Discrete & Computational Geometry

, Volume 14, Issue 4, pp 385–410 | Cite as

Almost tight upper bounds for the single cell and zone problems in three dimensions

  • D. Halperin
  • M. Sharif
Article

Abstract

We consider the problem of bounding the combinatorial complexity of a single cell in an arrangement ofn low-degree algebraic surface patches in 3-space. We show that this complexity isO(n2+ε), for any ε>0, where the constant of proportionality depends on ε and on the maximum degree of the given surfaces and of their boundaries. This extends several previous results, almost settles a 9-year-old open problem, and has applications to motion planning of general robot systems with three degrees of freedom. As a corollary of the above result, we show that the overall complexity of all the three-dimensional cells of an arrangement ofn low-degree algebraic surface patches, intersected by an additional low-degree algebraic surface patch σ (the so-calledzone of σ in the arrangement) isO(n2+ε), for any ε>0, where the constant of proportionality depends on ε and on the maximum degree of the given surfaces and of their boundaries.

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Copyright information

© Springer-Verlag New York Inc. 1995

Authors and Affiliations

  • D. Halperin
    • 1
  • M. Sharif
    • 2
    • 3
  1. 1.Robotics Laboratory, Department of Computer ScienceStanford UniversityStanfordUSA
  2. 2.School of Mathematical SciencesTel Aviv UniversityRamat AvivIsrael
  3. 3.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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