CGAL: Why is halfplane represented by six rays?

Question

I've just started playing with Nef polyhedrons on the plane - the simple program below creates a halfplane, defined by a line y=0, and then this halfplane is explored by the CGAL Explorer.

#include <iostream>

#include <CGAL/Exact_integer.h>
#include <CGAL/Extended_cartesian.h>
#include <CGAL/Nef_polyhedron_2.h>

using Kernel = CGAL::Extended_cartesian<CGAL::Exact_integer>;
using Polyhedron = CGAL::Nef_polyhedron_2<Kernel>;
using Line = Polyhedron::Line;

using std::cout;
using std::endl;

int main()
{
  const Polyhedron p(Line(0, 1, 0), Polyhedron::INCLUDED);  
  const auto ex = p.explorer();
  for (auto it = ex.vertices_begin(); it != ex.vertices_end(); ++it)
  {
    if (ex.is_standard(it))
    {
      cout << "Point: " << ex.point(it) << endl;
    }
    else
    {
      cout << "Ray:   " << ex.ray(it) << endl;
    }
  }
}

The program output:

Ray:   0 0 -1 -1
Ray:   0 0 -1 0
Ray:   0 0 -1 1
Ray:   0 0 1 -1
Ray:   0 0 1 0
Ray:   0 0 1 1

Why these six rays?

Answer 1

I'm answering my own question. According to these explanations from the CGAL online manual, each 2D polyhedron is bounded by an infinitely large frame, which is represented by four infinitely remoted vertices. These boundary vertices have extended coordinates (+infinity, +infinity), (+infinity, -infinity), (-infinity, +infinity) and (-infinity, -infinity). Such non-standard vertices in CGAL are represented by rays - for example, the point (+infinity, -infinity) is stored as a ray with beginning in the origin (0,0) and direction (1,-1).

So, a polyhedron, consisting of the single halfplane y>0, will have six non-standard vertices - four ones will belong to the frame, plus two ones, describing the line y=0. All its faces will look like this:

face 0, marked by 0
* no outer face cycle

face 1, marked by 0
* outer face cycle:
frame halfedge:    (0 0 -1 0) => (0 0 -1 -1)
frame halfedge:    (0 0 -1 -1) => (0 0 1 -1)
frame halfedge:    (0 0 1 -1) => (0 0 1 0)
internal halfedge: (0 0 1 0) => (0 0 -1 0)

face 2, marked by 1
* outer face cycle:
frame halfedge:    (0 0 -1 1) => (0 0 -1 0)
internal halfedge: (0 0 -1 0) => (0 0 1 0)
frame halfedge:    (0 0 1 0) => (0 0 1 1)
frame halfedge:    (0 0 1 1) => (0 0 -1 1)

Also please see the Figure 17.3 from the CGAL online manual.