Abstract: The contact projective structures are 2-graded parabolic geometries modeled on the family of lines tangent to the contact structure on the projectivization of a symplectic vector space. There will be described a construction, in the manner of T.Y. Thomas, associating an ambient affine connection to an appropriate family of paths in a contact manifold. From this ambient connection it is straightforward to build a canonical regular Cartan connection via the tractor formalism. This Cartan connection is normal if and only if a certain invariant torsion vanishes. A geometric application is an analogue for pseudo-hermitian structures of the classical Beltrami theorem.