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

Second degree surfaces are Liouville surfaces with Gauss curvature K=p(V)q(V). More precisely:

 
      (a-k)(b-k)(c-k)
K = - --------------- , a < b < c
      (k-u)^2 (k+v)^2

The first two pictures correspond to the case in which the parameter of the geodesic lies between a and b:

a <= C < b

If C=a then the geodesic will correspond to one of the parameter lines in the u direction.

As C approaches b, 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. If the starting point is not on one of the null-lines then the geodesic will pass through all the critical points but will never close.

The following two images correspond to the case in which the parameter of the geodesic is between b and c:

b < C <= c.

If C=c then the geodesic will correspond to one of the parameter lines in the v direction.