Now construction of transverse curves seems to work correctly. There are two caveats. One is that it runs really slow compared to the RP^2 algorithm, even though the search for separatrix segments is faster. The other one is that except for the first curve, every curve has another one that has the same length. The reason is probably that every curve in RP^2 has two lifts, so we get non-isotopic, but symmetric curves with respect to the hyper elliptic involution. If possible, it would be nice to choose one of them canonically.