Points such that the polyhedron is their convex hull.
Redundancies are allowed.
Vector (x0 x1 .. xd) represents a point in (d+1)-space (homogeneous coordinates.)
Affine points are identified by x0 > 0.
Points with x0 = 0 can be interpreted as rays.
polymake automatically normalizes each coordinate vector, dividing them by the first non-zero element.
The clients and rule subroutines can always assume that x0 is either 0 or 1.
Dual to INEQUALITIES.
Input section only. Ask for VERTICES if you want to compute a V-representation from an H-representation.
The i-th row is the normal vector of a hyperplane separating the i-th vertex from the rest ones.
This property is a by-product of redundant point elimination algorithm.
Dimension of the affine hull of the polyhedron = dimension of the polyhedron.
If the polytope is given purely combinatorically, it's the dimension of a minimal embedding space
deduced from the combinatorial structure.
Graph of the bounded subcomplex. Additionally each edge indicates the maximal dimension of a bounded
face containing it. Mainly used for visualization purposes.
Some invertible linear transformation that can be used to get back a previous coordinate repersentation of the polytope.
It operates from the right on point row vectors (e.g. in properties like POINTS,VERTICES,REL_INT_POINT);
its inverse operates from the left on hyperplane column vectors.
Unique names assigned to the VERTICES.
If specified, they are shown by visualization tools instead of vertex indices.
For a polytope build from scratch, you should create this property by yourself,
either manually in a text editor, or with a client program. If you build a polytope with a construction client
taking some other input polytope(s), you can create the labels automatically if you
call the client with a -relabel option. The exact format of the labels is dependent on the
construction, and is described by the corresponding client.
Vertex-facet incidence matrix, with rows corresponding to facets and columns
to vertices. Vertices and facets are numbered from 0 to N_VERTICES-1 rsp.
N_FACETS-1, according to their order in VERTICES rsp. FACETS.
The face lattice of the polytope organized as a directed graph.
Each node corresponds to some proper face of the polytope.
The nodes corresponding to the vertices and facets appear in the same order
as the elements of VERTICES and FACETS properties.
Two special nodes represent the whole polytope and the empty face.
All intermediate polytopes (with respect to the given insertion order) in the beneath-and-beyond algorithm are simplicial.
We have the implications: VERTICES in general position => ESSENTIALLY_GENERIC => SIMPLICIAL
Node connectivity of the GRAPH of the polytope, that is, the minimal number of nodes to be removed
from the graph such that the result is disconnected.
Condensed form of the flag vector, containing all entries indexed by sparse sets in {0,...,DIM-1}
in the following order: (1, f0, f1, f2, f02, f3, f03, f13, f4, f04, f14, f24, f024, f5,...).
Use Dehn-Sommerville equations, via user function N_FLAGS, to extend.
h-vector, defined via recursion on the face lattice of a polytope.
Coincides for simplicial (resp. simple) polytopes with the combinatorial definition
of the h-vector via shellings (resp. orderings of vertices).
Similar to VERTICES_IN_FACETS, but with columns corresponding to POINTS instead of VERTICES.
This property is a byproduct of convex hull computation algorithms.
It is discarded as soon as VERTICES_IN_FACETS is computed.
Similar to VERTICES_IN_FACETS, but with rows corresponding to INEQUALITIES instead of FACETS.
This property is a byproduct of convex hull computation algorithms.
It is discarded as soon as VERTICES_IN_FACETS is computed.
Reordered VERTICES_IN_FACETS for 2-d and 3-d polytopes.
Vertices are listed in the order of their appearance
when traversing the facet border counterclockwise seen from outside of the polytope.
For a 2-d polytope (which is a closed polygon), lists all vertices in the border traversing order.
Reordered DUAL_GRAPH for 3-d polytopes.
The neighbor facets are listed in the order corresponding to VIF_CYCLIC_NORMAL,
so that the first two vertices in VIF_CYCLIC_NORMAL make up the ridge to the first neighbor
facet and so on.
Approximate color encoding of objective function values.
For each vertex, the color is chosen by linear interpolation in HSV system
between fixed "maximal" and "minimal" colors, with the same ratio as the objective
function value has in relation to the global maximum and mininum on the entire polytope.
You can give a list of variable names here
(this amounts to naming the columns of the coordinate vectors for VERTICES).
This is of not much use for polymake. But this information will be
used by programs which convert polymake format to other formats (such as LP).
Since a Voronoi polyhedron is unbounded it must be artificially bounded for visualization purposes.
Allowed is any set of hyperplanes which makes the projection onto the last d-1 coordinates bounded.
By default, these are the vertical facets of a suitably scaled cube.
Not very useful for other unbounded polyhedra not related to Voronoi diagrams.
Graph of the Voronoi diagram of the SITES. The coordinates (homogeneous, before projection) are stored as node attributes.
The graph is truncated according to the VORONOI_BOUNDING_BOX. All vertices of the Voronoi diagram are visible (and represented in
the VORONOI_GRAPH) for the default VORONOI_BOUNDING_BOX.
Graph of the joint Voronoi diagram of the SITES and the vertices of Vor(SITES). The coordinates (homogeneous, before projection)
are stored as node attributes. The graph is truncated according to the VORONOI_BOUNDING_BOX. For the default VORONOI_BOUNDING_BOX it may happen
that some of the iterated Voronoi vertices are truncated. Use the client voronoi to produce proper iterated Voronoi diagrams.