This is a math problem whose solution would be great for physics, especially space flight planning.


Given two ellipses sharing a focus, could there be a pencil of ellipses sharing the same focus and tangent to the two given ellipses, or is there only one such ellipse? This problem has two versions: 2D (coplanar) and 3D (not coplanar).


Thanks for your answers.



Pencil of ellipses
It is a pencil (one parametric family). Instead of ellipses with one common focus,
consider ellipses with the same center. The reduction is achieved by the following:


Lemma. The map of the complex plane $z\mapsto z^2$ maps ellipses centered at $0$ to ellipses with one focus at zero, bijectively.


Now consider ellipses centered at $0$. Without loss of generality, we may assume that one of the two given ellipses $E\_1$ is a circle, another $E\_2$ arbitrary.
This can be achieved by a linear transformation of the plane $R^2$. Now choose an arbitrary line $L$ through the origin and stretch/compress $E\_1$ in the direction $L$. All stretched circles are tangent to the original circle $E\_1$ (at the two points of intersection of $E\_1$ with the line perpendicular to $L$). Choose the amount of stretch so that the stretched circle is tangent $E\_2$. The direction of the line $L\_1$ is the free parameter of the pencil. 


Proof of the lemma. Every ellipse centered at the origin (other than circle) can be parametrized 
as $t\mapsto c(re^{it}+r^{-1}e^{-it})$, $r>1$, the image of a circle $|z|=r$ under the Joukowski function. It has foci $2c,-2c$. Squaring gives
$t\mapsto c^2(r^2e^{2it}+r^{-2}e^{-2it}+2)$ which has one focus at $0$. Every ellipse with one focus at $0$ (other than circle) has this form, so it is a square of a uniquely defined ellipse with center at zero.


Remark. Of course this fact facilitates space missions to other planets:-) But one of the main things in planning a mission is the energy expense consideration. This makes some times for each given mission better than the other times.


Remark 2. The Lemma I learned from Arnold ("Huygens and Barrow...", Appendix I).


