Counting triangulations of some classes of subdivided convex polygons

TL;DR

This paper derives explicit formulas, generating functions, and asymptotic behaviors for counting triangulations of subdivided convex polygons, linking these results to minimal triangulation problems in computational geometry.

Contribution

It provides new explicit formulas and asymptotic analysis for triangulations of subdivided convex polygons, addressing a classical open problem.

Findings

Explicit formulas for triangulation counts

Generating functions for subdivided polygons

Asymptotic behavior as parameters grow

Abstract

We compute the number of triangulations of a convex $k$-gon each of whose sides is subdivided by $r-1$ points. We find explicit formulas and generating functions, and we determine the asymptotic behaviour of these numbers as $k$ and/or $r$ tend to infinity. We connect these results with the question of finding the planar set of points in general position that has the minimum possible number of triangulations - a well-known open problem from computational geometry.