  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.

Metric Geometry;Geometric Topology