Computation of the multi-chord distribution of convex and concave polygons

TL;DR

This paper derives analytical formulas and algorithms for the distribution of chord lengths in convex and concave polygons, improving computational efficiency and enabling analysis of complex polygon shapes.

Contribution

It provides new analytical expressions and algorithms for multi-chord distributions in convex and non-convex polygons, including non-simply connected shapes.

Findings

Analytical expressions for convex polygons' chord distributions.

Efficient algorithms for multi-chord distributions in concave polygons.

Applicability to non-simply connected polygons.

Abstract

Analytical expressions for the distribution of the length of chords corresponding to the affine invariant measure on the set of chords are given for convex polygons. These analytical expressions are a computational improvement over other expressions published in 2011. The correlation function of convex polygons can be computed from the results obtained in this work, because it is determined by the distribution of chords. An analytical expression for the multi-chord distribution of the length of chords corresponding to the affine invariant measure on the set of chords is found for non convex polygons. In addition we give an algorithm to find this multi-chord distribution which, for many concave polygons, is computationally more efficient than the said analytical expression. The results also apply to non simply connected polygons.