Does the geodesic on a surface $z = f(x,y)$ always trace out a straight line in the $xy$ plane? Let $z = f(x,y)$ be a surface. Let $(x_0, y_0, z_0)$ and $(x_1, y_1, z_1)$ be two points on that surface. Let $g(t) = \langle x(t), y(t), z(t)\rangle$ be a parameterization of the geodesic curve between the two points.  Is the following statement true?
Let $g_{xy} = \langle x(t), y(t), 0\rangle$ be the projection of the geodesic onto the $xy$ plane. Then $g_{xy}$ is the straight line defined by the points $(x_0, y_0)$ and $(x_1, y_1)$ on the $xy$ plane.

If the rigid surface containing the geodesic curve  rolls  in the plane of osculation with common normal without slipping then we have the same zero geodesic curvature in the plane as well as the surface, the trace on the plane is a straight line.
The instantaneous projection without any rolling is not in general a straight line.
The rolling contact line and projection of given line in fixed $(x-,y-) $ plane are two distinctly different lines.
