  A directed curve is a possibly singular curve with well-defined tangent lines
along the curve. Then the tangent surface to a directed curve is naturally
defined as the ruled surface by tangent geodesics to the curve, whenever any
affine connection is endowed with the ambient space. In this paper the local
diffeomorphism classification is completed for generic directed curves. Then it
turns out that the swallowtails and open swallowtails appear generically for
the classification on singularities of tangent surfaces.
