====== BigObject VoronoiDiagram ======
//from application [[..:tropical|tropical]]//\\
\\
Voronoi diagram with respect to the tropical metric in the tropical projective torus. Its combinatorics is controlled by a ''[[..:tropical:VoronoiDiagram#POLYTROPE_PARTITION |POLYTROPE_PARTITION]]''. See P. Criado, M. Joswig, P. Santos: Tropical bisectors and Voronoi diagrams, arXiv:1906.10950
? Example:
:: The following computes a tropical Voronoi diagram of three ''[[..:tropical:VoronoiDiagram#SITES |SITES]]'' in the tropical 3-torus.
:: > $T= new VoronoiDiagram(SITES=>[[-4,-4,0,0],[-3,0,2,0],[-2,-5,-2,0]]);
> print $T->POLYTROPE_PARTITION->size();
134
===== Properties =====
==== no category ====
{{anchor:ambient_dim:}}
? **''AMBIENT_DIM''**
:: Number of dimensions of the diagram. One less than the number of coordinates.
? Type:
:''[[..:common#Int |Int]]''
----
{{anchor:n_sites:}}
? **''N_SITES''**
:: Number of sites of the diagram.
? Type:
:''[[..:common#Int |Int]]''
----
{{anchor:polytrope_partition:}}
? **''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:
:''[[..:common#Array |Array]]<[[..:common#Pair |Pair]]<[[..:common#Matrix |Matrix]]<[[..:common#Rational |Rational]],[[..:common#NonSymmetric |NonSymmetric]]>,[[..:common#Matrix |Matrix]]<[[..:common#Int |Int]],[[..:common#NonSymmetric |NonSymmetric]]%%>>%%>''
? 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
>
----
{{anchor:sites:}}
? **''SITES''**
:: The sites of the tropical Voronoi diagram.
? Type:
:''[[..:common#Matrix |Matrix]]<[[..:common#Rational |Rational]],[[..:common#NonSymmetric |NonSymmetric]]>''
----
===== Methods =====
==== no category ====
{{anchor:visual:}}
? **''VISUAL''**
::UNDOCUMENTED
----