### Abstract

We show that the combinatorial complexity of all non-convex cells in an arrangement of n (possibly intersecting) triangles in 3-space is O(n^{7/3+δ}), for any δ>0, and that this bound is almost tight in the worst case. Our bound significantly improves a previous nearly cubic bound of Pach and Sharir. We also present a (nearly) worst-case optimal randomized algorithm for calculating a single cell of the arrangement, analyze some special cases of the problem where improved bounds (and better algorithms) can be obtained, and describe applications of our results to translational motion planning for polyhedra in 3-space.

Original language | English (US) |
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Title of host publication | Proceedings of the 4th Annual Symposium on Computational Geometry, SCG 1988 |

Publisher | Association for Computing Machinery, Inc |

Pages | 381-391 |

Number of pages | 11 |

ISBN (Electronic) | 0897912705, 9780897912709 |

DOIs | |

State | Published - Jan 6 1988 |

Event | 4th Annual Symposium on Computational Geometry, SCG 1988 - Urbana-Champaign, United States Duration: Jun 6 1988 → Jun 8 1988 |

### Other

Other | 4th Annual Symposium on Computational Geometry, SCG 1988 |
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Country | United States |

City | Urbana-Champaign |

Period | 6/6/88 → 6/8/88 |

### Fingerprint

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Proceedings of the 4th Annual Symposium on Computational Geometry, SCG 1988*(pp. 381-391). Association for Computing Machinery, Inc. https://doi.org/10.1145/73393.73432

**Triangles in space or Building (and analyzing) castles in the air.** / Aronov, Boris; Sharir, Micha.

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

*Proceedings of the 4th Annual Symposium on Computational Geometry, SCG 1988.*Association for Computing Machinery, Inc, pp. 381-391, 4th Annual Symposium on Computational Geometry, SCG 1988, Urbana-Champaign, United States, 6/6/88. https://doi.org/10.1145/73393.73432

}

TY - GEN

T1 - Triangles in space or Building (and analyzing) castles in the air

AU - Aronov, Boris

AU - Sharir, Micha

PY - 1988/1/6

Y1 - 1988/1/6

N2 - We show that the combinatorial complexity of all non-convex cells in an arrangement of n (possibly intersecting) triangles in 3-space is O(n7/3+δ), for any δ>0, and that this bound is almost tight in the worst case. Our bound significantly improves a previous nearly cubic bound of Pach and Sharir. We also present a (nearly) worst-case optimal randomized algorithm for calculating a single cell of the arrangement, analyze some special cases of the problem where improved bounds (and better algorithms) can be obtained, and describe applications of our results to translational motion planning for polyhedra in 3-space.

AB - We show that the combinatorial complexity of all non-convex cells in an arrangement of n (possibly intersecting) triangles in 3-space is O(n7/3+δ), for any δ>0, and that this bound is almost tight in the worst case. Our bound significantly improves a previous nearly cubic bound of Pach and Sharir. We also present a (nearly) worst-case optimal randomized algorithm for calculating a single cell of the arrangement, analyze some special cases of the problem where improved bounds (and better algorithms) can be obtained, and describe applications of our results to translational motion planning for polyhedra in 3-space.

UR - http://www.scopus.com/inward/record.url?scp=38049018141&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38049018141&partnerID=8YFLogxK

U2 - 10.1145/73393.73432

DO - 10.1145/73393.73432

M3 - Conference contribution

SP - 381

EP - 391

BT - Proceedings of the 4th Annual Symposium on Computational Geometry, SCG 1988

PB - Association for Computing Machinery, Inc

ER -