Voronoi diagrams with respect to criteria on vision information

Abstract

Voronoi diagrams for a set of geometric objects is a partition of the plane (or space in higher dimensions) into disjoint regions each dominated by some given object under a predetermined criterion. In this paper we are interested in various measures associated with criteria on goodness of an input line segment with respect to each point in the plane as the "point of view." These measures basically show how well a segment or information displayed on the segment can be seen from the point. Mathematically, the measures are defined in terms of the shapes of the triangle determined by the point and the line segment. We study the combinatorial and algorithmic complexities of those Voronoi diagrams We also study an associated optimization problem: find a point that maximizes the smallest measure value over the measures with respect to all the given line segments. We give sufficient conditions for an optimal point to lie on a Voronoi edge and present a heuristic optimization algorithm for those measures having this property.

On 2-Site Voronoi Diagrams Under Geometric Distance Functions

“…Consequently, geometric structures akin to 2-site Voronoi diagrams can arise as Voronoi diagrams of segments. This alternative approach was independently undertaken by Asano et al, and the "view angle" and "radius of circumscribing circle" distance functions reappeared in their works [2], [3] on Voronoi diagrams for segments soon after they had been proposed by Hodorkovsky [10] in the context of 2-site Voronoi diagrams. However, as Asano's et al research was originally motivated by mesh generation and improvement tasks, they were mostly interested in sets of segments representing edges of a simple polygon, and thus, non-intersecting (except, possibly, at the endpoints), what significantly alters the essence of the problem.…”