# Proof of $G^1$ continuity between ellipse segments

It's possible to draw an ellipse with a pen constrained by a loose, inflexible string and two nails in a board. Thinking about this, I wondered if it was possible to draw another shape that has some of the same features as a superellipse by using four nails positioned in a rectangle. This seems to work great, and gives very nice curves (although, qualitatively, they do not behave as much as superellipses as I assumed).

A natural expansion of this idea is to put a string around any number of nails positioned in a "convex" way. If you use $n>2$ nails, the resulting curve will consist of up to $2n$ ellipse segments joined at up to $2n$ points (depending on the tightness of the string).

My question is about the behavior of the curve at these points. Let's look at the following figure:

The nails are at A, B and C. Imagine your pen at D, and the string in red marked as AD and DC. You move the pen along the big ellipse towards F, where the string catches B and moves your pen over to the small ellipse where the pen continues to E.

What happens at F? I managed to figure out that the two ellipses do not share curvature at F. And surely, F has at least $G^0$ continuity. I'm convinced that all such points F along a curve like this with $n$ nails have $G^1$ continuity.

Is there an easy way to show this?