Coordinates for Polyhedra

• This page explains how the coordinates for objects of type Polytope<Coord> in application polytope work.

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.

As described above polyhedra in polymake are modelled as the intersection of a cone with the affine hyperplane defined by x₀=1. Hence, infinitely many cones give rise to the same polytope. The algorithms in polymake usually work with the homogenized cone homog(P) of a polyhedron. Hence, polymake takes care about the correct canonicalization of user input of polytope generators in the following way:
In order to construct homog(P), the cone defining the polyhedron is intersected with the hyperplane H₀: x₀=0. The rays defining the bounded part (R_b) and rays with x₀=0 (R₀) are just inherited. To obtain the rest of the generators for the unbounded part, it is necessary to carry out a “dual Fourier-Motzkin procedure”: Any two rays with different signs are linearly combined to a new ray that is contained in H₀. All these rays together with the rays in R_b and R₀ then define the homogenized cone homog(P).