Working in only two dimensions and assuming that the central body is at the origin of the coordinate system, given two points in space and knowing the transit time between those points, as well as the direction of motion, is it possible to calculate a body's orbit?
It seems to me that there should be enough information - at $T=0$, the object was at $(θ_0, r_0)$ and at $T = T_1$, $(θ_1, r_1)$.
Two points + knowing one focus gives a set of ellipses - but the additional information of transit time seems to me like it should be enough to narrow it down to 1-2 orbits in the general case, travelling in opposite directions if there are two. Is it possible? If so, how would I go about doing so (numeric approximations are fine)?