Counting and representing intersections among triangles in three dimensions

Esther Ezra; Micha Sharir

doi:10.1145/997817.997851

Counting and representing intersections among triangles in three dimensions

Esther Ezra, Micha Sharir

Research output: Contribution to conference › Paper › peer-review

1 Scopus citations

Abstract

We present an algorithm that efficiently counts all intersecting triples among a collection T of triangles in ℝ³ in nearly-quadratic time. This solves a problem posed by Pellegrini. Using a variant of the technique, one can represent the set of all κ triple intersections, in compact form, as the disjoint union of complete tripartite hypergraphs, which requires nearly-quadratic construction time and storage. Our approach also applies to any collection of convex planar objects of constant description complexity in ℝ³, with the same performance bounds. We also prove that this counting problem belongs to the 3SUM-hard family, and thus our algorithm is likely to be nearly optimal (since it is believed that 3SUM-hard problems cannot be solved in subquadratic time).

Original language	English
Pages	210-219
Number of pages	10
DOIs	https://doi.org/10.1145/997817.997851
State	Published - 2004
Externally published	Yes
Event	Proceedings of the Twentieth Annual Symposium on Computational Geometry (SCG'04) - Brooklyn, NY, United States Duration: 9 Jun 2004 → 11 Jun 2004

Conference

Conference	Proceedings of the Twentieth Annual Symposium on Computational Geometry (SCG'04)
Country/Territory	United States
City	Brooklyn, NY
Period	9/06/04 → 11/06/04

Bibliographical note

Funding Information:
Keywords: Triangles in three dimensions; Curve-sensitive cuttings; Counting intersections; Arrangements; 3SUM-hard problems ✩ Work on this paper has been supported by NSF Grants CCR-97-32101 and CCR-00-98246, by a grant from the US–Israeli Binational Science Foundation, by a grant from the Israel Science Fund, Israeli Academy of Sciences, for a Center of Excellence in Geometric Computing at Tel Aviv University, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. A preliminary version of this paper has appeared in Proc. 20th Annu. ACM Sympos. Comput. Geom., 2004, pp. 210– 219. * Corresponding author. E-mail addresses: [email protected] (E. Ezra), [email protected] (M. Sharir).

Funding

Keywords: Triangles in three dimensions; Curve-sensitive cuttings; Counting intersections; Arrangements; 3SUM-hard problems ✩ Work on this paper has been supported by NSF Grants CCR-97-32101 and CCR-00-98246, by a grant from the US–Israeli Binational Science Foundation, by a grant from the Israel Science Fund, Israeli Academy of Sciences, for a Center of Excellence in Geometric Computing at Tel Aviv University, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. A preliminary version of this paper has appeared in Proc. 20th Annu. ACM Sympos. Comput. Geom., 2004, pp. 210– 219. * Corresponding author. E-mail addresses: [email protected] (E. Ezra), [email protected] (M. Sharir).

Funders	Funder number
israel science fund
Israeli Academy of Sciences
National Science Foundation	CCR-00-98246, CCR-97-32101
Anacostia Community Museum
United States-Israel Binational Science Foundation
Tel Aviv University

Keywords

3SUM-hard problems
Arrangements
Counting intersections
Curve-sensitive cuttings
Triangles in three dimensions

Access to Document

10.1145/997817.997851

Cite this

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Counting and representing intersections among triangles in three dimensions

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Keywords

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