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


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.


Combinatorial Complexity Surface Patch Intersection Curve Relative Interior Lower Envelope 


  1. 1.
    P. K. Agarwal and M. Sharir, On the number of views of polyhedral terrains,Proc. 5th Canadian Conf. on Computational Geometry, 1993, pp. 55–60.Google Scholar
  2. 2.
    P. K. Agarwal, M. Sharir, and P. Shor, Sharp upper and lower bounds for the length of general Davenport-Schinzel sequences,J. Combin. Theory Ser. A 52 (1989), 228–274.MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    M. de Berg, Private communication, 1993.Google Scholar
  4. 4.
    M. de Berg, D. Halperin, M. Overmars, and M. van Kreveld, Sparse arrangements and the number of views of polyhedral scenes, Manuscript, 1991.Google Scholar
  5. 5.
    B. Chazelle, H. Edelsbrunner, L. Guibas, and M. Sharir, A singly exponential stratification scheme for real semi-algebraic varieties and its applications,Theoretical Comput. Sci. 84 (1991), 77–105.MATHCrossRefGoogle Scholar
  6. 6.
    K. Clarkson, H. Edelsbrunner, L. Guibas, M. Sharir, and E. Welzl, Combinatorial complexity bounds for arrangements of curves and spheres,Discrete Comput. Geom. 5 (1990), 99–160.MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    K. Clarkson and P. Shor, Applications of random sampling in computational geometry, II.Discrete Comput. Geom. 4 (1989), 387–421.MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    R. Cole and M. Sharir, Visibility problems for polyhedral terrains,J. Symbolic Comput. 7 (1989), 11–30.MathSciNetCrossRefGoogle Scholar
  9. 9.
    D. Halperin, Algorithmic Motion Planning via Arrangements of Curves and of Surfaces, Ph.D. Dissertation, Computer Science Department, Tel Aviv University, July 1992.Google Scholar
  10. 10.
    D. Halperin and M. Sharir, Near-quadratic bounds for the motion planning problem for a polygon in a polygonal environment,Proc. 34th Ann. Symp. on Foundations of Computer Science, 1993, pp. 382–391.Google Scholar
  11. 11.
    S. Hart and M. Sharir, Nonlinearity of Davenport-Schinzel sequences and of generalized path compression schemes,Combinatorica 6 (1986), 151–177.MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    S. L. Kleiman and D. Laksov, Schubert calculus,Amer. Math. Monthly 79 (1972), 1061–1082.MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    J. Pach and M. Sharir, The upper envelope of piecewise linear functions and the boundary of a region enclosed by convex plates: Combinatorial analysis,Discrete Comput. Geom. 4 (1989), 291–309.MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    M. Pellegrini, On lines missing polyhedral sets in 3-space,Proc. 9th ACM Symp. on Computational Geometry, 1993, pp. 19–28.Google Scholar
  15. 15.
    J. T. Schwartz and M. Sharir, On the two-dimensional Davenport-Schinzel problem,J. Symbolic Comput. 10 (1990), 371–393.MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    M. Sharir, Onk-sets in arrangements of curves and surfaces,Discrete Comput. Geom. 6 (1991), 593–613.MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    M. Sharir, Almost tight upper bounds for lower envelopes in higher dimensions, this issue, pp. 327–345.Google Scholar
  18. 18.
    M. Sharir and P. K. Agarwal,Davenport-Schinzel Sequences and Their Geometric Applications, Cambridge University Press, Cambridge, to appear.Google Scholar

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

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