[Math] Find minimum-area ellipse which encloses two ellipses

Question

I need an efficient algorithm to find the ellipse with the smallest possible area which encloses two given ellipses. The given ellipses are constrained to have coincident centers at the origin but can have any orientation. The problem is limited to two dimensions. Any ideas?

Answer 1

Yeah, there is a shear transformation that takes one of the ellipses to a circle. The least area ellipse enclosing the resulting figure is now evident by symmetry. Then use the inverse of the shear.

Now that I think of it, you can just shrink along the major axis of one of the ellipses and expand on the minor axis to get the circle.

All manipulations involved are with 2 by 2 matrices. The hypothesis of coincident centers is crucial.