Swap conditions for dynamic Voronoi diagrams for circles and line segments

Abstract

Dynamic maintenance of Voronoi diagrams for a set of disks moving independently in the plane along given trajectories is considered in this paper. The domain is limited by boundary represented by straight-line segments. Maintenance of a Voronoi diagram for moving objects (disks and/or line segments) over time requires calculation of topological events which occur when four objects arrive at positions where they are tangent to a common circle. Criteria for determination of such topological events for circles and line segments in the Euclidean metric have been dervided using a standard fractional transformation in complex plane. These criteria are represented in the form of polynomial algebraic equations, based on the coordinates and trajectories of the moving objects. These equations can normally only be solved using numerical methods.