Counting and representing intersections among triangles in three dimensions

Esther Ezra; Micha Sharir

doi:10.1016/j.comgeo.2005.02.003

Counting and representing intersections among triangles in three dimensions

Esther Ezra
, Micha Sharir

Tel Aviv University

Research output: Contribution to journal › Article › peer-review

14 Scopus citations

Abstract

We present an algorithm that efficiently counts all intersecting triples among a collection T of triangles in R ³ in nearly quadratic time. This solves a problem posed by Pellegrini [M. Pellegrini, On counting pairs of intersecting segments and off-line triangle range searching, Algorithmica 17 (1997) 380-398]. 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 planar objects of constant description complexity in R ³, 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 in the worst case.

Original language	English
Pages (from-to)	196-215
Number of pages	20
Journal	Computational Geometry: Theory and Applications
Volume	32
Issue number	3
DOIs	https://doi.org/10.1016/j.comgeo.2005.02.003
State	Published - Nov 2005
Externally published	Yes

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
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

10.1016/j.comgeo.2005.02.003

Cite this

@article{1ad742a4258145f9b49c576dff96404d,

title = "Counting and representing intersections among triangles in three dimensions",

abstract = "We present an algorithm that efficiently counts all intersecting triples among a collection T of triangles in R 3 in nearly quadratic time. This solves a problem posed by Pellegrini [M. Pellegrini, On counting pairs of intersecting segments and off-line triangle range searching, Algorithmica 17 (1997) 380-398]. 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 planar objects of constant description complexity in R 3, 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 in the worst case.",

keywords = "3sum-hard problems, Arrangements, Counting intersections, Curve-sensitive cuttings, Triangles in three dimensions",

author = "Esther Ezra and Micha Sharir",

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).",

year = "2005",

month = nov,

doi = "10.1016/j.comgeo.2005.02.003",

language = "אנגלית",

volume = "32",

pages = "196--215",

journal = "Computational Geometry: Theory and Applications",

issn = "0925-7721",

publisher = "Elsevier B.V.",

number = "3",

}

TY - JOUR

T1 - Counting and representing intersections among triangles in three dimensions

AU - Ezra, Esther

AU - Sharir, Micha

N1 - 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).

PY - 2005/11

Y1 - 2005/11

N2 - We present an algorithm that efficiently counts all intersecting triples among a collection T of triangles in R 3 in nearly quadratic time. This solves a problem posed by Pellegrini [M. Pellegrini, On counting pairs of intersecting segments and off-line triangle range searching, Algorithmica 17 (1997) 380-398]. 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 planar objects of constant description complexity in R 3, 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 in the worst case.

AB - We present an algorithm that efficiently counts all intersecting triples among a collection T of triangles in R 3 in nearly quadratic time. This solves a problem posed by Pellegrini [M. Pellegrini, On counting pairs of intersecting segments and off-line triangle range searching, Algorithmica 17 (1997) 380-398]. 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 planar objects of constant description complexity in R 3, 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 in the worst case.

KW - 3sum-hard problems

KW - Arrangements

KW - Counting intersections

KW - Curve-sensitive cuttings

KW - Triangles in three dimensions

UR - https://www.scopus.com/pages/publications/84867932233

U2 - 10.1016/j.comgeo.2005.02.003

DO - 10.1016/j.comgeo.2005.02.003

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:84867932233

SN - 0925-7721

VL - 32

SP - 196

EP - 215

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

IS - 3

ER -

Counting and representing intersections among triangles in three dimensions

Abstract

Bibliographical note

Funding

Keywords

Access to Document

Other files and links

Fingerprint

Cite this