Singular euclidean structures and riemann surfaces

Euclidean triangulated surface .Tl;M/ characterizes
a polyhedral metric with conical singularities associated with the vertices of the
triangulation. In this chapter we show that around any such a vertex we can introduce
complex coordinates in terms of which we can write down the conformal conical
metric, locally parametrizing the singular structure of .Tl;M/. This makes available
a powerful dictionary between 2–dimensional triangulations and complex geometry.