Piercing Pairwise Intersecting Convex Shapes in the Plane

Abstract

Let C be any family of pairwise intersecting convex shapes in a two dimensional Euclidean space. Let $\tau (C)$ denote the piercing number of C, that is, the minimum number of points required such that every shape in C contains at least one of these points. Define a shape to be $\alpha$-fat when the ratio of the radius of the smallest disk that encloses the shape over the radius of the largest disk that is enclosed in the shape is at most $\alpha$. Define $\alpha (C)$ to be the minimum value where each shape in C is $\alpha (C)$-fat. We prove that $\tau (C) \le 43.789\alpha (C) + 4 = O(\alpha (C))$ for any set C consisting of pairwise intersecting convex $\alpha$-fat shapes. This improves the previous best known upper-bound of $O(\alpha (C)^2)$. This result has a number of implications on other results concerning fat shapes, such as designing data structures with less complexity for 3-D vertical ray shooting and computing depth orders. Additionally, our results reduce the time complexity of the query time of these data structures. We also get better bounds for some restricted families of shapes. We show that $(5\sqrt{2}+2)\alpha (C)+ 25 + 5\sqrt{2} \le 9.072\alpha (C) + 32.072 = O(\alpha (C))$ piercing points are sufficient to pierce a set of arbitrarily oriented $\alpha$-fat rectangles. We also prove that $\tau (C) = 2$ when C is a set of pairwise intersecting homothets of regular hexagons. We show that the piercing number of a set of pairwise intersecting homothets of an arbitrary convex shape is at most 15. This improves the previous best upper-bound of 16. We also give an algorithm to calculate the exact location of the piercing points.

Research supported in part by NSERC.