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An ideal triangulation of a singular flat surface is a geodesic triangulationsuch 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 ofgeodesics having length $\leq L$ connecting them, where $L$ is any positivenumber, we prove that each singular flat surface has an ideal triangulationprovided that the surface has singular points when it has no boundarycomponents, or each of its boundary components has a singular point. Also, weprove that such a surface contains a finite number of geodesics which connectits singular points so that when we cut the surface through these arcs we get aflat disk with a non-singular interior.

Ideal Triangulation and Disk Unfolding of a Singular Flat Surface