Very amusing. Perhaps these curves are merely piecewise elliptical? Before doing any serious thinking, it's time to play...

For LiveGraphics3D:

Okay, let's figure it out. Heck, let's just compare it with the only ellipse it could possibly be. Let be the original parameterized ellipse, the inner fatpath (at a distance rw), and the ellipse that matches the fatpath at the extreme and values.

It looks close for rw=.2, ...

... but it aint no ellipse, Jasper.

Off course, this could be proved easily. (So do it.) And I wonder what fat values are necessary for the inner ellipse to cease to be smooth?