Ideal Triangulation and Disk Unfolding of a Singular Flat Surface

TL;DR

This paper proves the existence of ideal triangulations for certain singular flat surfaces and demonstrates how cutting along specific geodesics yields a non-singular flat disk, advancing understanding of their geometric structure.

Contribution

It establishes conditions under which singular flat surfaces admit ideal triangulations and describes a method to partition them into flat disks via geodesic cuts.

Findings

Existence of ideal triangulation for surfaces with singular points and boundary conditions.

Finite geodesic connections between singular points.

Method to cut surfaces into non-singular flat disks.

Abstract

An ideal triangulation of a singular flat surface is a geodesic triangulation such that its vertex set is equal to the set of singular points of the surface. Using the fact that each pair of points in a surface has a finite number of geodesics having length $\leq L$ connecting them, where $L$ is any positive number, we prove that each singular flat surface has an ideal triangulation provided that the surface has singular points when it has no boundary components, or each of its boundary components has a singular point. Also, we prove that such a surface contains a finite number of geodesics which connect its singular points so that when we cut the surface through these arcs we get a flat disk with a non-singular interior.