The problem of calculating the curves which one sees in discrete point sets, reminds me of an idea that I once had in relation to the Euclidean TSP: those curves seem likely to be related to the ordering on the optimal tour - and I wanted to calculate them.
I called the task of finding those structures to calculate the "fingerprint" of a graph and, I wanted to find a solution that works for general TSP instances.
My solution looks like that:
1.) find for every pair of vertices $(u,v)$ a third vertex $w$, that minimizes the detour when going from $u$ to $w$ and then from $w$ to $v$ instead of going directly from $u$ to $v$; this corresponds to minimizing $dist(u,w)+dist(w,v)-dist(u,v)$ with respect to $w$.
2.) the previous step yields equivalency classes of edges, whose minimal detour leads over the same node.
3.) select from each equivalency class the shortest edge and connect its end points to the vertex, over which the minimal detour leads. Let's call that detour the vertex detour of the vertex over which it leads.
4.) define a graph $F$, whose vertices correspond to the detours and whose edges connect pairs from the vertex detours, if the respective detours overlap, i.e. are of the form $(r,t,s)$ and $(t,s,u)$
having identified the maximal connected components of $F$, it is easy to check various assumptions about the nature of the curves or about the reliability of one's visual perception.
The description of my method may be flawed; for this I apologize and ask for response.