We construct for every half-translation surface satisfying the topological Veech dichotomy a tessellation of the Poincare upper half plane generalising the Farey tessellation for a flat torus. By construction, the Veech group stabilizes this tessellation. As a consequence, we get a bound on the volume of the corresponding Teichm\"uller curve for a lattice surface (Veech surface). There is a natural graph underlying this tessellation on which the affine group acts by automorphisms. We provide algorithms to determine a `coarse' fundamental domain and a generating set for the Veech group based on this graph. We also show that this graph has infinite diameter and is Gromov hyperbolic.