  We investigate the SL(2,R) invariant geodesic curves with the as- sociated
invariant distance function in parabolic geometry. Parabolic geom- etry
naturally occurs in the study of SL(2,R) and is placed in between the elliptic
and the hyperbolic (also known as the Lobachevsky half-plane and 2- dimensional
Minkowski half-plane space-time) geometries. Initially we attempt to use
standard methods of finding geodesics but they lead to degeneracy in this
setup. Instead, by studying closely the two related elliptic and hyperbolic
geometries we discover a unified approach to a more exotic and less obvious
parabolic case. With aid of common invariants we describe the possible dis-
tance functions that turn out to have some unexpected, interesting properties.
