Side Disks of a Spherical Great Polygon

TL;DR

This paper investigates the intersection properties of disks constructed from arcs between vertices on a circle, establishing bounds on disjoint pairs and relating their intersection graph to convex polygon triangulations.

Contribution

It introduces a novel analysis of side disks in spherical polygons, providing bounds on disjoint pairs and linking their intersection graph to polygon triangulations.

Findings

Number of disjoint disk pairs is between (n-2)(n-3)/2 and n(n-3)/2.

Intersection graph is a subgraph of a convex n-gon triangulation.

Provides geometric bounds and graph-theoretic relations for side disks.

Abstract

Take a circle and mark $n\in\mathbb{N}$ points on it designated as vertices. For any arc segment between two consecutive vertices which does not pass through any other vertex, there is a disk centered at its midpoint and has its end points on the boundary. We analyze intersection behaviour of these disks and show that the number of disjoint pairs among them is between $\frac{(n-2)(n-3)}{2}$ and $\frac{n(n-3)}{2}$ and their intersection graph is a subgraph of a triangulation of a convex $n$-gon.