given a finite sample of a smooth vector field, i.e. a set $$S=\{(p_i,v_i)\mid i\in \{1,\dots, n\}, p_i,v_i\in\mathbb{R}^2\}$$ where the $p_i$ are the points at which we have a vector $v_i$, how do I construct an integral curved through this vector field. I would guess there is no guarantee in general that such an integral curve would be unique, so I'm interested in a general method where I can supply an optimisation function and the set $S$, and from that compute an integral curve minimizing the optimisation function.
I would also be interested in ways of computing not an integral curve through such a field, but a transversal curve through such a field.
Any suggestions for papers/algorithms/methods of solving this problem would be appreciated.