The Overlay of Lower Envelopes and Its Applications

Let F and 9 be two collections of a total of n bivariate algebraic functions of constant maximum degree. The minimization diagrams of F, 9 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 9 is O(n2+E), for any £, > O. This result has several applications: (i) a nearquadratic 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.