Minimum Separating Distance Between Two Ellipsoids

Question

Suppose I have two ellipsoids in three dimensions, defined by:

$Q_{0} = (X-v_{0})^{T}A_{0}(X-v_{0})-1 = 0$

$Q_{1} = (X-v_{1})^{T}A_{1}(X-v_{1})-1 = 0$

Is there a technique to find the minimum distance and direction that $v_{0}$ must be displaced from $v_{1}$ such that the ellipsoids are not overlapping?

Answer 1

Partial answer:

Without loss of generality, take $v_0 = 0$. Consider the minimization problem $$ \min X^TA_0 X \quad \text{s.t.} \quad (X - v_1)^TA_1 (X-v_1) = 1. $$ The two ellipsoids overlap if this minimum is $1$ or smaller. Our goal is ultimately to find the $v$ of smallest magnitude for which this minimum is exactly $1$. With Lagrange multipliers, we end up with the equation $$ 2A_0X = \lambda(2 A_1X - A_1v_1) \implies 2(\lambda A_1 - A_0)X = A_1 v_1 \implies\\ 2(\lambda I - A_1^{-1}A_0)X = v_1. $$

Answer 2

The closed-form formula exists for two ellipses(2D). It's pretty long. You can find it here. I am not sure if there exists a closed-form formula for two Ellipsoids but for the iterative solution you can use this.