Let me define each half space i as:


$${H\_i}:{c\_i}{\bf{x}} \le {b\_i}$$
The intersection of all such ${H\_i}$ gives a polyhedron (bounded or not). Suppose I am interested in if ${H\_i}$ is active (corresponding to a facet) in such polyhedron and if so, what is all of its vertices and rays, since we know that such each facet in a polyhedron must be in the form of
$$Conv(V) + Rays(R)$$
for some set of vertices $V$ and some sets of $Rays$. Is there any well established codes (like qhull) that handles this type of problems well? Thank you.



algorithms and tools available for a particular polytope computation
I've had luck with [polymake](https://polymake.org) and I think its basic functions can easily do all you ask for in your question. I think [this tutorial page](https://polymake.org/doku.php/tutorial/apps_polytope) covers all you want (except getting the rays -- for that [this page](https://polymake.org/doku.php/tutorial/face_lattice_tutorial) should help).


