The overlay of lower envelopes and its applications

Let ℱ and G be two collections of a total of n (possibly partially defined) bivariate, algebraic functions of constant maximum degree. The minimization diagrams of ℱ, G are the planar maps obtained by the xy-projections of the lower envelopes of ℱ, G, respectively. We show that the combinatorial complexity of the overlay of the minimization diagrams of ℱ and of G is O(n^2+ε), for any ε > 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.