How to extract the curve network from the parametrization?

1. Remove degenerate triangles from the parametrization, then apply QEx and extract the edges that we want.
2. Apply QEx to the shifted parametrization, then somehow extract a "skeleton". The screenshots below is approximately what I mean. Works well for simple input, but for more compilcated data it's not so great.
3. We could somehow define an implicit function on the parametrization? Then extract what we need (I don't like this idea very much)
4. Adrien's suggestion: trace the curve network in the parametrized mesh, then fit Bezier curves.

One thing I observed and I think will be important to consider is where the mesh is cut. Right now, using libigl, the cuts are defined implicitly by combing the field, and therefore I cannot control their placement. It seems to me we should avoid cutting along the contour and always cut in the direction which is orthogonal to the contour.