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Discrete & Computational Geometry

, Volume 15, Issue 1, pp 1–13 | Cite as

The overlay of lower envelopes and its applications

  • P. K. Agarwal
  • O. Schwarzkopf
  • M. Sharir
Article

Abstract

Let\(\mathcal{F}\) and\(\mathcal{G}\) be two collections of a total ofn (possibly partially defined) bivariate, algebraic functions of constant maximum degree. The minimization diagrams of\(\mathcal{F}, \mathcal{G}\) are the planar maps obtained by the xy-projections of the lower envelopes of\(\mathcal{F}, \mathcal{G}\), respectively. We show that the combinatiorial complexity of the overlay of the minimization diagrams of\(\mathcal{F}\) and of\(\mathcal{G}\) is O(n2+ɛ), for any ɛ>0. This result has several applications: (i) a near-quadratic upper bound on the complexity of the region in 3-space enclosed between the lower envelope of one such collection of functions and the upper envelope of another collection; (ii) an efficient and simple divide-and-conquer algorithm for constructing lower envelopes in three dimensions; and (iii) a near-quadratic upper bound on the complexity of the space of all plane transversals of a collection of simply shaped convex sets in three dimensions.

Keywords

Computational Geometry Combinatorial Complexity Function Graph Lower Envelope Bivariate Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc 1996

Authors and Affiliations

  • P. K. Agarwal
    • 1
  • O. Schwarzkopf
    • 2
  • M. Sharir
    • 3
    • 4
  1. 1.Department of Computer ScienceDuke UniversityDurhamUSA
  2. 2.Department of Computer ScienceUtrecht UniversityUtrechtThe Netherlands
  3. 3.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael
  4. 4.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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