# 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).

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.

• Thanks, I know that the planets will move while a craft is traveling between them. I wanted to create a static solution and then iterate and find an approximate solution. This is for a game so a rough sketch will do. So far I couldn't find any solution – thalador Feb 18 '16 at 21:09
• @thalador: A simple way to find one solution would be to reflect $F$ across the perpendicular bisector of $UV$, producing $G$. It just won't be the only solution. – Henning Makholm Feb 18 '16 at 21:31
• @henning-makolm Unfortunately your idea doesn't seem to lead to a ellipse that contains U and V, which is necessary in my use-case – thalador Mar 7 '16 at 17:29
• @thalador: Then you must be doing something wrong. Can you provide a concrete example where it doesn't work for you? – Henning Makholm Mar 7 '16 at 18:04
• @henning-makolm I made a quick sketch (imgur.com/9uvzYeA). Which G do you refer to if you say "reflect"? Maybe that's why it doesn't work for me – thalador Mar 7 '16 at 18:41