BigObject VoronoiDiagram [polymake wiki]

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====== 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.
  :: <code perl> > $T= new VoronoiDiagram(SITES=>[[-4,-4,0,0],[-3,0,2,0],[-2,-5,-2,0]]);
 > print $T->POLYTROPE_PARTITION->size();
 134
</code>
===== 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.
    :: <code perl> > $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
 >
</code>


----
{{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


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