# Almost tight upper bounds for lower envelopes in higher dimensions

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## Abstract

We consider the problem of bounding the combinatorial complexity of the lower envelope of*n* surfaces or surface patches in*d*-space (*d*≥3), all algebraic of constant degree, and bounded by algebraic surfaces of constant degree. We show that the complexity of the lower envelope of*n* such surface patches is*O*(*n*^{d−1+∈}), for any ∈>0; the constant of proportionality depends on ∈, on*d*, on*s*, the maximum number of intersections among any*d*-tuple of the given surfaces, and on the shape and degree of the surface patches and of their boundaries. This is the first nontrivial general upper bound for this problem, and it almost establishes a long-standing conjecture that the complexity of the envelope is*O*(*n*^{d-2}λ_{ q }(*n*)) for some constant*q* depending on the shape and degree of the surfaces (where λ_{ q }(*n*) is the maximum length of (*n, q*) Davenport-Schinzel sequences). We also present a randomized algorithm for computing the envelope in three dimensions, with expected running time*O*(*n*^{2+∈}), and give several applications of the new bounds.

## Keywords

Voronoi Diagram Surface Patch Algebraic Surface Intersection Curve Relative Interior## References

- 1.P. K. Agarwal, B. Aronov, and M. Sharir, Computing envelopes in four dimensions with applications,
*Proc. 10th ACM Symp. on Computational Geometry*, 1994, pp. 348–358.Google Scholar - 2.P. K. Agarwal, O. Schwarzkopf, and M. Sharir, The overlay of lower envelopes and its applications, Manuscript, 1993.Google Scholar
- 3.P. K. Agarwal and M. Sharir, On the number of views of polyhedral terrains,
*Proc. 5th Canadian Conference on Computational Geometry*, 1993, pp. 55–61. (To appear in*Discrete Comput. Geom.*)Google Scholar - 4.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.zbMATHMathSciNetCrossRefGoogle Scholar - 5.J. Bochnak, M. Coste, and M-F. Roy,
*Géométrie Algébrique Réelle*, Springer-Verlag, Berlin, 1987.zbMATHGoogle Scholar - 6.J. D. Boissonnat and K. Dobrindt, On-line randomized construction of the upper envelope of triangles and surface patches in
**R**^{3}, Manuscript, 1993.Google Scholar - 7.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 - 8.B. Chazelle, H. Edelsbrunner, L. Guibas, and M. Sharir, Diameter, width, closest line pair, and parametric searching,
*Discrete Comput. Geom.***10**(1993), 183–196.zbMATHMathSciNetCrossRefGoogle Scholar - 9.B. Chazelle, H. Edelsbrunner, L. Guibas, M. Sharir, and J. Stolfi, Lines in space: combinatorics and algorithms,
*Algorithmica*, to appear.Google Scholar - 10.K. Clarkson, New Applications of random sampling in computational geometry,
*Discrete Comput. Geom.***2**(1987), 195–222.zbMATHMathSciNetCrossRefGoogle Scholar - 11.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.zbMATHMathSciNetCrossRefGoogle Scholar - 12.K. Clarkson and P. Shor, Applications of random sampling in computational geometry, II,
*Discrete Comput. Geom.***4**(1989), 387–421.zbMATHMathSciNetCrossRefGoogle Scholar - 13.G. E. Collins, Quantifier elimination for real closed fields by cylindric algebraic decomposition,
*Proc. 2nd GI Conf. on Automata Theory and Formal Languages*, Springer-Verlag, Berlin, 1975, pp. 134–183.Google Scholar - 14.M. de Berg, K. Dobrindt, and O. Schwarzkopf, On lazy randomized incremental construction,
*Proc. 26th ACM Symp. on Theory of Computing*, 1994, pp. 105–114.Google Scholar - 15.H. Edelsbrunner and R. Seidel, Voronoi diagrams and arrangements,
*Discrete Comput. Geom.***1**(1986), 25–44.zbMATHMathSciNetCrossRefGoogle Scholar - 16.J.-J. Fu and R.C.T. Lee, Voronoi diagrams of moving points in the plane,
*Internat. J. Comput. Geom. Appl.***1**(1991), 23–32.zbMATHMathSciNetCrossRefGoogle Scholar - 17.L. Guibas, J. Mitchell, and T. Roos, Voronoi diagrams of moving points in the plane
*Proc. 17th Internat. Workshop on Graph-Theoret. Concepts in Computer Science*, Lecture Notes in Computer Science, Vol. 570, Springer-Verlag, Berlin, 1991, pp. 113–125.CrossRefGoogle Scholar - 18.D. Halperin and M. Sharir, New bounds for lower envelopes in three dimensions, with applications to visibility in terrains, this issue, pp. 313–326.Google Scholar
- 19.D. Halperin and M. Sharir, Near-quadratic bounds for the motion planning problem for a polygon in a polygonal environment,
*Proc. 34th IEEE Symp. on Foundations of Computer Science*, 1993, pp. 382–391.Google Scholar - 20.D. Halperin and M. Sharir, Almost tight upper bounds for the single cell and zone problems in three dimensions,
*Proc. 10th ACM Symp. on Computational Geometry*, 1994, pp. 11–20.Google Scholar - 21.S. Hart and M. Sharir, Nonlinearity of Davenport-Schinzel sequences and of generalized path compression schemes.
*Combinatorica***6**(1986), 151–177.zbMATHMathSciNetCrossRefGoogle Scholar - 22.R. Hartshorne,
*Algebraic Geometry*, Springer-Verlag, New York, 1977.zbMATHCrossRefGoogle Scholar - 23.D. Haussler and E. Welzl, ε-nets and simplex range queries,
*Discrete Comput. Geom.***2**(1987), 127–151.zbMATHMathSciNetCrossRefGoogle Scholar - 24.J. Matoušek, Approximations and optimal geometric divide-and-conquer,
*Proc. 23rd ACM Symp. on Theory of Computing*, 1991, pp. 506–511.Google Scholar - 25.S. Mohaban and M. Sharir, Ray shooting amidst spheres in three dimensions and related problems, manuscript, 1993.Google Scholar
- 26.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.zbMATHMathSciNetCrossRefGoogle Scholar - 27.M. Pellegrini, On lines missing polyhedral sets in 3-space,
*Proc. 9th ACM Symp. on Computational Geometry*, 1993, pp. 19–28.Google Scholar - 28.J. T. Schwartz and M. Sharir, On the two-dimensional Davenport-Schinzel problem,
*J. Symbolic Comput.***10**(1990), 371–393.zbMATHMathSciNetCrossRefGoogle Scholar - 29.M. Sharir, On
*k*-sets in arrangements of curves and surfaces,*Discrete Comput. Geom.***6**(1991), 593–613.zbMATHMathSciNetCrossRefGoogle Scholar - 30.M. Sharir and P. K. Agarwal,
*Davenport-Schinzel Sequences and Their Geometric Applications*, Cambridge University Press, Cambridge, to appear.Google Scholar