Approximating the k-level in three-dimensional plane arrangements

Sariel Har-Peled, Haim Kaplan, Micha Sharir

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

Abstract

Let H be a set of n non-vertical planes in three dimensions, and let r < n be a parameter. We give a construction that approximates the (n/r)-level of the arrangement A(H) of H by a terrain consisting of O(r/ε3) triangular faces, which lies entirely between the levels n/r and (1 + ε)n/r. The proof does not use sampling, and exploits techniques based on planar separators and various structural properties of levels in three-dimensional arrangements and of planar maps. This leads to conceptually cleaner constructions of shallow cuttings in three dimensions. On the way, we get two other results that are of independent interest: (a) We revisit an old result of Bambah and Rogers (J Lond Math Soc 1(3):304-314, 1952) about triangulating a union of convex pseudo-disks, and provide an alternative proof that yields an efficient algorithmic implementation. (b) We provide a new construction of cuttings in two dimensions.

Original languageEnglish
Title of host publicationA Journey through Discrete Mathematics
Subtitle of host publicationA Tribute to Jiri Matousek
Pages467-503
Number of pages37
ISBN (Electronic)9783319444796
DOIs
StatePublished - 1 Jan 2017

All Science Journal Classification (ASJC) codes

  • General Computer Science
  • General Economics,Econometrics and Finance
  • General Business,Management and Accounting
  • General Mathematics

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