Aspect-ratio Voronoi diagram and its complexity bounds

Abstract

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 $\mathrm{O}(n^{2+\epsilon })$ upper bound and $\mathrm{\Omega }(n^2)$ lower bound on the complexity, where ε 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.