The maximum number of intersections of two polygons

TL;DR

This paper explores the maximum intersections between two polygons with odd numbers of vertices, proposing a conjecture that the maximum is (p-1)(q-1)+2 for simple polygons.

Contribution

It formulates and investigates a conjecture on the maximum intersections of two simple polygons with odd vertices, extending known results to more complex cases.

Findings

Conjecture that maximum intersections is (p-1)(q-1)+2 for odd p and q.

Analysis of cases where polygons are not simple or have even vertices.

Identification of the difficulty in proving the conjecture for simple polygons with odd vertices.

Abstract

We investigate the maximum number of intersections between two polygons with p and q vertices, respectively, in the plane. The cases where p or q is even or the polygons do not have to be simple are quite easy and already known, but when p and q are both odd and both polygons are simple, the problem is more difficult. The conjectured maximum is (p-1)(q-1)+2 for all odd p and q.