THE PREDICATES FOR THE EXACT VORONOI DIAGRAM OF ELLIPSES UNDER THE EUCLIDIEAN METRIC

Abstract

This article examines the computation of the Delaunay graph and its dual Voronoi diagram of a set of ellipses in the Euclidean plane. We propose the first complete methods, under the exact computation paradigm, for the predicates of an incremental algorithm: κ1 decides which one of two given ellipses is closest to a given exterior point; κ2 decides the position of a query ellipse relative to an external bitangent line of two given ellipses; κ3 decides the position of a query ellipse relative to a Voronoi circle of three given ellipses; κ4 determines the type of conflict between a Voronoi edge, defined by four given ellipses, and a query ellipse. The article is restricted to non-intersecting ellipses, but the extension to arbitrary ones is possible. The ellipses are input in parametric representation, i.e., constructively in terms of their axes, center and rotation. For κ1 and κ2 we derive algebraic conditions optimal in terms of the degree of the algebraic numbers involved, and provide efficient implementations in C++ . For κ3 we compute a tight bound on the number of complex tritangent circles and design an exact symbolic-numeric algorithm, which is implemented in MAPLE. This essentially answers κ4 as well. We conclude with further work on lifting the condition of non-intersecting ellipses.