# Generalized Voronoi Diagrams and Lie Sphere Geometry

## Document Type

Presentation Abstract

## Presentation Date

10-16-2023

## Abstract

The classical Voronoi diagram for a set S of points in the Euclidean plane is the subdivision of the plane into Voronoi cells, one for each point in the set. The Voronoi cell for a point p is the set of points in the plane that have p as the closest point in S. This notion is so fundamental that it arises in a multitude of contexts, both in theoretical mathematics and in the real world.

The notion of Voronoi diagram may be expanded by changing the underlying geometry, by allowing the sites to be sets rather than points, by weighting sites, by subdividing the domain based on farthest point rather than closest point, or by subdividing the domain based on which k sites are closest.

"Lie sphere geometry” can be used to describe many such "generalized Voronoi diagrams."

In this talk, we give overviews of generalized Voronoi diagrams and Lie sphere geometry, and we describe how they are related.

## Recommended Citation

Payne, Tracy, "Generalized Voronoi Diagrams and Lie Sphere Geometry" (2023). *Colloquia of the Department of Mathematical Sciences*. 667.

https://scholarworks.umt.edu/mathcolloquia/667

## Additional Details

October 16, 2023 at 3:00 p.m. Math 103