  A Schnyder wood is an orientation and coloring of the edges of a planar map
satisfying a simple local property. We propose a generalization of Schnyder
woods to graphs embedded on the torus with application to graph drawing. We
prove several properties on this new object. Among all we prove that a graph
embedded on the torus admits such a Schnyder wood if and only if it is an
essentially 3-connected toroidal map. We show that these Schnyder woods can be
used to embed the universal cover of an essentially 3-connected toroidal map on
an infinite and periodic orthogonal surface. Finally we use this embedding to
obtain a straight-line flat torus representation of any toroidal map in a
polynomial size grid.
