Simple Euclidean arrangements with one (>=5)-gon

TL;DR

This paper investigates Euclidean arrangements of pseudolines with a single polygon of five or more sides, deriving formulas for the number of triangles and quadrilaterals, and establishing conditions for stretchability.

Contribution

It provides a formula for counting polygons in arrangements with one large polygon and proves stretchability under certain adjacency conditions.

Findings

Number of triangles in the arrangement is n - k.

Number of quadrilaterals is k + n(n-5)/2.

Arrangements with all pseudolines adjacent to the polygon are stretchable.

Abstract

Let L be a simple Euclidean arrangement of n pseudolines. It is shown that if L has exactly one (>=5)=gon P, and k is the number of edges of P that are adjacent to an unbounded cell of the subarrangement of L induced by the pseudolines in P, then L has exactly n-k triangles and k+n(n-5)/2 quadrilaterals. We also prove that if each pseudoline of L is adjacent to P then L is stretchable.