# The overlay of lower envelopes and its applications

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## Abstract

Let\(\mathcal{F}\) and\(\mathcal{G}\) be two collections of a total of*n* (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(n^{2+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## References

- 1.P. Agarwal and M. Sharir, On the number of views of polyhedral terrains,
*Discrete Comput. Geom.***12**(1994), 177–182.zbMATHCrossRefMathSciNetGoogle Scholar - 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.M. Atallah, Some dynamic computational geometry problems.
*Comput. Math. Appl.***11**(1985), 1171–1181.zbMATHCrossRefMathSciNetGoogle Scholar - 4.J. D. Boissonnat and K. Dobrindt, On-line randomized construction of the upper envelope of triangles and surface patches in ℝ
^{3}, Tech. Report 1878, INRIA, Sophia-Antipolis, 1993.Google Scholar - 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.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.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.zbMATHCrossRefMathSciNetGoogle Scholar - 8.K. Clarkson and P. Shor, Applications of random sampling in computational geometry, II,
*Discrete Comput. Geom.***4**(1989), 387–421.zbMATHCrossRefMathSciNetGoogle Scholar - 9.M. de Berg, K. Dobrindt, and O. Schwarzkopf, On lazy randomized incremental construction,
*Discrete Comput. Geom.***14**(1995), 261–286.zbMATHCrossRefMathSciNetGoogle Scholar - 10.H. Edelsbrunner,
*Algorithms in Combinatorial Geometry*, Springer-Verlag, Berlin, 1987.zbMATHGoogle Scholar - 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.H. Edelsbrunner and M. Sharir, The maximum number of ways to stab
*n*convex nonintersecting objects in the plane is 2n–2,*Discrete Comput. Geom.***5**(1990), 35–42.zbMATHCrossRefMathSciNetGoogle Scholar - 13.J. Goodman, R. Pollack, and R. Wenger, Geometric transversal theory, in
*New Trends in Discrete and Computational Geometry*(J. Pach, ed.), Springer-Verlag, New York, 1993, pp. 163–198.Google Scholar - 14.J. Goodman, R. Pollack, and R. Wenger, Bounding the number of geometric permutations induced by
*k*-transversals,*Proc. 10th Ann. Symp. on Computational Geometry*, 1994, pp. 192–197.Google Scholar - 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.S. Hart and M. Sharir, Nonlinearity of Davenport-Schinzel sequences and of generalized path compression schemes,
*Combinatorica***6**(1986), 151–177.zbMATHCrossRefMathSciNetGoogle Scholar - 17.J. Heintz, T. Recio, and M.-F. Roy, Algorithms in real algebraic geometry and applications to computational geometry, in
*Discrete 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.M. Sharir, On
*k*-sets in arrangements of curves and surfaces,*Discrete Comput. Geom.***6**(1991), 593–613.zbMATHCrossRefMathSciNetGoogle Scholar - 19.M. Sharir, Almost tight upper bounds for lower envelopes in higher dimensions,
*Discrete Comput. Geom.***12**(1994), 327–345.zbMATHCrossRefMathSciNetGoogle Scholar - 20.M. Sharir and P. Agarwal,
*Davenport-Schinzel Sequences and Their Geometric Applications*, Cambridge University Press, New York, 1995.zbMATHGoogle Scholar