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

Abstract

Analytical expressions for the distribution of the length of chords corre- sponding to the a?ne invariant measure on the set of chords are given for convex polygons. These analytical expressions are a computational improve- ment over other expressions published in 2009 and 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 a?ne invariant measure on the set of chords is found for non convex polygons. In addition we give an algorithm to ?nd this multi-chord distribution which, for many concave polygons, is computation- ally more e?cient than the said analytical expression. The results also apply to non simply connected polygons.