A singular euclidean structure on a closed surface is a metric which is locally isometric to the euclidean plane except for a finite number of cone singularities. We descibe a canonical cellutation of the space of singular euclidean structures on a given surface with a given number of cone points. The cells are convex euclidean polyhedra. We also obtain a cellution of the Teichmuller space (times a simplex) of the same surface with the same number of punctures, by exactly the same polyhedral complex. This gives a canonical identification of these spaces, and hence a new proof that the space of singular euclidean structures is topologically a ball.
J. London Math. Soc. 44 (1991) 553-565.