Liouville Surfaces with K=p(U)q(V)

One way to visualize geodesics on a surface is to use a graph of the Gauss curvature (u,v, K(u,v)).

The first case corresponds to the geodesics with parameter 1 <= C < 2 on a ellipsoid:

In this case the geodesic passes through the four critical points. In our case this corresponds to C = 2 and on the ellipsoids this geodesic represents the major ellipse through the umbilic points.

The last case corresponds to the case in which 2 < C <= 3: