Non-crossing chords of a polygon with forbidden positions

TL;DR

This paper explores the structure of non-crossing chords in polygons, introduces the reduced Euler characteristic to characterize convexity, and extends the analysis to forbidden positions, revealing new geometric insights.

Contribution

It introduces the reduced Euler characteristic as a tool to analyze polygon diagonals and extends existing theories to include forbidden positions, providing a complete solution to a problem by G. C. Shephard.

Findings

Reduced Euler characteristic characterizes polygon convexity.

The shape of polygons is determined by forbidden diagonals.

Generalized Catalan numbers appear in the enumeration.

Abstract

In this paper, we systematically study non-crossing chords of simple polygons in the plane. We first introduce the reduced Euler characteristic of a family of line-segments, and subsequently investigate the structure of the diagonals and epigonals of a polygon. Interestingly enough, the reduced Euler characteristic of a subfamily of diagonals and epigonals characterizes the geometric convexity of polygons. In particular, an alternative and complete answer is given for a problem proposed by G. C. Shephard. Meanwhile, we extend such research to non-crossing diagonals and epigonals with forbidden positions in some appropriate sense. We prove that the reduced Euler characteristic of diagonals with forbidden positions only depends on the information involving convex partitions by those forbidden diagonals, and it determines the shapes of polygons in a surprising way. Incidentally, some kinds of generalized Catalan's numbers naturally arise.