Constructing the exact Voronoi diagram of arbitrary lines in three-dimensional space: With fast point-location

Michael Hemmer; Ophir Setter; Dan Halperin

doi:10.1007/978-3-642-15775-2_34

Constructing the exact Voronoi diagram of arbitrary lines in three-dimensional space: With fast point-location

Michael Hemmer^*, Ophir Setter, Dan Halperin

^*Corresponding author for this work

School of Computer Science

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

We introduce a new, efficient, and complete algorithm, and its exact implementation, to compute the Voronoi diagram of lines in space. This is a major milestone towards the robust construction of the Voronoi diagram of polyhedra. As we follow the exact geometric-computation paradigm, it is guaranteed that we always compute the mathematically correct result. The algorithm is complete in the sense that it can handle all configurations, in particular all degenerate ones. The algorithm requires O(n^3+ε ) time and space, where n is the number of lines. The Voronoi diagram is represented by a data structure that permits answering point-location queries in O(log² n) expected time. The implementation employs the Cgal packages for constructing arrangements and lower envelopes together with advanced algebraic tools.

Original language	English
Title of host publication	Algorithms, ESA 2010 - 18th Annual European Symposium, Proceedings
Pages	398-409
Number of pages	12
Edition	PART 1
DOIs	https://doi.org/10.1007/978-3-642-15775-2_34
State	Published - 2010
Event	18th Annual European Symposium on Algorithms, ESA 2010 - Liverpool, United Kingdom Duration: 6 Sep 2010 → 8 Sep 2010

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Number	PART 1
Volume	6346 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	18th Annual European Symposium on Algorithms, ESA 2010
Country/Territory	United Kingdom
City	Liverpool
Period	6/09/10 → 8/09/10

Keywords

CGAL
Computational Geometry
Lower Envelopes
Point Location
Robust Geometric Computing
Voronoi Diagrams

Access to Document

10.1007/978-3-642-15775-2_34

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Hemmer, M., Setter, O., & Halperin, D. (2010). Constructing the exact Voronoi diagram of arbitrary lines in three-dimensional space: With fast point-location. In Algorithms, ESA 2010 - 18th Annual European Symposium, Proceedings (PART 1 ed., pp. 398-409). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6346 LNCS, No. PART 1). https://doi.org/10.1007/978-3-642-15775-2_34

Hemmer, Michael ; Setter, Ophir ; Halperin, Dan. / Constructing the exact Voronoi diagram of arbitrary lines in three-dimensional space : With fast point-location. Algorithms, ESA 2010 - 18th Annual European Symposium, Proceedings. PART 1. ed. 2010. pp. 398-409 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); PART 1).

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abstract = "We introduce a new, efficient, and complete algorithm, and its exact implementation, to compute the Voronoi diagram of lines in space. This is a major milestone towards the robust construction of the Voronoi diagram of polyhedra. As we follow the exact geometric-computation paradigm, it is guaranteed that we always compute the mathematically correct result. The algorithm is complete in the sense that it can handle all configurations, in particular all degenerate ones. The algorithm requires O(n3+ε ) time and space, where n is the number of lines. The Voronoi diagram is represented by a data structure that permits answering point-location queries in O(log2 n) expected time. The implementation employs the Cgal packages for constructing arrangements and lower envelopes together with advanced algebraic tools.",

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author = "Michael Hemmer and Ophir Setter and Dan Halperin",

note = "Funding Information: This work has been supported in part by the Israel Science Foundation (grant no. 236/06), by the German-Israeli Foundation (grant no. 969/07), and by the Hermann Minkowski–Minerva Center for Geometry at Tel Aviv University.; null ; Conference date: 06-09-2010 Through 08-09-2010",

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Hemmer, M, Setter, O & Halperin, D 2010, Constructing the exact Voronoi diagram of arbitrary lines in three-dimensional space: With fast point-location. in Algorithms, ESA 2010 - 18th Annual European Symposium, Proceedings. PART 1 edn, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), no. PART 1, vol. 6346 LNCS, pp. 398-409, 18th Annual European Symposium on Algorithms, ESA 2010, Liverpool, United Kingdom, 6/09/10. https://doi.org/10.1007/978-3-642-15775-2_34

Constructing the exact Voronoi diagram of arbitrary lines in three-dimensional space : With fast point-location. / Hemmer, Michael; Setter, Ophir; Halperin, Dan.

Algorithms, ESA 2010 - 18th Annual European Symposium, Proceedings. PART 1. ed. 2010. p. 398-409 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6346 LNCS, No. PART 1).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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Hemmer M, Setter O, Halperin D. Constructing the exact Voronoi diagram of arbitrary lines in three-dimensional space: With fast point-location. In Algorithms, ESA 2010 - 18th Annual European Symposium, Proceedings. PART 1 ed. 2010. p. 398-409. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); PART 1). doi: 10.1007/978-3-642-15775-2_34

Constructing the exact Voronoi diagram of arbitrary lines in three-dimensional space: With fast point-location

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Keywords

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