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

2026-03-20

AI Summary

This study provides a complete characterization of the topological structure of Voronoi diagrams for four lines in general position in three-dimensional Euclidean space. By integrating combinatorial geometric analysis, spherical mapping to encode unbounded features, and an exhaustive search algorithm, the work achieves the first full classification of this problem. A key contribution is the introduction of a locally insertable and removable “full twist” mechanism, which establishes a one-to-one correspondence between nearest- and farthest-point Voronoi diagrams. The authors prove that the number of vertices is always an even integer between 0 and 8, inclusive, and that all such values are realizable. In the absence of a full twist, each type of Voronoi diagram admits exactly 15 distinct realizable topologies, and any configuration can be generated from a base topology through the application of full twists.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Machine Learning: Learning with Manifolds
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

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• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures
• Social Networks and Social Media: Social media analysis through the lenses of networks

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.

Problem

• Voronoi diagram
• 3D lines
• structural classification
• general position
• Euclidean metric

Innovation

• Voronoi diagram
• twist
• 3D line arrangement
• topological classification
• computational geometry