On Intersecting Polygons

03/20/2023

∙

by   Kada Williams, et al.

∙

0

∙

share

Consider two regions in the plane, bounded by an n-gon and an m-gon, respectively. At most how many connected components can there be in their intersection? This question was asked by Croft. We answer this asymptotically, proving the bounds ⌊m/2⌋·⌊n/2⌋≤ f(n,m)≤⌊m/2⌋·n/2 + m/2 where f(n,m) denotes the maximal number of components and m≤ n. Furthermore, we give an exact answer to the related question of finding the maximal number of components if the m-gon is required to be convex: ⌊m+n-2/2⌋ if n≥ m+2 and n-2 otherwise.