On the computation of zone and double zone diagrams

TL;DR

This paper explores the computation of zone and double zone diagrams in various geometric spaces, extending existing methods beyond Euclidean planes and providing algorithms for their approximation.

Contribution

It introduces a generalized iterative method for computing zone diagrams in a broad class of spaces, including spheres and convex normed spaces, with convergence guarantees.

Findings

The iterative method converges to a double zone diagram in the generalized setting.

A simple algorithm enables approximate Voronoi diagram computation in these spaces.

Topological properties of Voronoi cells are established, such as boundary characteristics.

Abstract

Classical objects in computational geometry are defined by explicit relations. Several years ago the pioneering works of T. Asano, J. Matousek and T. Tokuyama introduced "implicit computational geometry", in which the geometric objects are defined by implicit relations involving sets. An important member in this family is called "a zone diagram". The implicit nature of zone diagrams implies, as already observed in the original works, that their computation is a challenging task. In a continuous setting this task has been addressed (briefly) only by these authors in the Euclidean plane with point sites. We discuss the possibility to compute zone diagrams in a wide class of spaces and also shed new light on their computation in the original setting. The class of spaces, which is introduced here, includes, in particular, Euclidean spheres and finite dimensional strictly convex normed spaces. Sites of a general form are allowed and it is shown that a generalization of the iterative method suggested by Asano, Matousek and Tokuyama converges to a double zone diagram, another implicit geometric object whose existence is known in general. Occasionally a zone diagram can be obtained from this procedure. The actual (approximate) computation of the iterations is based on a simple algorithm which enables the approximate computation of Voronoi diagrams in a general setting. Our analysis also yields a few byproducts of independent interest, such as certain topological properties of Voronoi cells (e.g., that in the considered setting their boundaries cannot be "fat").