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

Evanthia Papadopoulou, Zeyu Wang

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.

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

Abstract

We consider the Voronoi diagram of lines in 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 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.
Paper Structure (12 sections, 19 theorems, 1 equation, 11 figures, 1 table)

This paper contains 12 sections, 19 theorems, 1 equation, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Contribution.
  3. Preliminaries
  4. Algebraic framework and geometric reinterpretation
  5. Some combinatorial results on Voronoi diagrams of lines in $\mathbb{R}^3$
  6. Twists
  7. Full twists
  8. Local modification.
  9. Partial twists and further properties
  10. Identifying the Voronoi topologies by exhaustive search
  11. Properties of configurations
  12. A search algorithm, proving \ref{['thm:topologywithoutfulltwist', 'thm:topology']}

Key Result

Theorem 1

The Voronoi diagram of four lines in general position in $\mathbb{R}^{3}$ (both the nearest and farthest diagram) has 15 distinct topologies, assuming there are no full twists among the trisectors. The distinct topologies of the nearest and the farthest diagrams are in one-to-one correspondence. Fur

Figures (11)

  • Figure 1: A projected trisector on a projected bisector. Branches are shown in blue and the asymptotes are shown in red. The $\mathrm{NVD}$ faces are shown in green and the $\mathrm{FVD}$ faces in orange.
  • Figure 3: Left: a partial twist. Right: a full twist. Different colors denote different trisectors.
  • Figure 5: Left: a 3D view of a full twist. Middle: illustration for the proof of \ref{['lem:fulltwist']}. Faces $F_1$ and $F_2$ are highlighted, intersecting at the black segment which is not an edge of the diagram. Right: the unique structure of the $\mathrm{NVD}(L)$ around the full twist.
  • Figure 6: Left: local $\mathrm{NVD}$/$\mathrm{FVD}$ structure; right: the modified diagram after adding a full twist.
  • Figure 7: A local section of a trisector system, illustrating the setup of \ref{['lem:trisystem1']}.
  • ...and 6 more figures

Theorems & Definitions (22)

  • Theorem 1
  • Theorem 2
  • Lemma 2
  • Lemma 2: Everett2009
  • Lemma 2
  • Lemma 2
  • Remark 3
  • Lemma 3
  • Lemma 3
  • Theorem 4
  • ...and 12 more