## Abstract

We introduce (1/r)-cuttings for collections of surfaces in 3-space that are sensitive to an additional collection of curves. Specifically, let S be a set of n surfaces in ℝ^{3} of constant description complexity, and let C be a set of m curves in ℝ^{3} of constant description complexity. Let 1 ≤ r ≤ min{m, n} be a given parameter. We show the existence of a (1/r)-cutting Ξ of S of size O(r^{3+ε}), for any ε > 0, such that the number of crossings between the curves of C and the cells of Ξ is O(m^{1+ε}r). The latter bound improves, by roughly a factor of r, the bound that can be obtained for cuttings based on vertical decompositions. We view curve-sensitive cuttings as a powerful tool that is potentially useful in various scenarios that involve curves and surfaces in three dimensions. As a preliminary application, we use the construction to obtain a bound of O(m^{1/2+ε}n^{2+ε}), for any ε > 0, on the complexity of the multiple zone of m curves in the arrangement of n surfaces in 3-space.

Original language | English |
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Pages | 136-143 |

Number of pages | 8 |

DOIs | |

State | Published - 2003 |

Event | Nineteenth Annual Symposium on Computational Geometry - san Diego, CA, United States Duration: 8 Jun 2003 → 10 Jun 2003 |

### Conference

Conference | Nineteenth Annual Symposium on Computational Geometry |
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Country/Territory | United States |

City | san Diego, CA |

Period | 8/06/03 → 10/06/03 |

## Keywords

- Arrangements
- Curves and surfaces
- Cuttings
- Many cells
- Zone