We characterize functions that are the directed X-ray
of a planar convex body from a source that is a positive distance
from the body. In addition to a concavity condition the necessary
and sufficient conditions involve the structure of points of zero
curvature and a priori estimates for derivatives of the directed
X-ray near supporting rays and points of zero curvature. The
techniques employed also lead to explicit methods for constructing
families of planar convex bodies with a common directed X-ray.
