Lower Envelopes of Surface Patches in 3-Space

Let Σ be a collection of n surface patches, each being the graph of a partially defined semi-algebraic function of constant description complexity, and assume that any triple of them intersect in at most s = 2 points. We show that the complexity of the lower envelope of the surfaces in Σ is O(n² log^6+ε n), for any ε > 0. This almost settles a long-standing open problem posed by Halperin and Sharir [26], thirty years ago, who showed the nearly-optimal albeit weaker bound of O(n² · 2^c√log ⁿ) on the complexity of the lower envelope, where c > 0 is some constant. Our approach is fairly simple and is based on hierarchical cuttings and gradations, as well as a simple charging scheme. We extend our analysis to the case s > 2, under a “favorable cross section” assumption, in which case we show that the bound on the complexity of the lower envelope is O(n² log^11+ε n), for any ε > 0. Incorporating these bounds with the randomized incremental construction algorithms of Boissonnat and Dobrindt [16], we obtain efficient constructions of lower envelopes of surface patches with the above properties, whose overall expected running time is O(n²polylogn), as well as efficient data structures that support point location queries in their minimization diagrams in O(log² n) expected time.