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 R 3 of constant description complexity, and let C be a set of m curves in R 3 of constant description complexity. Let 1= r = min{ m,n } be a given parameter. We show the existence of a (1/ r )-cutting ? of S of size O ( r 3+e ), for any e > 0 , such that the number of crossings between the curves of C and the cells of ? is O ( m 1+e ). 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+e n 2+e ), for any e > 0 , on the complexity of the multiple zone of m curves in the arrangement of n surfaces in 3-space