## 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 |
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Pages (from-to) | 196-215 |

Number of pages | 20 |

Journal | Computational Geometry: Theory and Applications |

Volume | 32 |

Issue number | 3 |

DOIs | |

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: estere@post.tau.ac.il (E. Ezra), michas@post.tau.ac.il (M. Sharir).

## Keywords

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