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(r3+ε ), for any ε > 0, such that the number of crossings between the curves of C and the cells of Ξ is O(m1+ε r). 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(m1/2+ε n2+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...

• Type: Conference
• Year: 2003
• Where: COMPGEOM
• Authors: Vladlen Koltun, Micha Sharir
• Keywords: COMPGEOM 2003, Constant Description Complexity, Curve-sensitive Cuttings, Surfaces