A curve is the intersection of two ellipsoids that are given by
$ (r - C_1)^T Q_1 (r - C_1) = 1 $
$ (r - C_2)^T Q_2 (r - C_2) = 1 $
I'd like to find the points on this curve of intersection that are at a minimum/maximum distance from the origin.
My attempt:
Building a Lagrange multiplier function
$ g(r , \lambda_1, \lambda_2) = r^T r + \lambda_1 \left( (r - C_1)^T Q_1 (r - C_1) - 1 \right) + \lambda_2 \left( (r - C_2)^T Q_2 (r - C_2) - 1\right) $
it follows that the gradient is
$ \nabla_r g = 2 \bigg( r + \lambda_1 Q_1 (r - C_1) + \lambda_2 Q_2 (r - C_2) \bigg) = \mathbb{0}$
In addition to this equation we have the two equations of the two ellipsoids.
There are $5$ unkowns which are the three coordinates of $r$ and $\lambda_1$ and $\lambda_2$.
That's where I am at. The task now is to solve these equations for the unknowns.
Any help would be appreciated. Thank you all.
Following the analysis given in the question, we have the constraint
$ r + \lambda_1 Q_1 (r - C_1) + \lambda_2 Q_2 (r - C_2) ) = 0 $
This is a vector equation, and what it says is that $r$ is a linear combination of $ Q_1 (r - C_1) $ and $ Q_2 (r - C_2) $.
Therefore, the following determinant holds true
$ \begin{vmatrix} r && Q_1 (r - C_1) && Q_2 (r - C_2) \end{vmatrix} = 0 $
or, equivalent
$ r \cdot \bigg( Q_1 (r - C_1) \times Q_2 (r - C_2) \bigg) = 0 $
Combined with $r^T Q_1 r = 1 $ and $r^T Q_2 r = 1 $, these three equations will have a finite number of solutions. Evaluating $r^T r$ for each we can determine the minimum and the maximum.
I've worked an example in this Sage worksheet.
The real solutions are shown at the end of the list of all solutions.