Spatial tessellations: concepts and applications of Voronoi diagrams
Spatial tessellations: concepts and applications of Voronoi diagrams
Almost tight upper bounds for the single cell and zone problems in three dimensions
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
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This Letter first defines an aspect ratio of a triangle by the ratio of the longest side over the minimal height. Given a set of line segments, any point p in the plane is associated with the worst aspect ratio for all the triangles defined by the point and the line segments. When a line segment s"i gives the worst ratio, we say that p is dominated by s"i. Now, an aspect-ratio Voronoi diagram for a set of line segments is a partition of the plane by this dominance relation. We first give a formal definition of the Voronoi diagram and give O(n^2^+^@e) upper bound and @W(n^2) lower bound on the complexity, where @e is any small positive number. The Voronoi diagram is interesting in itself, and it also has an application to a problem of finding an optimal point to insert into a simple polygon to have a triangulation that is optimal in the sense of the aspect ratio.