### 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 |

### Publication series

Name | Proceedings of the 4th Annual Symposium on Computational Geometry, SCG 1988 |
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### 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 |

### ASJC Scopus subject areas

- Geometry and Topology

## Fingerprint Dive into the research topics of 'Triangles in space or Building (and analyzing) castles in the air'. Together they form a unique fingerprint.

## Cite this

Aronov, B., & Sharir, M. (1988). Triangles in space or Building (and analyzing) castles in the air. In

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