Résumé
By a result of W. P. Thurston, the moduli space of flat metrics on the sphere with n cone singularities of prescribed positive curvatures is a complex hyperbolic orbifold of dimension n − 3. The Hermitian form comes from the area of the metric. Using geometry of Euclidean polyhedra, we observe that this spacehas a natural decomposition into real hyperbolic convex polyhedra of dimensions n-3 and \leq (n-1)/2.By a result of W. Veech, the moduli space of flat metrics on a compact surface with cone singularities of prescribed negative curvatures has a foliation whose leaves have a local structure of complex pseudo-spheres. The complex structure comes again from the area of the metric. The form can be degenerate; its signature depends on the curvatures prescribed. Using polyhedral surfaces in Minkowski space, we show that this moduli space has a natural decomposition into sphericalconvex polyhedra.