How do I find the orbit of a moving point object given 3 past passing positions and the tangents at these passing positions, and given that the orbit is known to be an ellipse?
For 2D/3D cases the ellipse has 5/7 degrees of freedom. You say you have 3 points and their "tangents". A single 2D/3D point gives 3/5 equations. 3 points give 9/15 equations.
So that in both 2D and 3D cases you have an overdetermined equation system (actually it's overdetermined even with 2 points).
In the general case it may be solved for instance by one of those methods.