Sweeping arrangements of curves

Abstract

We consider arrangements of curves that intersect palrwise in at most k points. We show that a curve can sweep any such arrangement and maintain the k-intersection property if and only if k equals 1 or 2. We apply thii result to an eclectic set of problems: finding boolean formula for polygons with curved edges, counting triangles and digons in arrangements of pseudocircles, and finding extension curves for arrangements. We also discuss implementing the sweep.