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


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.


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.


  1. 1.
    P. Agarwal and M. Sharir, On the number of views of polyhedral terrains,Discrete Comput. Geom. 12 (1994), 177–182.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    P. Agarwal, M. Sharir, and P. Shor, Shorp upper and lower bounds for the length of general Davenport-Schinzel sequences.J. Combin. Theory Ser. A 52 (1989), 228–274.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    M. Atallah, Some dynamic computational geometry problems.Comput. Math. Appl. 11 (1985), 1171–1181.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    J. D. Boissonnat and K. Dobrindt, On-line randomized construction of the upper envelope of triangles and surface patches in ℝ3, Tech. Report 1878, INRIA, Sophia-Antipolis, 1993.Google Scholar
  5. 5.
    S. Cappell, J. E. Goodman, J. Pach, R. Pollack, M. Sharir, and R. Wenger, Common tangents and common transversals,Adv. in Math. 106 (1994), 198–215.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    B. Chazelle, H. Edelsbrunner, L. Guibas, and M. Sharir, A singly exponential stratification scheme for real semi-algebraic varieties and its applications,Proc. 16th Internat. Colloq. on Automata, Languages, and Programming, 1989, pp. 179–193.Google Scholar
  7. 7.
    K. Clarkson, H. Edelsbrunner, L. Guibas, M. Sharir, and E. Welzl, Combinatorial complexity bounds for arrangements of curves and spheresDiscrete Comput. Geom. 5 (1990), 99–160.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    K. Clarkson and P. Shor, Applications of random sampling in computational geometry, II,Discrete Comput. Geom. 4 (1989), 387–421.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    M. de Berg, K. Dobrindt, and O. Schwarzkopf, On lazy randomized incremental construction,Discrete Comput. Geom. 14 (1995), 261–286.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    H. Edelsbrunner,Algorithms in Combinatorial Geometry, Springer-Verlag, Berlin, 1987.zbMATHGoogle Scholar
  11. 11.
    H. Edelsbrunner, L. Guibas, and M. Sharir, The upper envelope of piecewise linear functions: Algorithms and applications,Discrete Comput. Geom. 4 (1989), 311–336.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    H. Edelsbrunner and M. Sharir, The maximum number of ways to stabn convex nonintersecting objects in the plane is 2n–2,Discrete Comput. Geom. 5 (1990), 35–42.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    J. Goodman, R. Pollack, and R. Wenger, Geometric transversal theory, inNew Trends in Discrete and Computational Geometry (J. Pach, ed.), Springer-Verlag, New York, 1993, pp. 163–198.Google Scholar
  14. 14.
    J. Goodman, R. Pollack, and R. Wenger, Bounding the number of geometric permutations induced byk-transversals,Proc. 10th Ann. Symp. on Computational Geometry, 1994, pp. 192–197.Google Scholar
  15. 15.
    D. Halperin and M. Sharir New bounds for lower envelopes in three dimensions, with applications to visibility in terrains,Discrete Comput. Geom. 12 (1994), 313–326.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    S. Hart and M. Sharir, Nonlinearity of Davenport-Schinzel sequences and of generalized path compression schemes,Combinatorica 6 (1986), 151–177.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    J. Heintz, T. Recio, and M.-F. Roy, Algorithms in real algebraic geometry and applications to computational geometry, inDiscrete and Computational Geometry: Papers from the DIMACS Special Year (J. E. Goodman, R. Pollack, and W. Steiger, eds.), AMS Press, Providence, RI, 1991, pp. 137–163.Google Scholar
  18. 18.
    M. Sharir, Onk-sets in arrangements of curves and surfaces,Discrete Comput. Geom. 6 (1991), 593–613.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    M. Sharir, Almost tight upper bounds for lower envelopes in higher dimensions,Discrete Comput. Geom. 12 (1994), 327–345.zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    M. Sharir and P. Agarwal,Davenport-Schinzel Sequences and Their Geometric Applications, Cambridge University Press, New York, 1995.zbMATHGoogle Scholar

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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