Generalized Voronoi Diagrams and Lie Sphere Geometry

TL;DR

This paper employs Lie sphere geometry to unify and extend the understanding of various generalized Voronoi diagrams, including classical, power, Apollonius, and medial axes, through convex hull and algebraic approaches.

Contribution

It introduces a Lie geometric framework that generalizes and unifies multiple types of Voronoi diagrams, linking them to convex hull problems and algebraic structures.

Findings

Unified geometric description of generalized Voronoi diagrams

Algorithms for computing these diagrams and their complexity analysis

Extension of classical diagrams to include spheres and half-spaces

Abstract

We use Lie sphere geometry to describe two large categories of generalized Voronoi diagrams that can be encoded in terms of the Lie quadric, the Lie inner product, and polyhedra. The first class consists of diagrams defined in terms of extremal spheres in the space of Lie spheres, and the second class includes minimization diagrams for functions that can be expressed in terms of affine functions on a higher-dimensional space. These results unify and generalize previous descriptions of generalized Voronoi diagrams as convex hull problems. Special cases include classical Voronoi diagrams, power diagrams, order $k$ and farthest point diagrams, Apollonius diagrams, medial axes, and generalized Voronoi diagrams whose sites are combinations of points, spheres and half-spaces. We describe the application of these results to algorithms for computing generalized Voronoi diagrams and find the complexity of these algorithms.