# Voronoi diagram

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## Voronoi diagram

(mathematics, graphics)
(After G. Voronoi) For a set S of points in the Euclidean plane, the partition Vor(S) of the plane into the voronoi polygons associated with the members of S. Vor(S) is the dual of the Delaunay triangulation of S.
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The Dirichlet domain of [GAMMA] is based on the bisectors of some chosen center and its images by [GAMMA].
We call them double Dirichlet domains. In some sense characterizing a group acting discontinuously on hyperbolic space by having a double Dirichlet domain comes down to study 'how close' the group is to a hyperbolic reflection group.
This is in fact both a Ford domain and a Dirichlet domain with center ti for every t > 0.
A Dirichlet fundamental domain which has multiple centers is called a double Dirichlet Domain.
It is easy to see that if [tau][GAMMA][[tau].sup.-1] has a double Dirichlet domain, [GAMMA] has a double Dirichlet domain.
As F is a also a Dirichlet domain, F [intersection] [[gamma].sup.-1](F) [subset or equal to] [[summation].sub.[gamma]].
From Corollary 4.4, it follows that that all examples given in Section VII.3 in the book of Elstrodt, Grunewald and Mennicke  are groups whose Ford domain is also a Dirichlet domain. Note that this does not follow immediately from the results of .
First note that the side-pairing transformations of a Dirichlet domain of center j in the case of a Bianchi group are not uniquely determined, as the group has a non-trivial stabilizer of j, namely [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
The goal is to obtain [L.sup.[infinity]] estimates for the DNWR error [w.sup.(k)] (t) in the case of [theta] = 1/2 and a > b, i.e., when the Dirichlet domain is larger.
Here, we assume that a < b, i.e., the Dirichlet domain is smaller than the Neumann domain.
In the second case, we repeat the experiment, except that we swap the roles of the two subdomains, so that the Dirichlet domain is now smaller than the Neumann one.
Thus, chapters discuss sequentially the complex plane, transformations of the plane, hyperbolic geometry, elliptic geometry, expansion of geometries to different curvature scales, the topology of surfaces, the relationship of topology of surfaces to geometry, the Gauss-Bonnet formula, quotient spaces, and the Dirichlet domain. The final chapter then turns to the present state of research in cosmic topology, discussing possible shapes of the universe, cosmic crystallography, and circles in the sky.

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