On the Distance Between two Ellipses in $\mathbb{R}^3$

Abstract

We prove that in the general case the number of critical points of the distance function between two ellipses in $\mathbb{R}^3$ equals to 4, 6, 8, 10, 12, 14 or 16. As an example, the distance between the nowadays orbits of Neptune and Pluto has six critical points: one maximum two minima and three saddle points. The global minimum is 2.503 au (astronomical units), while the global maximum is 79.111 au. If we ignore the perturbations, then in year 21103 AD the distance between Neptune and Pluto would be 2.527 au.