TY - GEN

T1 - The overlay of lower envelopes in three dimensions and its applications

AU - Agarwal, Pankaj K.

AU - Schwarzkopf, Otfried

AU - Sharir, Micha

N1 - Publisher Copyright:
© 1995 ACM.

PY - 1995/9/1

Y1 - 1995/9/1

N2 - Let F and G be two collections of a total of n bivariate (possibly partially-defined) algebraic functions of constant maximum degree. The minimization diagrams of F, G are the planar subdivisions obtained by the projections of the lower envelopes of F, G respectively, onto the xy-plane. We show that the combinatorial complexity of the overlay of the minimization diagrams of F and G is O(n2+∈), for any ∈ > 0 (the actual bound that we prove is somewhat stronger). This result has several applications: (i) an O(n2+∈) upper bound on the complexity of the region in ℝ3 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 combinatorial complexity of the space of plane transversals of n compact convex simply-shaped sets in ℝ3.

AB - Let F and G be two collections of a total of n bivariate (possibly partially-defined) algebraic functions of constant maximum degree. The minimization diagrams of F, G are the planar subdivisions obtained by the projections of the lower envelopes of F, G respectively, onto the xy-plane. We show that the combinatorial complexity of the overlay of the minimization diagrams of F and G is O(n2+∈), for any ∈ > 0 (the actual bound that we prove is somewhat stronger). This result has several applications: (i) an O(n2+∈) upper bound on the complexity of the region in ℝ3 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 combinatorial complexity of the space of plane transversals of n compact convex simply-shaped sets in ℝ3.

UR - http://www.scopus.com/inward/record.url?scp=0040358077&partnerID=8YFLogxK

U2 - 10.1145/220279.220299

DO - 10.1145/220279.220299

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AN - SCOPUS:0040358077

T3 - Proceedings of the Annual Symposium on Computational Geometry

SP - 182

EP - 189

BT - Proceedings of the 11th Annual Symposium on Computational Geometry, SCG 1995

PB - Association for Computing Machinery

T2 - 11th Annual Symposium on Computational Geometry, SCG 1995

Y2 - 5 June 1995 through 7 June 1995

ER -