  Suppose that the surfaces K0 and Kr are the boundaries of two convex,
complete, connected C^2 bodies in R^3. Assume further that the (Euclidean)
distance between any point x in Kr and K0 is always r (r > 0). For x in Kr, let
{\Pi}(x) denote the nearest point to x in K0. We show that the projection {\Pi}
preserves geodesics in these surfaces if and only if both surfaces are
concentric spheres or co-axial round cylinders. This is optimal in the sense
that the main step to establish this result is false for C^{1,1} surfaces.
Finally, we give a non-trivial example of a geodesic preserving projection of
two C^2 non-constant distance surfaces. The question whether for any C^2 convex
surface S0, there is a surface S whose projection to S0 preserves geodesics is
open.
