An approximate arrangement algorithm for semi-algebraic curves

Victor Milenkovic, Elisha Sacks

Research output: Chapter in Book/Report/Conference proceedingConference contribution

4 Scopus citations

Abstract

We present an arrangement algorithm for plane curves. The inputs are (1) continuous, compact, x-monotone curves and (2) a module that computes approximate crossing points of these curves. There are no general position requirements. We assume that the crossing module output is e accurate, but allow it to be inconsistent, meaning that three curves are in cyclic y order over an x interval. The curves are swept with a vertical line using the crossing module to compute and process sweep events. When the sweep detects an inconsistency, the algorithm breaks the cycle to obtain a linear order. We prove correctness in a realistic computational model of the crossing module. The number of vertices in the output is V = 2n + N + min(3kn, n2/2) and the running time is O(Vlogn) for n curves with N crossings and k inconsistencies. The output arrangement is realizable by curves that are 0(ε + knε) close to the input curves, except in kn neighborhoods of the curve tails. The accuracy can be guaranteed everywhere by adding tiny horizontal extensions to the segment tails, but without the running time bound. An implementation is described for semi-algebraic curves based on a numerical equation solver. Experiments show that the extensions only slightly increase the running time and have little effect on the error. On challenging data sets, the number of inconsistencies is at most 3N, the output accuracy is close to ε, and the running time is close to that of the standard, non-robust floating point sweep.

Original languageEnglish (US)
Title of host publicationProceedings of the Twenty-Second Annual Symposium on Computational Geometry 2006, SCG'06
PublisherAssociation for Computing Machinery (ACM)
Pages237-245
Number of pages9
ISBN (Print)1595933409, 9781595933409
DOIs
StatePublished - 2006
Event22nd Annual Symposium on Computational Geometry 2006, SCG'06 - Sedona, AZ, United States
Duration: Jun 5 2006Jun 7 2006

Publication series

NameProceedings of the Annual Symposium on Computational Geometry
Volume2006

Other

Other22nd Annual Symposium on Computational Geometry 2006, SCG'06
CountryUnited States
CitySedona, AZ
Period6/5/066/7/06

Keywords

  • Arrangements
  • Robust computational geometry

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Computational Mathematics

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