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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?

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In an observational situation you generally do not have the tangents, nor full three dimensional data on the positions. Instead you have two angular degrees of freedom and a time hash for each observation. –  dmckee Feb 17 '12 at 20:18

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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.

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Although I do not fully understand "freedom", for 2D case , are 3 positions and 2 tangents enough and non overloaded? –  seven_swodniw Feb 17 '12 at 18:32
Suggestion to the answer(v1): Replace the word overloaded with overdetermined. –  Qmechanic Feb 17 '12 at 18:39
Sorry , I must have read your answer . Forget the previous my comment. –  seven_swodniw Feb 17 '12 at 18:40
I think that when counting degrees of freedom you should include the time dimension. –  dmckee Feb 17 '12 at 20:15
@Qmechanic: corrected, thanks –  valdo Feb 21 '12 at 21:52

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