Construct ellipse from two arbitrary points and a given focal point Can an ellipse be constructed from these three given points: * Focal point $\mathrm F$ * Two arbitrary points $\mathrm U$, $\mathrm V$ lying on the ellipse The background is a orbital maneuver between two celestial bodies. The celestial bodies circulate around the same focal point (e.g. the sun), hence a transfer orbit (the ellipse) has to have the same focal point.

No, that is not enough. The most we can say is that the _other_ focus $G$ must satisfy $$ |UG| - |VG| = |FV| - |FU| $$ where the right-hand side is constant.

The locus of the possible $G$s will in general be one branch of a hyperbola.

(Once you decide where $G$ is, a single point one the ellipse is enough to determine it, of course).

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Note that the problem as stated is not directly applicable to finding transfer orbits anyway. You can construct any number of ellipses with the sun as a focus that connect the positions of Earth and Mars _now_ , but the time it takes to traverse those trajectories will not be the same, so not all of them will actually hit Mars at the time Mars is at the fixed endpoints.