  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
space has a natural decomposition into real hyperbolic convex polyhedra of
dimensions $n-3$ and $\leq \frac{1}{2}(n-1)$.
  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 spherical convex polyhedra.
