Trace a surface curved with a line

I am trying to “trace” a curve surface with a continuous line in both directions (vertically and horizontally). The drawing shows my goals: trace with one line a surface which​ is curve and the starting point and the end point is the same.
For now I manage to do it with two different lines (one for each directions), but the points are inside the surface and not in the boundary (first problem), they are two different lines instead of one (second problem), the starting point and the end point should be the same. I subdivide the curve in point in order to get the point on the surface to connects and to be able to change the density.
Do you know how to do that?

path surface (12.3 KB)

Hello your questions are interesting but you began many discussions without finishing them or answering people that gives you some.hints (me for example). Will you make another thread with the same questions with holes ? More clarity direction … Will be appreciated

Thanks for the reply.
Sorry, I am new and I did not know that is possible to end the discussion. The question is different and the solution that you post is not really about solving my problem. I manage to create that, the problem is combining two lines path together.