On the Union of Cylinders in Three Dimensions

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Abstract

We show that the combinatorial complexity of the union of n infinite cylinders in ℝ3, having arbitrary radii, is O(n2+ε), for any ε>0; the bound is almost tight in the worst case, thus settling a conjecture of Agarwal and Sharir (Discrete Comput. Geom. 24:645-685, 2000), who established a nearly-quadratic bound for the restricted case of nearly congruent cylinders. Our result extends, in a significant way, the result of Agarwal and Sharir (Discrete Comput. Geom. 24:645-685, 2000), in particular, a simple specialization of our analysis to the case of nearly congruent cylinders yields a nearly-quadratic bound on the complexity of the union in that case, thus significantly simplifying the analysis in Agarwal and Sharir (Discrete Comput. Geom. 24:645-685, 2000). Finally, we extend our technique to the case of "cigars" of arbitrary radii (that is, Minkowski sums of line-segments and balls) and show that the combinatorial complexity of the union in this case is nearly-quadratic as well. This problem has been studied in Agarwal and Sharir (Discrete Comput. Geom. 24:645-685, 2000) for the restricted case where all cigars have (nearly) equal radii. Based on our new approach, the proof follows almost verbatim from the analysis for infinite cylinders and is significantly simpler than the proof presented in Agarwal and Sharir (Discrete Comput. Geom. 24:645-685, 2000).

Original languageEnglish
Pages (from-to)45-64
Number of pages20
JournalDiscrete and Computational Geometry
Volume45
Issue number1
DOIs
StatePublished - Jan 2011
Externally publishedYes

Bibliographical note

Funding Information:
Part of the work on this paper was performed at the Department of Computer Science, Duke University, Durham, NC 27708-0129, USA, and has been supported by NSF under grants CNS-05-40347, CFF-06-35000, and DEB-04-25465, by ARO grants W911NF-04-1-0278 and W911NF-07-1-0376, and by an NIH grant 1P50-GM-08183-01. A preliminary version of this paper has appeared in Proc. 49th Annu. IEEE Sympos. Found. Comput. Sci. 2008.

Keywords

  • (1/r)-cuttings
  • Geometric arrangements
  • Lower envelopes
  • Union of simply-shaped bodies

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