2 Search Results for "Jüttler, Bert"

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DOI: 10.4230/LIPIcs.SoCG.2026.84

The Voronoi Diagram of Four Lines in ℝ³

Authors: Evanthia Papadopoulou and Zeyu Wang

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

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)

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@InProceedings{papadopoulou_et_al:LIPIcs.SoCG.2026.84,
  author =	{Papadopoulou, Evanthia and Wang, Zeyu},
  title =	{{The Voronoi Diagram of Four Lines in \mathbb{R}³}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{84:1--84:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.84},
  URN =		{urn:nbn:de:0030-drops-258916},
  doi =		{10.4230/LIPIcs.SoCG.2026.84},
  annote =	{Keywords: Voronoi diagram, lines, three dimensions, structural properties}
}

Document

DOI: 10.4230/LIPIcs.ISAAC.2017.7

Voronoi Diagrams for Parallel Halflines and Line Segments in Space

Authors: Franz Aurenhammer, Bert Jüttler, and Günter Paulini

Published in: LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

Abstract

We consider the Euclidean Voronoi diagram for a set of $n$ parallel halflines in 3-space. A relation of this diagram to planar power diagrams is shown, and is used to analyze its geometric and topological properties. Moreover, an easy-to-implement space sweep algorithm is proposed that computes the Voronoi diagram for parallel halflines at logarithmic cost per face. Previously only an approximation algorithm for this problem was known. Our method of construction generalizes to Voronoi diagrams for parallel line segments, and to higher dimensions.

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Franz Aurenhammer, Bert Jüttler, and Günter Paulini. Voronoi Diagrams for Parallel Halflines and Line Segments in Space. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 7:1-7:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{aurenhammer_et_al:LIPIcs.ISAAC.2017.7,
  author =	{Aurenhammer, Franz and J\"{u}ttler, Bert and Paulini, G\"{u}nter},
  title =	{{Voronoi Diagrams for Parallel Halflines and Line Segments in Space}},
  booktitle =	{28th International Symposium on Algorithms and Computation (ISAAC 2017)},
  pages =	{7:1--7:10},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-054-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{92},
  editor =	{Okamoto, Yoshio and Tokuyama, Takeshi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.7},
  URN =		{urn:nbn:de:0030-drops-82157},
  doi =		{10.4230/LIPIcs.ISAAC.2017.7},
  annote =	{Keywords: Voronoi diagram, line segments, space-sweep algorithm}
}

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2 Search Results for "Jüttler, Bert"

The Voronoi Diagram of Four Lines in ℝ³

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Voronoi Diagrams for Parallel Halflines and Line Segments in Space

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