We consider a polynomial vector field F in the plane. We begin by showing that the trajectories of F may be piecewise trivialized. In fact this result holds in far greater generality than polynomial F, namely we need only assume that F is definable in an o-minimal structure to derive the trivialization. Given this we may abstract the situation for such a vector field F by attaching to it a set of first order logical axioms T_F that describe the geometric behavior of the trajectories. Under appropriate finiteness conditions for the closed trajectories of F we show that T_F is a set of axioms of finite rank, where rank is interpreted in the recently developed context of thorn forking. We also have a converse, namely T_F being of the appropriate rank implies the finiteness conditions for closed trajectories of F alluded to above. Finally we point out how the logical considerations relevant to the theory T_F lead to numerous questions in the general model theory of densely ordered structures. This is joint work with Patrick Speissegger.