Each polyhedron can be written as the Minkowski sum of a convex polytope (spanned by points a,b,c,...) and a cone (generated by the rays r,s,t,...). If the polyhedron is bounded, i.e. it is a polytope, then the cone is empty.

Suppose our polyhedron lives in a d-space V.

In order to obtain a unified view on the polytope and the cone section of a polyhedron, we embed V as an affine subspace of a (d+1)-space W such that the image of V contains the point (1,0,0,...,0) and it is parallel to the subspace spanned by the last d unit vectors.

Points from both sections can now be identified with infinite rays through the origin in W. Facets are identified with the a hyperplane containing the image of the facet in V and the origin in W. This hyperplane is represented by a normal vector.

Note that a facet defining hyperplane is not uniquely determined if the polyhedron is not full-dimensional.

A vertex is incident with a facet if and only if the scalar product of their representatives in W is zero.

The polytope point a=(ay,az) is modelled as the infinite ray from the origin (0,0,0) through the point (1,ay,az), i.e. the set of all non-negative multiples of (1,ay,az). The cone point r=(ry,rz) becomes the ray through (0,ry,rz).

The facet containing the points a and c is represented by an (oriented) normal vector of the (hyper-)plane spanned by a, c, and the origin. If d=2, as in the picture, the normal vector can be computed as the cross product of a and c. The normal vector will be oriented such that it points towards the interior of the polyhedron.

According to this model two points in W are identical to `polymake` if they differ by a positive multiple. In particular, for a polytope point in the input data it is not required that the first coordinate is 1; it just has to be some positive number.

`polymake` is not able to handle unbounded polyhedra which contain an affine line. The combinatorial structure of such not pointed polyhedra cannot be embedded into the combinatorial structure of some bounded polytope. A polyhedron is not of this kind if and only if there is a linear transformation of W that maps the polyhedron to a bounded polytope. In this case there is a unique minimal representation of the polyhedron as the Minkowski sum of a polytope and a cone. `polymake` treats all the points (= rays) of this minimal representation as vertices.