Curve-sensitive cuttings

We introduce (1/r)-cuttings for collections of surfaces in 3-space, such that the cuttings are sensitive to an additional collection of curves. Specifically, let S be a set of n surfaces and let C be a set of m curves in ℝ ³, all of constant description complexity. Let 1 ≤ r ≤ min{m, n} be a given parameter. We show the existence of a (1/r)-cutting Θ 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 Θ is O(mr ^1+ε). 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 for 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 ^1/2n ^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. After the conference publication of this paper [V. Koltun and M. Sharir, Proceedings of the 19th ACM Symposium on Computational Geometry, 2003, pp. 136-143], curve-sensitive cuttings were applied to derive an algorithm for efficiently counting triple intersections among planar convex objects in three dimensions [E. Ezra and M. Sharir, Proceedings of the 20th ACM Symposium on Computational Geometry, 2004, pp. 210-219], and we expect additional applications to arise in the future.