Get a mesh_3 'off' from Projection_traits_xy_3 constrained delaunay - c++

Question

I've computed a 2D constrained delaunay triangulation from 2.5D data using the projection_traits_xy_3 [1]. Now I would like to get a mesh that I can visualize.
I've managed to do that with 3d delaunay following the manual[2], how could I achieve it with a 2.5D CDT ?
[...]
typedef CGAL::Projection_traits_xy_3<K> Gt;
typedef CGAL::Constrained_Delaunay_triangulation_2<Gt, Tds> CDT;
[...]
CDT cdt;
cdt.insert(points.begin(),points.end());
[...]
¿?
[...]
std::ofstream out(outdir + "out.off");
Polyhedron output_mesh;
CGAL::output_surface_facets_to_polyhedron(¿?, output_mesh);
out << output_mesh;
[1] http://pastebin.com/HzAwrnW5
[2] http://doc.cgal.org/latest/Point_set_processing_3/index.html#chappoint_set_processing_3
http://doc.cgal.org/latest/Surface_reconstruction_points_3/

Answer 1

Here comes pseudocode forwriting it into an off file
cout << "OFF\n" << cdt.number_of_vertices()
<< " " << cdt.number_of_faces() << " 0" << std::endl;
std::map<vertex_handle,int> indices;
int counter = 0;
for all finite vertices v {
cout << v->point() <<std::endl;
indices.insert(v, counter++);
}
for all finite faces f {
cout << "3 " << indices[f->vertex(0)]
<< " " << indices[f->vertex(1)]
<< " " << indices[f->vertex(2)] << std::endl;
}

Answer 2

From #Andreas suggestion:
Here comes the code for writing it into an off file
std::ofstream outstream("output.off");
outstream << "OFF\n" << cdt.number_of_vertices()
<< " " << cdt.number_of_faces() << " 0" << std::endl;
std::map<CDT::Vertex_handle,int> indices;
int counter = 0;
for(CDT::Finite_vertices_iterator it = cdt.finite_vertices_begin(); it != cdt.finite_vertices_end(); ++it)
{
outstream << it->point() << std::endl;
indices.insert(std::pair<CDT::Vertex_handle,int>(it, counter++));
}
for(CDT::Finite_faces_iterator it = cdt.finite_faces_begin(); it != cdt.finite_faces_end(); ++it)
{
outstream << "3 " << indices[it->vertex(0)]
<< " " << indices[it->vertex(1)]
<< " " << indices[it->vertex(2)] << std::endl;
}