  Contact projective structures have been profoundly studied by D.J.F. Fox. He
associated to a contact projective structure a canonical projective structure
on the same manifold. We interpret Fox' construction in terms of the equivalent
parabolic (Cartan) geometries, showing that it is an analog of Fefferman's
construction of a conformal structure associated to a CR structure. We show
that, on the level of Cartan connections, this Fefferman--type construction is
compatible with normality if and only if the initial structure has vanishing
contact torsion. This leads to a geometric description of the paths that have
to be added to the contact geodesics of a contact projective structure in order
to obtain the subordinate projective structure. They are exactly the chains
associated to the contact projective structure, which are analogs of the
Chern-Moser chains in CR geometry. Finally, we analyze the consequences for the
geometry of chains and prove that a chain-preserving contactomorphism must be a
morphism of contact projective structures.
