to the direction
will now form a curved surface, no longer a plane as in Euclidean space. A surface of this sort formed by geodesics is called a geodesic surface; and the one we are considering will be the geodesic surface passing through
and perpendicular to the direction
at
. We see, then, that in a Euclidean space the geodesic surfaces are planes, just as the geodesics are straight lines; whereas in a non-Euclidean space the geodesic surfaces are curved, just as the geodesics are curves.