Dual mesh

[stla]

Hello,

Starting with the code found here I implemented the dual mesh with CGAL. It could be a nice example for the doc.

// centroid of a face of the mesh
EPoint3 centroid(EMesh3& mesh, EMesh3::Face_index f) {
  EK::FT x = 0;
  EK::FT y = 0;
  EK::FT z = 0;
  int nvertices = 0;
  for(EMesh3::Vertex_index v :
           vertices_around_face(mesh.halfedge(f), mesh)) {
    nvertices++;
    EPoint3 p = mesh.point(v);
    x += p.x();
    y += p.y();
    z += p.z();
  }
  return EPoint3(x / nvertices, y / nvertices, z / nvertices);
}

EMesh3 dualMesh(EMesh3& mesh) {
  EMesh3 dualmesh;
  // property to remember new vertices per face
  auto fvertex = 
    mesh.add_property_map<EMesh3::Face_index, EMesh3::Vertex_index>(
      "f:vertex", EMesh3::null_vertex()).first;
  // for each face add its centroid to the dual mesh
  for(EMesh3::Face_index f : mesh.faces()) {
    fvertex[f] = dualmesh.add_vertex(centroid(mesh, f));
  }
  // add new face for each vertex
  for(EMesh3::Vertex_index v : mesh.vertices()) {
    std::vector<EMesh3::Vertex_index> vs;
    for(EMesh3::Face_index f : faces_around_target(mesh.halfedge(v), mesh)) {
      vs.push_back(fvertex[f]);
    }
    dualmesh.add_face(vs);
  }
  return dualmesh;
}

Icosphere and its dual mesh:

[stla]

Hmm but this definition of the dual polyhedron is wrong, it yields non-coplanar faces. It should be right for Platonic solids, perhaps.