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Construction of Voronoi diagrams in the plane by using maps. (English) Zbl 0715.68085

Summary: An algorithm by Guibas and Stolfi (1985) constructs, for a finite set S of n sites in the plane, a triangulation T(S) of S that strictly refines the Delaunay diagram Del(S) when there exists a circle passing through at least four points of S and none of the sites is contained in its interior. For this triangulation T(S) there exist isometries \({\mathcal T}\) such that \({\mathcal T}(T(S))\neq T({\mathcal T}(S))\). The Voronoi diagram Vor(S) is the straight-line dual of Del(S) and the substitution of T(S) into Del(S) leads to needless calculus, indeed to the creation of imaginary vertices for Vor(S). We present here a variant of the Guibas and Stolfi algorithm that determines, with the same complexity, Del(S) and not a triangulation, and uses a simpler data structure. We also give approximation tests of collinearity and cocircularity so that for any similitude \({\mathcal T}\) of the plane, Del(\({\mathcal T}(S))={\mathcal T}(Del(S))\).

MSC:

68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
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