Curve-sensitive cuttings

We introduce (1/r)-cuttings for collections of surfaces in 3-space that are sensitive to an additional collection of curves. Specifically, let S be a set of n surfaces in ℝ³ of constant description complexity, and let C be a set of m curves in ℝ³ of constant description complexity. Let 1 ≤ r ≤ min{m, n} be a given parameter. We show the existence of a (1/r)-cutting Ξ of S of size O(r^3+ε), for any ε > 0, such that the number of crossings between the curves of C and the cells of Ξ is O(m^1+εr). The latter bound improves, by roughly a factor of r, the bound that can be obtained for cuttings based on vertical decompositions. We view curve-sensitive cuttings as a powerful tool that is potentially useful in various scenarios that involve curves and surfaces in three dimensions. As a preliminary application, we use the construction to obtain a bound of O(m^1/2+εn^2+ε), for any ε > 0, on the complexity of the multiple zone of m curves in the arrangement of n surfaces in 3-space.