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The Voronoi Diagram of Four Lines in ℝ³

Authors Evanthia Papadopoulou , Zeyu Wang

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Voronoi diagram
  • lines
  • three dimensions
  • structural properties

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Abstract

We consider the Voronoi diagram of lines in ℝ³ 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 twist, which is a pair of consecutive intersections among trisector branches; only two types of twists are possible, so-called full and 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 ℝ³ 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.

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Evanthia Papadopoulou and Zeyu Wang. The Voronoi Diagram of Four Lines in ℝ³. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 84:1-84:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.84

Author Details

Evanthia Papadopoulou
  • Faculty of Informatics, Università della Svizzera italiana (USI), Lugano, Switzerland
Zeyu Wang
  • Faculty of Informatics, Università della Svizzera italiana (USI), Lugano, Switzerland

Funding

Supported by the Swiss National Science Foundation (SNF), project 200021E_201356.

Acknowledgements

We thank Dr. Martin Suderland for early discussions and valuable comments related to Sections 2, 3, and for two Matlab-based visualisation tools, which allowed us to visualize Γ(FVD(L)), Figure 6, and Figure 13. We also thank anonymous reviewers for comments that helped improve the presentation of this paper, the general position assumption, and Lemma 3.

Supplementary Materials

References

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