  We derive the equations of chains for path geometries on surfaces by solving
the equivalence problem of a related structure: sub-Riemannian geometry of
signature $(1,1)$ on a contact 3-manifold. This approach is significantly
simpler than the standard method of solving the full equivalence problem for
path geometry. We then use these equations to give a characterization of
projective path geometries in terms of their chains (the chains projected to
the surface coincide with the paths) and study the chains of four examples of
homogeneous path geometries. In one of these examples (horocycles in the
hyperbolic planes) the projected chains are bicircular quartics.
