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 picture corresponds to the case in which the parameter of the geodesic is such that 3 < C <=4:

As C approaches 3, the geodesic approaches the critical points. If the starting point of the geodesic is on one of the null-lines than the geodesic will be closed and will go through all four critical points:

The first picture corresponds to the case in which the parameter of the geodesic is such that 2 <= C <=3: