The Voronoi Diagram of Four Lines in $\mathbb{R}^3$

TL;DR

This paper classifies the structure of Voronoi diagrams of four lines in three-dimensional space, revealing the possible topologies, their properties, and how they can be transformed through local modifications called twists.

Contribution

It provides a complete classification of Voronoi diagrams for four lines in general position in $\,\mathbb{R}^3$, introducing the concept of twists and their role in diagram transformations.

Findings

Number of vertices in the diagram ranges from 0 to 8 and all are realizable.

Two types of twists, full and partial, are identified as key structural elements.

Farthest and nearest diagrams have 15 topologies in the absence of full twists.

Abstract

We consider the Voronoi diagram of lines in $\mathbb{R}^3$ under the Euclidean metric, and give a full classification of its structure in the base case of four lines in general position. We first show that the number of vertices in the Voronoi diagram of four lines in general position is always even, between 0 and 8, and all such numbers can be realized. We identify a key structure for the diagram formation, called a \emph{twist}, which is a pair of consecutive intersections among trisector branches; only two types of twists are possible, so-called \emph{full} and \emph{partial} twists. A full twist is a purely local structure, which can be inserted or removed without affecting the rest of the diagram. Assuming no full twists, the nearest and the farthest Voronoi diagrams of four lines, each have 15 distinct topologies, which are in one-to-one correspondence; the two-dimensional faces are all unbounded, and the total number of vertices is at most six. The unbounded features of the farthest diagram, encoded in a two-dimensional spherical map, are also in one-to-one correspondence. The identified topologies are all realizable. Any Voronoi diagram of four lines in general position in $\mathbb{R}^3$ can be obtained from one of these topologies by inserting full twists; each twist induces a bounded face of exactly two vertices in both the nearest and farthest diagrams. We obtain the classification by an exhaustive search algorithm using some new structural and combinatorial observations of line Voronoi diagrams.