Discrete & Computational Geometry

, Volume 12, Issue 3, pp 313–326 | Cite as

New bounds for lower envelopes in three dimensions, with applications to visibility in terrains

  • D. Halperin
  • M. Sharir
Article

Abstract

We consider the problem of bounding the complexity of the lower envelope ofn surface patches in 3-space, all algebraic of constant maximum degree, and bounded by algebraic arcs of constant maximum degree, with the additional property that the interiors of any triple of these surfaces intersect in at most two points. We show that the number of vertices on the lower envelope ofn such surface patches is\(O(n^2 \cdot 2^{c\sqrt {\log n} } )\), for some constantc depending on the shape and degree of the surface patches. We apply this result to obtain an upper bound on the combinatorial complexity of the “lower envelope” of the space of allrays in 3-space that lie above a given polyhedral terrainK withn edges. This envelope consists of all rays that touch the terrain (but otherwise lie above it). We show that the combinatorial complexity of this ray-envelope is\(O(n^3 \cdot 2^{c\sqrt {\log n} } )\) for some constantc; in particular, there are at most that many rays that pass above the terrain and touch it in four edges. This bound, combined with the analysis of de Berget al. [4], gives an upper bound (which is almost tight in the worst case) on the number of topologically different orthographic views of such a terrain.

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Copyright information

© Springer-Verlag New York Inc. 1994

Authors and Affiliations

  • D. Halperin
    • 1
  • M. Sharir
    • 1
    • 2
  1. 1.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  3. 3.Robotics Laboratory, Computer Science DepartmentStanford UniversityStanfordUSA

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