# Voronoi Cells in Metric Algebraic Geometry of Plane Curves

## TL;DR

This paper explores how Voronoi and Delaunay cells of plane curves relate to their metric geometry, providing algebraic descriptions and methods to approximate key features from finite samples.

## Contribution

It proves that Voronoi and Delaunay cells of plane curves can be approximated as limits of sampled cells, linking discrete samples to continuous metric features.

## Key findings

- Voronoi and Delaunay cells can be obtained as limits of sampled cells.
- Provides algebraic equations for medial axis, curvature, evolute, bottlenecks, and reach.
- Offers formulas for degrees of algebraic varieties representing these features.

## Abstract

Voronoi cells of varieties encode many features of their metric geometry. We prove that each Voronoi or Delaunay cell of a plane curve appears as the limit of a sequence of cells obtained from point samples of the curve. We use this result to study metric features of plane curves, including the medial axis, curvature, evolute, bottlenecks, and reach. In each case, we provide algebraic equations defining the object and, where possible, give formulas for the degrees of these algebraic varieties. We show how to identify the desired metric feature from Voronoi or Delaunay cells, and therefore how to approximate it by a finite point sample from the variety.