Joining ellipses with matching tangents

from blog Tony Finch's blog, | ↗ original
In my previous entry I wrote about constructing a four-point egg, using curcular arcs that join where their tangents are at 45°. I wondered if I could do something similar with ellipses. As before, I made an interactive ellipse workbench to experiment with the problem. I got something working, but I have questions… What I wanted to do is swing a...