Available versions of this document: latest release, release 4.13, release 4.12, release 4.11, release 4.10, release 4.9, release 4.8, release 4.7, release 4.6, release 4.5, release 4.4, release 4.3, release 4.2, release 4.1, release 4.0, release 3.6, release 3.5, nightly master
Reference documentation for older polymake versions: release 3.4, release 3.3, release 3.2
BigObject VoronoiDiagram
from application tropical
Voronoi diagram with respect to the tropical metric in the tropical projective torus. Its combinatorics is controlled by a POLYTROPE_PARTITION
. See P. Criado, M. Joswig, P. Santos: Tropical bisectors and Voronoi diagrams, arXiv:1906.10950
Properties
no category
-
AMBIENT_DIM
Number of dimensions of the diagram. One less than the number of coordinates.
- Type:
-
N_SITES
Number of sites of the diagram.
- Type:
-
POLYTROPE_PARTITION
Representation of the tropical Voronoi diagram. Each such polyhedron is a domain in which the distance to the set of sites $S$ is a minimum of linear functions. This list of regions is represented as an array of pairs of matrices. The first matrix in each pair represents the region itself (a polytrope) as a shortest path matrix. The second matrix (the labels) gives the index of the site $s\in S$ with maximum $s_j-s_i$ such that the cone $\{x:x_i-s_i<= x_k-s_k <= x_j-s_j \forall k\in [d+1]\}$ intersects this cell (or $-1$ if no such index exists). Then, in this region, $dist(x,S)$ is a minimum of the linear functions $(x_j-s_j)-(x_i-s_i)$ for each $s$ labelled with $(i,j)$.
- Type:
- Example:
Here is one polytrope cell.
> $T= new VoronoiDiagram(SITES=>[[-4,-4,0,0],[-3,0,2,0],[-2,-5,-2,0]]); > print $T->POLYTROPE_PARTITION->[0]; <0 inf inf inf -4 0 2 0 -5 inf 0 inf -4 inf inf 0 > <-1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 >
-
SITES
The sites of the tropical Voronoi diagram.
- Type: