  One may characterize the projective geometry \(\Pi\) on an  affine manifold either in terms of an equivalence class of geodesic  curves under arbitrary parameter   diffeomorphisms[39],  thereby eliminating all the parameter descriptions and hence all  possible notions of distance along the curves satisfying   (6),[40]  or one may take the process of autoparallelism of directions  as fundamental in defining a projective structure. 