Curve-sensitive cuttings

Vladlen Koltun, Micha Sharir

Research output: Contribution to conferencePaperpeer-review

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 ℝ3 of constant description complexity, and let C be a set of m curves in ℝ3 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(r3+ε), for any ε > 0, such that the number of crossings between the curves of C and the cells of Ξ is O(m1+εr). 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(m1/2+εn2+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 languageEnglish
Pages136-143
Number of pages8
DOIs
StatePublished - 2003
EventNineteenth Annual Symposium on Computational Geometry - san Diego, CA, United States
Duration: 8 Jun 200310 Jun 2003

Conference

ConferenceNineteenth Annual Symposium on Computational Geometry
Country/TerritoryUnited States
Citysan Diego, CA
Period8/06/0310/06/03

Keywords

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

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