### Abstract

A triangle is said to be δ-fat if its smallest angle is at least δ>0. A connected component of the complement of the union of a family of triangles is called hole. It is shown that any family of δ-fat triangles in the plane determines at most O (n/δ log 2/δ) holes. This improves on some earlier bounds of Efrat, Rote, Sharir, Matousek et al. Solving a problem of Agarwal and Bern, we also give a general upper bound for the number of holes determined by n triangles in the plane with given angles. As a corollary, we obtain improved upper bounds for the boundary complexity of the union of fat polygons in the plane, which, in turn, leads to better upper bounds for the running times of some known algorithms for motion planning, for finding a separator line for a set of segments, etc.

Original language | English |
---|---|

Title of host publication | Annual Symposium on Foundations of Computer Science - Proceedings |

Publisher | IEEE |

Pages | 423-431 |

Number of pages | 9 |

Publication status | Published - 2000 |

Event | 41st Annual Symposium on Foundations of Computer Science (FOCS 2000) - Redondo Beach, CA, USA Duration: Nov 12 2000 → Nov 14 2000 |

### Other

Other | 41st Annual Symposium on Foundations of Computer Science (FOCS 2000) |
---|---|

City | Redondo Beach, CA, USA |

Period | 11/12/00 → 11/14/00 |

### Fingerprint

### ASJC Scopus subject areas

- Hardware and Architecture

### Cite this

*Annual Symposium on Foundations of Computer Science - Proceedings*(pp. 423-431). IEEE.

**On the boundary complexity of the union of fat triangles.** / Pach, Janos; Tardos, G.

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

*Annual Symposium on Foundations of Computer Science - Proceedings.*IEEE, pp. 423-431, 41st Annual Symposium on Foundations of Computer Science (FOCS 2000), Redondo Beach, CA, USA, 11/12/00.

}

TY - GEN

T1 - On the boundary complexity of the union of fat triangles

AU - Pach, Janos

AU - Tardos, G.

PY - 2000

Y1 - 2000

N2 - A triangle is said to be δ-fat if its smallest angle is at least δ>0. A connected component of the complement of the union of a family of triangles is called hole. It is shown that any family of δ-fat triangles in the plane determines at most O (n/δ log 2/δ) holes. This improves on some earlier bounds of Efrat, Rote, Sharir, Matousek et al. Solving a problem of Agarwal and Bern, we also give a general upper bound for the number of holes determined by n triangles in the plane with given angles. As a corollary, we obtain improved upper bounds for the boundary complexity of the union of fat polygons in the plane, which, in turn, leads to better upper bounds for the running times of some known algorithms for motion planning, for finding a separator line for a set of segments, etc.

AB - A triangle is said to be δ-fat if its smallest angle is at least δ>0. A connected component of the complement of the union of a family of triangles is called hole. It is shown that any family of δ-fat triangles in the plane determines at most O (n/δ log 2/δ) holes. This improves on some earlier bounds of Efrat, Rote, Sharir, Matousek et al. Solving a problem of Agarwal and Bern, we also give a general upper bound for the number of holes determined by n triangles in the plane with given angles. As a corollary, we obtain improved upper bounds for the boundary complexity of the union of fat polygons in the plane, which, in turn, leads to better upper bounds for the running times of some known algorithms for motion planning, for finding a separator line for a set of segments, etc.

UR - http://www.scopus.com/inward/record.url?scp=0034513685&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034513685&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:0034513685

SP - 423

EP - 431

BT - Annual Symposium on Foundations of Computer Science - Proceedings

PB - IEEE

ER -