Closest Distance Between Two Ellipses

Yesterday, I needed to solve this.  My first thought was that it couldn't be that hard.  I figured that I could go online and find a solution, but I wanted to solve it myself.

I went on a walk to think about it.  The more I thought about it, the more challenging it seemed.  Still, I figured there might be some quality of an ellipse that made it easy.

To solve the closest distance between two ellipses, I thought I'd get an equation for the distance from a point to an ellipse and optimize it.

 I thought that would be easy enough.  I came up with what I thought was an interesting way to solve that problem.  I used a theorem made by a mathematician name Lagrange.

I ended up with a mess.  It turned out that I needed to solve this horrible 4th degree polynomial.  It really felt like the wrong direction.  When I got to the 4th degree polynomial I thought a bunch of stuff would fall out and I'd end up with a simple answer and think, "Cool.  I used Lagrange."

I got frustrated with myself.  I went on another walk.  I went to a little fishing hole by a bridge.  I had my pencil with me.  I started reworking the equations on a light pole.  It was still ugly.

I came back home and work on it again.  And it still looked ugly.

I woke this morning and worked it again.  It seems like I was stuck with that stupid fourth degree polynomial.  So I broke down and looked thru a couple books.  No answer.

So, I said screw it, I'll google it.  I couldn't believe chapter 5 in this article.  There it was.  That mess.  And to my chagrin, a numerical technique to solve it... what I thought I'd have to do.

Still, I'm not convinced.