold_cone_helper.h>
#include <CGAL/Surface_mesh_parameterization/IO/File_off.h>
#include <CGAL/Surface_mesh_parameterization/orbifold_enums.h>
#include <CGAL/Surface_mesh_parameterization/Error_code.h>
#include <CGAL/Surface_mesh_parameterization/orbifold_shortest_path.h>
#include <CGAL/Weights/tangent_weights.h>
#include <CGAL/Weights/cotangent_weights.h>
#include <CGAL/assertions.h>

#include <CGAL/assertions.h>
#include <CGAL/circulator.h>
#include <CGAL/Default.h>
#include <CGAL/Timer.h>
#include <CGAL/use.h>

#if defined(CGAL_EIGEN3_ENABLED)
#include <CGAL/Eigen_solver_traits.h>
#ifdef CGAL_SMP_USE_SPARSESUITE_SOLVERS
#include <Eigen/UmfPackSupport>
#endif
#endif

#include <boost/array.hpp>
#include <boost/tuple/tuple.hpp>
#include <boost/type_traits/is_same.hpp>

#include <unordered_map>
#include <unordered_set>
#include <cmath>
#include <fstream>
#include <iostream>
#include <string>
#include <vector>
#include <utility>

/// \file Orbifold_Tutte_parameterizer_3.h

// @todo checks that cones are different, are on seams, seam is one connected
//       component
// @todo Should the order of cones provided in entry matter ? Map the first cone
//       to [-1, -1] for example ?

namespace CGAL {

namespace Surface_mesh_parameterization {

/// \ingroup PkgSurfaceMeshParameterizationOrbifoldHelperFunctions
///
/// reads a serie of cones from an input stream. Cones are passed as an
/// integer value that is the index of a vertex handle in the mesh tm`, using
/// the vertex index property map `vpmap` for correspondency.
///
/// \attention The mesh is here `tm`, it is the base mesh of the `CGAL::Seam_mesh`
///            that is passed in input, <i>not</i> the seam mesh itself.
///
/// \tparam TriangleMesh A triangle mesh, model of `FaceListGraph` and `HalfedgeListGraph`.
/// \tparam VertexIndexMap must be a model of `ReadablePropertyMap` with
///                        `boost::graph_traits<TriangleMesh>::%vertex_descriptor` as key type and
///                        a unique integer as value type.
/// \tparam ConeOutputIterator a model of `OutputIterator` with value type
///                            `boost::graph_traits<TriangleMesh>::%vertex_descriptor`.
///
/// \param tm the triangular mesh to be parameterized
/// \param in the input stream
/// \param vpmap an initialized vertex index map
/// \param out the output iterator
///
/// \pre The number of cones must match the chosen \link PkgSurfaceMeshParameterizationEnums Orbifold_type \endlink.
/// \pre No two cones correspond to the same vertex (all cones have different index).
///
/// \return The corresponding vertex descriptors are output, in the same order as the input integers, in `out`.
///         The function checks if the input is valid (no duplicate, correct number of cones) and returns an `Error_code`.
template<typename TriangleMesh, typename VertexIndexMap, typename ConeOutputIterator>
Error_code read_cones(const TriangleMesh& tm, std::ifstream& in, VertexIndexMap vpmap, ConeOutputIterator out)
{
  typedef typename boost::graph_traits<TriangleMesh>::vertex_descriptor TM_vertex_descriptor;
  typedef typename boost::graph_traits<TriangleMesh>::vertex_iterator   TM_vertex_iterator;

  std::vector<int> cones;
  cones.reserve(4);
  int cone_index;
  while(in >> cone_index)
    cones.push_back(cone_index);

#ifdef CGAL_SMP_ORBIFOLD_DEBUG
  std::cout << "Input cones: ";
  for(std::size_t i=0; i<cones.size(); ++i)
    std::cout << cones[i] << " ";
  std::cout << std::endl;
#endif

  if(cones.size() < 3 || cones.size() > 4) {
    std::cerr << "Error: Not enough or too many input cones" << std::endl;
    return ERROR_WRONG_PARAMETER;
  }

  if(!internal::are_cones_unique(cones)) {
    std::cerr << "Error: The input cones are not unique" << std::endl;
    return ERROR_WRONG_PARAMETER;
  }

  // Locate the cones in the underlying mesh 'tm'
  std::vector<TM_vertex_descriptor> cone_vds_in_tm(cones.size()); // need this to keep the correct order

  // Since the cones are unique, we only need to loop all the vertices once
  TM_vertex_iterator vit, end;
  boost::tie(vit, end) = vertices(tm);
  for(; vit!=end; ++vit) {
    for(std::size_t i=0; i<cones.size(); ++i) {
      TM_vertex_descriptor vd = *vit;
      if(vpmap[vd] == cones[i])
        cone_vds_in_tm[i] = vd;
    }
  }

  CGAL_postcondition_code(for(std::size_t i=0; i<cones.size(); ++i))
  CGAL_postcondition(cone_vds_in_tm[i] != TM_vertex_descriptor());

  for(std::size_t i=0; i<cones.size(); ++i)
    *out++ = cone_vds_in_tm[i];

  return OK;
}

/// \ingroup PkgSurfaceMeshParameterizationOrbifoldHelperFunctions
///
/// Same as above, using the default indexation of the vertices of `tm`: vertices
/// are numbered from `0` to `num_vertices(tm)-1`, in the order that they appear
/// while calling `vertices(tm)`.
template<typename TriangleMesh, typename ConeOutputIterator>
Error_code read_cones(const TriangleMesh& tm, std::ifstream& in, ConeOutputIterator out)
{
  typedef typename boost::graph_traits<TriangleMesh>::vertex_descriptor TM_vertex_descriptor;
  typedef typename boost::graph_traits<TriangleMesh>::vertex_iterator   TM_vertex_iterator;

  std::unordered_map<TM_vertex_descriptor, int> m;
  int counter = 0;

  TM_vertex_iterator vit, end;
  boost::tie(vit, end) = vertices(tm);
  for(; vit!=end; ++vit)
  {
    TM_vertex_descriptor vd = *vit;
    m[vd] = counter++;
  }

  return read_cones(tm, in, boost::make_assoc_property_map(m), out);
}

/// \ingroup PkgSurfaceMeshParameterizationOrbifoldHelperFunctions
///
/// Same as above, but from a file instead of a stream.
template<typename TriangleMesh, typename VertexIndexMap, typename ConeOutputIterator>
Error_code read_cones(const TriangleMesh& tm, const char* filename, VertexIndexMap vpmap, ConeOutputIterator out)
{
  std::ifstream in(filename);
  return read_cones(tm, in, vpmap, out);
}

/// \ingroup PkgSurfaceMeshParameterizationOrbifoldHelperFunctions
///
/// Same as above, but from a file instead of a stream. The default indexation
/// of the vertices of `tm` is used: vertices are numbered from `0` to `num_vertices(tm)-1`,
/// in the order that they appear while calling `vertices(tm)`.
template<typename TriangleMesh, typename ConeOutputIterator>
Error_code read_cones(const TriangleMesh& tm, const char* filename, ConeOutputIterator out)
{
  std::ifstream in(filename);
  return read_cones(tm, in, out);
}

/// \ingroup PkgSurfaceMeshParameterizationOrbifoldHelperFunctions
///
/// locates the cones on the seam mesh (that is, find the corresponding seam mesh
/// `vertex_descriptor`) and mark them with a tag to indicate whether the cone is a
/// simple cone or a duplicated cone (see \link PkgSurfaceMeshParameterizationEnums Cone_type \endlink).
///
/// \attention The cones must be ordered: the first and last cones are the extremities of the seam.
///
/// \tparam SeamMesh is the same mesh that is passed to the parameterizer. It is an object of type`CGAL::Seam_mesh`,
///                  but is passed here as a template parameter for convenience, to avoid
///                  having to pass the multiple template parameters of the class `CGAL::Seam_mesh`.
/// \tparam ConeInputBidirectionalIterator must be a model of `BidirectionalIterator`
///                  with value type `boost::graph_traits<SeamMesh::Triangle_mesh>::%vertex_descriptor`.
/// \tparam ConeMap must be a model of `AssociativeContainer`
///                 with `boost::graph_traits<SeamMesh>::%vertex_descriptor` as key type and
///                 \link PkgSurfaceMeshParameterizationEnums Cone_type \endlink as value type.
///
/// \param mesh the seam mesh
/// \param first, beyond the range of cones, as vertex descriptors of the base mesh.
/// \param cones an object of type `ConeMap`. Cones will be stored in this container
///              as vertex descriptors of the seam mesh, along with their associated cone types.
template<typename SeamMesh, typename ConeInputBidirectionalIterator, typename ConeMap>
bool locate_cones(const SeamMesh& mesh,
                  ConeInputBidirectionalIterator first, ConeInputBidirectionalIterator beyond,
                  ConeMap& cones)
{
  typedef typename SeamMesh::Triangle_mesh                                 Triangle_mesh;

  typedef typename boost::graph_traits<Triangle_mesh>::vertex_descriptor   TM_vertex_descriptor;
  typedef typename boost::graph_traits<SeamMesh>::vertex_descriptor        vertex_descriptor;

  // property map to go from TM_vertex_descriptor to Point_3
  typedef typename internal::Kernel_traits<Triangle_mesh>::PPM             PM_PPM;
  const PM_PPM pm_ppmap = get(boost::vertex_point, mesh.mesh());

  // property map to go from vertex_descriptor to Point_3
  typedef typename internal::Kernel_traits<SeamMesh>::PPM                  PPM;
  const PPM ppmap = get(boost::vertex_point, mesh);

  for(vertex_descriptor vd : vertices(mesh)) {
    for(ConeInputBidirectionalIterator cit=first; cit!=beyond; ++cit) {
      ConeInputBidirectionalIterator last = (--beyond)++;

      TM_vertex_descriptor smvd = *cit;
      if(get(ppmap, vd) == get(pm_ppmap, smvd)) { // same geometric position
        Cone_type ct;
        if(cit == first)
          ct = First_unique_cone;
        else if(cit == last)
          ct = Second_unique_cone;
        else
          ct = Duplicated_cone;

        cones.insert(std::make_pair(vd, ct));
      }
    }
  }

  return internal::check_cone_validity(mesh, first, beyond, cones);
}

/// \ingroup PkgSurfaceMeshParameterizationOrbifoldHelperFunctions
///
/// Same as above, but the cones are <i>not</i> ordered and we thus use seam mesh
/// information to determine which cones are extremities of the seam (so-called
/// <i>unique cones</i>) or not (so-called <i>duplicate cones</i>).
template<typename SeamMesh, typename ConeInputBidirectionalIterator, typename ConeMap>
bool locate_unordered_cones(const SeamMesh& mesh,
                            ConeInputBidirectionalIterator first, ConeInputBidirectionalIterator beyond,
                            ConeMap& cones)
{
  CGAL_precondition(cones.empty());
  CGAL_precondition(std::distance(first, beyond) == 3 || std::distance(first, beyond) == 4);

  typedef typename SeamMesh::Triangle_mesh                                 Triangle_mesh;

  typedef typename boost::graph_traits<Triangle_mesh>::vertex_descriptor   TM_vertex_descriptor;
  typedef typename boost::graph_traits<SeamMesh>::vertex_descriptor        vertex_descriptor;
  typedef typename boost::graph_traits<SeamMesh>::halfedge_descriptor      halfedge_descriptor;

  // find a vertex on the seam
  vertex_descriptor vertex_on_seam;
  for(vertex_descriptor vd : vertices(mesh)) {
    if(mesh.has_on_seam(vd)) {
      vertex_on_seam = vd;
      break;
    }
  }

  CGAL_assertion(vertex_on_seam != vertex_descriptor());

  // property map to go from TM_vertex_descriptor to Point_3
  typedef typename internal::Kernel_traits<Triangle_mesh>::PPM           PM_PPM;
  const PM_PPM pm_ppmap = get(boost::vertex_point, mesh.mesh());

  // property map to go from vertex_descriptor to Point_3
  typedef typename internal::Kernel_traits<SeamMesh>::PPM                PPM;
  const PPM ppmap = get(boost::vertex_point, mesh);

  bool first_cone_met = false;

  // walk on the seam and mark if we encounter a cone
  vertex_descriptor end = vertex_on_seam;
  do {
    ConeInputBidirectionalIterator current = first;
    for(; current!=beyond; ++current) {
      TM_vertex_descriptor smvd = *current;
      if(get(ppmap, vertex_on_seam) == get(pm_ppmap, smvd)) { // the seam mesh vertex is a cone
        // We have encountered a cone. Must check if the cone is a Unique_cone
        // or a Duplicated_cone.

        // A check is to look at the sources of two halfedges with the same direction
        // on either side of the seam. If the sources are the same, it's a Unique_cone;
        // if they differ, it's a duplicated_cone.

        halfedge_descriptor hd = halfedge(vertex_on_seam, mesh);
        halfedge_descriptor other_hd = opposite(hd, mesh);

        // little trick to go from border halfedge on one side of the seam to
        // interior halfedge on the other side of the seam
        CGAL_assertion(other_hd.seam);
        other_hd.seam = false;

        Cone_type ct;
        if(target(hd, mesh) == source(other_hd, mesh)) {
          if(first_cone_met)
            ct = Second_unique_cone;
          else {
            ct = First_unique_cone;
            first_cone_met = true;
          }
        } else {
          ct = Duplicated_cone;
        }

        cones.insert(std::make_pair(vertex_on_seam, ct));
      }
    }

    // Move to the next vertex_descriptor on the seam
    vertex_on_seam = source(halfedge(vertex_on_seam, mesh), mesh);
    CGAL_assertion(mesh.has_on_seam(vertex_on_seam));

  } while(vertex_on_seam != end);

  return internal::check_cone_validity(mesh, first, beyond, cones);
}

/// \ingroup  PkgSurfaceMeshParameterizationMethods
///
/// The class `Orbifold_Tutte_parameterizer_3` implements <em>Orbifold Tutte Planar
/// Embeddings</em> \cgalCite{aigerman2015orbifold}.
///
/// This is a borderless parameterization. A one-to-one mapping is guaranteed.
///
/// The main function of the class `Orbifold_Tutte_parameterizer_3` is `parameterize()`,
/// to which the user provides a `Seam_mesh` with marked edges (the seams)
/// and a set of vertices of the mesh (the cones). The choice of cones influences
/// the resulting parameterization, but not the choice of the seam path between these cones.
///
/// Some helper functions related to the class `Orbifold_Tutte_parameterizer_3`
/// (for example to read and compute paths between cones) can be found
/// \link PkgSurfaceMeshParameterizationOrbifoldHelperFunctions here \endlink.
///
/// The example \ref Surface_mesh_parameterization/orbifold.cpp "orbifold.cpp"
/// shows how to select cones on the input mesh and automatically construct
/// the seams and the cones on the `Seam_mesh`.
///
/// \attention The global function `CGAL::Surface_mesh_parameterization::parameterize()` cannot be used
/// with this parameterizer. Users should use this class's member function `parameterize()` instead.
///
/// \tparam SeamMesh must be a `Seam_mesh`, with underlying mesh any model of `FaceListGraph` and `HalfedgeListGraph`.
///
/// \tparam SolverTraits_ must be a model of `SparseLinearAlgebraTraits_d`.<br>
///         <b>%Default:</b> If \ref thirdpartyEigen "Eigen" 3.1 (or greater) is available
///         and `CGAL_EIGEN3_ENABLED` is defined, then an overload of `Eigen_solver_traits`
///         is provided as default parameter:
/// \code
///   CGAL::Eigen_solver_traits<
///           Eigen::SparseLU<Eigen_sparse_matrix<double>::EigenType> >
/// \endcode
///         Moreover, if SparseSuite solvers are available, which is greatly preferable for speed,
///         then the default parameter is:
/// \code
///   CGAL::Eigen_solver_traits<
///           Eigen::UmfPackLU<Eigen_sparse_matrix<double>::EigenType> >
/// \endcode
///
/// \sa \ref PkgSurfaceMeshParameterizationOrbifoldHelperFunctions
///
template < typename SeamMesh,
           typename SolverTraits_ = Default>
class Orbifold_Tutte_parameterizer_3
{
public:
#ifndef DOXYGEN_RUNNING
  #if !defined(CGAL_EIGEN3_ENABLED)
  CGAL_static_assertion_msg(!(boost::is_same<SolverTraits_, Default>::value),
                            "Error: You must either provide 'SolverTraits_' or link CGAL with the Eigen library");
  #endif

  typedef typename Default::Get<
    SolverTraits_,
  #if defined(CGAL_EIGEN3_ENABLED)
    #ifdef CGAL_SMP_USE_SPARSESUITE_SOLVERS
      CGAL::Eigen_solver_traits<
        Eigen::UmfPackLU<Eigen_sparse_matrix<double>::EigenType> >
    #else
      CGAL::Eigen_solver_traits<
        Eigen::SparseLU<Eigen_sparse_matrix<double>::EigenType> >
    #endif
  #else
    SolverTraits_ // no parameter provided, and Eigen is not enabled: so don't compile!
  #endif
  >::type                                                     Solver_traits;
#else
  /// Solver traits type
  typedef SolverTraits_                                       Solver_traits;
#endif

  /// Mesh halfedge type
  typedef typename boost::graph_traits<SeamMesh>::halfedge_descriptor  halfedge_descriptor;

private:
  typedef typename boost::graph_traits<SeamMesh>::vertex_descriptor    vertex_descriptor;
  typedef typename boost::graph_traits<SeamMesh>::face_descriptor      face_descriptor;

  typedef typename boost::graph_traits<SeamMesh>::vertex_iterator      vertex_iterator;
  typedef typename boost::graph_traits<SeamMesh>::face_iterator        face_iterator;

  // Solver traits subtypes:
  typedef typename Solver_traits::Vector                               Vector;
  typedef typename Solver_traits::Matrix                               Matrix;

  // Kernel subtypes
  typedef typename internal::Kernel_traits<SeamMesh>::Kernel        Kernel;
  typedef typename internal::Kernel_traits<SeamMesh>::PPM           PPM;
  typedef typename Kernel::FT                                       NT;
  typedef typename Kernel::Vector_2                                 Vector_2;
  typedef typename Kernel::Vector_3                                 Vector_3;
  typedef typename Kernel::Point_2                                  Point_2;
  typedef typename Kernel::Point_3                                  Point_3;

  const Orbifold_type orb_type;
  const Weight_type weight_type;

private:
  // Check input's correctness.
  template<typename ConeMap>
  Error_code check_cones(ConeMap cmap) const
  {
    if(orb_type == Parallelogram) {
      if(cmap.size() != 6) {
        std::cerr << "Using orb_type '" << get_orbifold_type(orb_type)
                  << "' requires 4 vertices marked as cones (thus 6 in the seam mesh)" << std::endl;
        std::cerr << "currently: " << cmap.size() << std::endl;
        return ERROR_WRONG_PARAMETER;
      }
    } else if(cmap.size() != 4){ // orb_type == Square, Diamond, Triangle
      std::cerr << "Using orb_type '" << get_orbifold_type(orb_type)
                << "' requires 3 vertices marked as cones (thus 4 in the seam mesh)" << std::endl;
      std::cerr << "currently: " << cmap.size() << std::endl;
      return ERROR_WRONG_PARAMETER;
    }

    return OK;
  }

  // Compute the number of linear constraints in the system.
  int number_of_linear_constraints(const SeamMesh& mesh) const
  {
    if(orb_type == Parallelogram) {
      // number of constraints for orb I, II, III is the number of seam edges
      // and 3 constrained cones.
      return 3 + static_cast<int>(mesh.number_of_seam_edges());
    }
    else { // orb_type == Square, Diamond, Triangle
      // number of constraints for orb I, II, III is the number of seam edges
      // and 2 constrained cones.
      return 2 + static_cast<int>(mesh.number_of_seam_edges());
    }
  }

  // Adds a positional constraint on a vertex x_ind, so that x_ind * w = rhs.
  void addConstraint(Matrix& M, Vector& B, int& id_r, int id_c, double w, Point_2 rhs) const
  {
    M.set_coef(2*id_r, 2*id_c, w, true /*new_coef*/);
    M.set_coef(2*id_r + 1, 2*id_c + 1, w, true /*new_coef*/);

    // Since we are filling the big system M.Xf = B with
    // ( L A' ) ( Xf ) = ( C )
    // ( A 0  ) ( Xf ) = ( 0 )
    // we already add the transposed here:
    M.set_coef(2*id_c, 2*id_r, w, true /*new_coef*/);
    M.set_coef(2*id_c + 1, 2*id_r + 1, w, true /*new_coef*/);

    B[2*id_r] = rhs.x();
    B[2*id_r + 1] = rhs.y();

    ++id_r; // current line index in A is increased
  }

  // Adds constraints so that T * x_sinds = x_tinds, where T is a 2x2
  // matrix, and the Transformation T is modified to affine from
  // linear by requiring that T * x_si - x_ti = T * x_s1 - x_t1.
  void addTransConstraints(int s0, int t0, int s, int t,
                           int& id_r,
                           const std::vector<double>& T,
                           Matrix& M, Vector& B) const
  {
    // Everything is duplicated since we are filling the big system M.Xf = B with
    // ( L A' ) ( Xf ) = ( C )
    // ( A 0  ) ( Xf ) = ( 0 )

    // Iterate on both rows ot the 2x2 matrix T
    for(int vert_ind=0; vert_ind<2; ++vert_ind) {
      // building up the equations by summing up the terms

      // Matlab lines are commented for comparison.
      // Matlab fills together 2*x-1 and 2*x, but C++ fills 2*x and 2*x+1,
      // as everything (including loops!) starts at 0 and not 1.

      // <T(vert_ind,:), x_si>
      // obj.A(end+1, 2*sinds(ind)+[-1,0]) = T(vert_ind,:);
      M.set_coef(2*id_r + vert_ind, 2*s, T[2 * vert_ind], true /*new_coef*/);
      M.set_coef(2*id_r + vert_ind, 2*s + 1, T[2 * vert_ind + 1], true /*new_coef*/);

      M.set_coef(2*s, 2*id_r + vert_ind, T[2 * vert_ind], true /*new_coef*/);
      M.set_coef(2*s + 1, 2*id_r + vert_ind, T[2 * vert_ind + 1], true /*new_coef*/);

      // -<T(vert_ind,:), x_s1>
      // obj.A(end, 2*sinds(1)+[-1,0]) = obj.A(end, 2*sinds(1)+[-1,0]) - T(vert_ind,:);
      M.add_coef(2*id_r + vert_ind, 2*s0, - T[2 * vert_ind]);
      M.add_coef(2*id_r + vert_ind, 2*s0 + 1, - T[2 * vert_ind + 1]);

      M.add_coef(2*s0, 2*id_r + vert_ind, - T[2 * vert_ind]);
      M.add_coef(2*s0 + 1, 2*id_r + vert_ind, - T[2 * vert_ind + 1]);

      //  - x_ti
      // obj.A(end, 2*tinds(ind)+vert_ind-2) = obj.A(end, 2*tinds(ind)+vert_ind-2)-1;
      M.add_coef(2*id_r + vert_ind, 2*t + vert_ind, -1);

      M.add_coef(2*t + vert_ind, 2*id_r + vert_ind, -1);

      // + x_t1
      // obj.A(end, 2*tinds(1)+vert_ind-2) = obj.A(end, 2*tinds(1)+vert_ind-2)+1;
      M.add_coef(2*id_r + vert_ind, 2*t0 + vert_ind, 1);

      M.add_coef(2*t0 + vert_ind, 2*id_r + vert_ind, 1);

      // left hand side is zero
      // obj.b=[obj.b; 0];
      B[2*id_r + vert_ind] = 0;
    }

    ++id_r; // current line index in M is increased
  }

  // Add the constraints from a seam segment to the linear system.
  void constrain_seam_segment(const std::vector<std::pair<int, int> >& seam_segment,
                              NT ang, int& current_line_id_in_M,
                              Matrix& M, Vector& B) const
  {
    // check that if there is a common vertex, it is at the beginning
    const bool is_reversed = (seam_segment.back().first == seam_segment.back().second);

    if(is_reversed) {
      ang *= -1;
    }

    // The rotation matrix according to the angle 'ang'. Put in a vector
    // because we need to access it later and Matrix does not provide read access...
    std::vector<double> R(4);
    R[0] = std::cos(2 * CGAL_PI / ang);
    R[1] = - std::sin(2 * CGAL_PI / ang);
    R[2] = std::sin(2 * CGAL_PI / ang);
    R[3] = std::cos(2 * CGAL_PI / ang);

    const int s0 = is_reversed ? seam_segment.back().first : seam_segment.front().first;
    const int t0 = is_reversed ? seam_segment.back().second : seam_segment.front().second;

    typename std::vector<std::pair<int, int> >::const_iterator it = seam_segment.begin(),
                                                               end = seam_segment.end();

    // ignore the first entry of the seam segment (they correspond to a constrained point)
    if(is_reversed)
      --end;
    else
      ++it;

    for(; it!=end; ++it) {
      const int s = it->first;
      const int t = it->second;
      CGAL_assertion(s != t);

      // sending s to t (and _not_ t to s !)
      addTransConstraints(t0, s0, t, s, current_line_id_in_M, R, M, B);
    }
  }

  // Computes the rotational constraint on the border of the mesh.
  // Cone constraints are also added.
  template<typename ConeMap,
           typename VertexIndexMap>
  void AddRotationalConstraint(const SeamMesh& mesh,
                               const ConeMap& cmap,
                               VertexIndexMap vimap,
                               Matrix& M, Vector& B) const
  {
    // positions of the cones in the plane
    typedef std::vector<Point_2>                      Point_container;
    const Point_container& tcoords =
      internal::get_cones_parameterized_coordinates<Point_container>(orb_type);

    // angles at the cones
    typedef std::vector<NT>                           Angle_container;
    const Angle_container& angs = internal::get_angles_at_cones<Angle_container>(orb_type);

    // The index of the line in M that we are filling next.

    // Since we are filling the big system M.Xf = B with
    // ( L A' ) ( Xf ) = ( C )
    // ( A 0  ) ( Xf ) = ( 0 )
    // we do not start at 0, but at the first line below the matrix L
    // (note that this should thus be 2*num_vertices, but in the filling functions
    // we use 2*line_number to fill two at the time...)
    int current_line_id_in_M = static_cast<int>(num_vertices(mesh));
    CGAL_postcondition_code(int initial_line_id = current_line_id_in_M;)

    // Initialize some variables used in the seam walk
    int start_cone_index = -1; // index of the beginning of the seam
    vertex_descriptor start_cone;
    internal::find_start_cone(cmap, vimap, start_cone, start_cone_index);
    CGAL_postcondition(start_cone != vertex_descriptor() && start_cone_index != -1);

    // parameterize the initial cone
    addConstraint(M, B, current_line_id_in_M, start_cone_index,
                  1. /*entry in M*/, tcoords[0]);

    // by property of the seam mesh, the canonical halfedge that points to start_cone
    // is on the seam, and is not on the border
    const halfedge_descriptor hd = halfedge(start_cone, mesh);
    CGAL_precondition(mesh.has_on_seam(hd));
    halfedge_descriptor bhd = opposite(hd, mesh);
    CGAL_precondition(is_border(bhd, mesh));

    // points between two cones, and the corresponding points on the opposite side of the seam
    std::vector<std::pair<int, int> > seam_segment;
    std::size_t segment_index = 0; // counting the segments (3 max)

    // Go through the seam, marking rotation and cone constraints
    while(true) { // breaking at the last cone
      // Get the two halfedges on each side of the stream
      const halfedge_descriptor hd1 = bhd; // only for clarity

      // the non-border halfedge with same vertices (in the underlying mesh of the seam
      // mesh) as bhd is simply bhd with the 'seam' boolean set to false
      const halfedge_descriptor hd2(bhd, false /* not on seam*/);

      // Compute the corresponding indices
      const vertex_descriptor hd1_source = source(hd1, mesh);
      const vertex_descriptor hd2_source = source(hd2, mesh);
      const int hd1s_index = get(vimap, hd1_source);
      const int hd2s_index = get(vimap, hd2_source);

      // If orbifold type IV and it is second cone in flattening, add constraint
      if(orb_type == Parallelogram && cmap.find(hd1_source) != cmap.end()
                                   && segment_index == 1) {
        addConstraint(M, B, current_line_id_in_M, hd1s_index,
                      1. /*entry in M*/, tcoords[1]);
      }

      // Add the pair to the seam segment
      seam_segment.push_back(std::make_pair(hd1s_index, hd2s_index));

      // Check if we have reached a cone
      const vertex_descriptor bhd_target = target(bhd, mesh);
      typename ConeMap::const_iterator is_in_map = cmap.find(bhd_target);
      if(is_in_map != cmap.end()) {
        // add the target to finish the seam segment
        const vertex_descriptor hd1_target = target(hd1, mesh);
        const vertex_descriptor hd2_target = target(hd2, mesh);
        const int hd1t_index = get(vimap, hd1_target);
        const int hd2t_index = get(vimap, hd2_target);

        seam_segment.push_back(std::make_pair(hd1t_index, hd2t_index));

        CGAL_assertion(segment_index < angs.size());
        NT ang = angs[segment_index];
        constrain_seam_segment(seam_segment, ang, current_line_id_in_M, M, B);

        // Check if we have reached the end of the seam
        if(is_in_map->second == Second_unique_cone) {
          CGAL_assertion(hd1_target == hd2_target);
        // the last cone of the seam is constrained
          addConstraint(M, B, current_line_id_in_M, hd1t_index,
                        1. /*entry in M*/, tcoords.back());
          break;
        }

        seam_segment.clear();
        ++segment_index;
      }

      // move to the next halfedge couple (walking on the border of the seam)
      bhd = next(bhd, mesh);
      CGAL_postcondition(mesh.has_on_seam(bhd) && is_border(bhd, mesh));
    }

    CGAL_postcondition(current_line_id_in_M - initial_line_id == number_of_linear_constraints(mesh));
  }

  // Computes the coefficients of the mean value Laplacian matrix for the edge.
  // `ij` in the face `ijk`
  void fill_mvc_matrix(const Point_3& pi, int i,
                       const Point_3& pj, int j,
                       const Point_3& pk, int k, Matrix& M) const
  {
    // For MVC, the entry of M(i,j) is - [ tan(gamma_ij/2) + tan(delta_ij)/2 ] / |ij|
    // where gamma_ij and delta_ij are the angles at i around the edge ij

    // This function computes the angle alpha at i, and add
    // -- M(i,j) += tan(alpha / 2) / |ij|
    // -- M(i,k) += tan(alpha / 2) / |ik|
    // -- M(i,i) -= M(i,j) + M(i,k)

    // The other parts of M(i,j) and M(i,k) will be added when this function
    // is called from the neighboring faces of F_ijk that share the vertex i

    // @fixme unefficient: lengths are computed (and inversed!) twice per edge

    // Set w_i_base: - tan(alpha / 2)
    const Point_3& p = pk;
    const Point_3& q = pi;
    const Point_3& r = pj;
    const CGAL::Weights::Tangent_weight<NT> tangent_weight(p, q, r);

    // Set w_ij in matrix
    const NT w_ij = tangent_weight.get_w_r();
    M.add_coef(2*i, 2*j, w_ij);
    M.add_coef(2*i + 1, 2*j + 1, w_ij);

    // Set w_ik in matrix
    const NT w_ik = tangent_weight.get_w_p();
    M.add_coef(2*i, 2*k, w_ik);
    M.add_coef(2*i + 1, 2*k + 1, w_ik);

    // Add to w_ii (w_ii = - sum w_ij)
    const NT w_ii = - w_ij - w_ik;

    M.add_coef(2*i, 2*i, w_ii);
    M.add_coef(2*i + 1, 2*i + 1, w_ii);
  }

  // Compute the mean value Laplacian matrix.
  template<typename VertexIndexMap>
  void mean_value_laplacian(const SeamMesh& mesh,
                            VertexIndexMap vimap,
                            Matrix& M) const
  {
    const PPM ppmap = get(vertex_point, mesh);

    for(face_descriptor fd : faces(mesh)) {
      const halfedge_descriptor hd = halfedge(fd, mesh);

      const vertex_descriptor vd_i = target(hd, mesh);
      const vertex_descriptor vd_j = target(next(hd, mesh), mesh);
      const vertex_descriptor vd_k = source(hd, mesh);
      const Point_3& pi = get(ppmap, vd_i);
      const Point_3& pj = get(ppmap, vd_j);
      const Point_3& pk = get(ppmap, vd_k);
      const int i = get(vimap, vd_i);
      const int j = get(vimap, vd_j);
      const int k = get(vimap, vd_k);

      fill_mvc_matrix(pi, i, pj, j, pk, k, M);
      fill_mvc_matrix(pj, j, pk, k, pi, i, M);
      fill_mvc_matrix(pk, k, pi, i, pj, j, M);
    }
  }

  // Compute the system weights using a Cotangent Laplacian.
  template<typename VertexIndexMap>
  void cotangent_laplacien(SeamMesh& mesh,
                           VertexIndexMap vimap,
                           Matrix& M) const
  {
    const PPM pmap = get(vertex_point, mesh);
    for (const halfedge_descriptor hd : halfedges(mesh)) {

      const vertex_descriptor vi = source(hd, mesh);
      const vertex_descriptor vj = target(hd, mesh);
      const int i = get(vimap, vi);
      const int j = get(vimap, vj);

      if (i > j) continue;
      const CGAL::Weights::Cotangent_weight<SeamMesh> cotangent_weight;
      const NT w_ij = NT(2) * cotangent_weight(hd, mesh, pmap);

      // ij
      M.set_coef(2*i, 2*j, w_ij, true /* new coef */);
      M.set_coef(2*i + 1, 2*j + 1, w_ij, true /* new coef */);

      // ji
      M.set_coef(2*j, 2*i, w_ij, true /* new coef */);
      M.set_coef(2*j + 1, 2*i + 1, w_ij, true /* new coef */);

      // ii
      M.add_coef(2*i, 2*i, - w_ij);
      M.add_coef(2*i + 1, 2*i + 1, - w_ij);

      // jj
      M.add_coef(2*j, 2*j, - w_ij);
      M.add_coef(2*j + 1, 2*j + 1, - w_ij);
    }
  }

  // Copy the solution into the UV property map.
  template <typename VertexIndexMap, typename VertexUVMap>
  void assign_solution(const SeamMesh& mesh,
                       const Vector& X,
                       VertexUVMap uvmap,
                       const VertexIndexMap vimap) const
  {
    CGAL_assertion(X.dimension() == static_cast<int>(2 * num_vertices(mesh)));

    for(vertex_descriptor vd : vertices(mesh)) {
      const int index = get(vimap, vd);
      const NT u = X(2*index);
      const NT v = X(2*index + 1);

      put(uvmap, vd, Point_2(u, v));
    }
  }

  // Solves the linear system.
  template <typename VertexUVMap,
            typename VertexIndexMap>
  Error_code computeFlattening(const SeamMesh& mesh,
                               const Matrix& M, const Vector& B,
                               VertexUVMap uvmap, VertexIndexMap vimap) const
  {
    CGAL_precondition(M.row_dimension() == M.column_dimension());
    CGAL_precondition(M.row_dimension() == B.dimension());

    const int big_n = M.row_dimension();
    const std::size_t n = 2 * num_vertices(mesh);

    NT D;
    Vector Xf(big_n);

    CGAL::Timer task_timer;
    task_timer.start();

    Solver_traits solver;
    if(!solver.linear_solver(M, B, Xf, D)) {
      std::cerr << "Could not solve linear system" << std::endl;
      return ERROR_CANNOT_SOLVE_LINEAR_SYSTEM;
    }
    CGAL_assertion(D == 1.0);

    Vector X(n);
    for(std::size_t i=0; i<n; ++i) {
      X[i] = Xf[i];
    }

#ifdef CGAL_SMP_ORBIFOLD_DEBUG
    std::ofstream outf("matrices/X.txt");
    for(std::size_t i=0; i<n; ++i) {
      outf << X[i] << " ";
      outf << X[++i] << '\n';
    }
#endif

    assign_solution(mesh, X, uvmap, vimap);
    return OK;
  }

public:
  /// computes a one-to-one mapping from a triangular 3D surface mesh
  /// to a piece of the 2D space.
  /// The mapping is piecewise linear (linear in each triangle).
  /// The result is the (u,v) pair image of each vertex of the 3D surface.
  ///
  /// \tparam ConeMap must be a model of `AssociativeContainer`
  ///                 with key type `boost::graph_traits<Seam_mesh>::%vertex_descriptor` and
  ///                 \link PkgSurfaceMeshParameterizationEnums Cone_type \endlink as value type.
  /// \tparam VertexUVmap must be a model of `ReadWritePropertyMap` with
  ///         `boost::graph_traits<Seam_mesh>::%vertex_descriptor` as key type and
  ///         %Point_2 (type deduced from `Seam_mesh` using `Kernel_traits`)
  ///         as value type.
  /// \tparam VertexIndexMap must be a model of `ReadablePropertyMap` with
  ///         `boost::graph_traits<Seam_mesh>::%vertex_descriptor` as key type and
  ///         a unique integer as value type.
  ///
  /// \param mesh a `Seam_mesh` parameterized by any model of a `FaceListGraph` and `HalfedgeListGraph`
  /// \param bhd a halfedge on the border of the seam mesh
  /// \param cmap a mapping of the `vertex_descriptor`s of `mesh` that are cones
  ///             to their respective \link PkgSurfaceMeshParameterizationEnums Cone_type \endlink
  ///             classification.
  /// \param uvmap an instanciation of the class `VertexUVmap`.
  /// \param vimap an instanciation of the class `VertexIndexMap`.
  ///
  /// \pre `mesh` must be a triangular mesh.
  /// \pre The underlying mesh of `mesh` is a topological ball.
  /// \pre The vertices must be indexed (vimap must be initialized).
  /// \pre The cones are vertices of `mesh` and their number is adapted to
  ///      the orbifold type (4 for types I, II or III and 6 for type IV).
  /// \pre The seam edges form a set of segments that contains the different cones,
  ///      and starts and ends at two different cones.
  /// \pre The seam edges form a set of segments that is homotopic to a
  ///      line. Specifically, the paths between cones must not self intersect
  ///      or intersect other paths (see Figure below).
  ///
  /// \cgalFigureBegin{Surface_mesh_parameterizationfigorbifold, orbifold_path.svg}
  /// Invalid (left) and valid (right) seam paths. Seam edges are drawn in teal.
  /// Cones are marked in yellow and blue.
  /// \cgalFigureEnd
  template<typename ConeMap,
           typename VertexIndexMap,
           typename VertexUVMap>
  Error_code parameterize(SeamMesh& mesh,
                          halfedge_descriptor bhd,
                          const ConeMap& cmap,
                          VertexUVMap uvmap,
                          VertexIndexMap vimap) const
  {
    CGAL_precondition(is_valid_polygon_mesh(mesh));
    CGAL_precondition(is_triangle_mesh(mesh));
    CGAL_USE(bhd);

    Error_code status;

    status = check_cones(cmap);
    if(status != OK) {
      return status;
    }

    const int lcn = number_of_linear_constraints(mesh);
    const int nbVertices = static_cast<int>(num_vertices(mesh));

    // To avoid concatenating matrices, we will write everything directly
    // in the big system that we will (eventually solve). It is:

    //  M.Xf = B with
    // ( L A' ) ( Xf ) = ( C )
    // ( A 0  ) ( Xf ) = ( 0 )

    // where:
    // -- L is a (2 * nbVertices, 2 * nbVertices) Laplacian matrix
    // -- A is a (2 * lcn,        2 * nbVertices) constraint matrix
    // -- C is a (2 * lcn) right hand side vector

    // Thus M is a (2 * (nbVertices + lcn), 2 * (nbVertices + lcn)) matrix
    const int total_size = 2 * (lcn + nbVertices);
    Matrix M(total_size, total_size);
    Vector B(total_size);

    // %%%%%%%%%%%%%%%%%%%%%%%
    //   Boundary conditions
    // %%%%%%%%%%%%%%%%%%%%%%%

    // add rotational constraints
    AddRotationalConstraint(mesh, cmap, vimap, M, B);

    // %%%%%%%%%%%%%%%%%%%%
    //  Energy (Laplacian)
    // %%%%%%%%%%%%%%%%%%%%
    if(weight_type == Cotangent)
      cotangent_laplacien(mesh, vimap, M);
    else // weight_type == Mean_value
      mean_value_laplacian(mesh, vimap, M);

#ifdef CGAL_SMP_ORBIFOLD_DEBUG
    std::ofstream outM("matrices/M.txt");
    outM.precision(20);
    outM << total_size << " " << total_size << std::endl;

  #ifdef CGAL_EIGEN3_ENABLED
    for(int k=0; k<M.eigen_object().outerSize(); ++k) {
      for(typename Eigen::SparseMatrix<double>::InnerIterator
                                            it(M.eigen_object(), k); it; ++it) {
        outM <<  it.row() << " " << it.col() << " " << it.value() << '\n';
      }
    }
  #else // !CGAL_EIGEN3_ENABLED
    for(int k=0; k<M.row_dimension(); k++)
      for(int l=0; l<M.column_dimension(); l++)
        outM << k << " " << l << " " << M.get_coef(k, l) << '\n';
  #endif

    std::ofstream outB("matrices/B.txt");
    outB.precision(20);
    outB << total_size << std::endl;
    outB << B << std::endl;
#endif // CGAL_SMP_ORBIFOLD_DEBUG

    // compute the flattening by solving the boundary conditions
    // while satisfying the convex combination property with L
    status = computeFlattening(mesh, M, B, uvmap, vimap);
    if(status != OK)
      return status;

#ifdef CGAL_SMP_ORBIFOLD_DEBUG
    std::ofstream out("orbifold_result.off");
    IO::output_uvmap_to_off(mesh, bhd, uvmap, out);
#endif

    return OK;
  }

public:
  /// Constructor of the parameterizer. The arguments allow to select
  /// the desired orbifold and weight types.
  Orbifold_Tutte_parameterizer_3(const Orbifold_type orb_type = Square,
                                 const Weight_type weight_type = Cotangent)
    :
      orb_type(orb_type),
      weight_type(weight_type)
  { }
};

} // namespace Surface_mesh_parameterization

} // namespace CGAL

#include <CGAL/enable_warnings.h>

#endif // CGAL_SURFACE_MESH_PARAMETERIZATION_ORBIFOLD_TUTTE_PARAMETERIZER_3_H
