The coloured figure is an ellipse. It can be changed by sliding its top apex or one of its foci (both in yellow).

The curve in black is a smoothly connected string of four circular arcs, such that its apeces are those of the ellipse. I call it a pseudo-ellipse. The pseudo-ellipse can be changed by sliding the red point, and can be anything between a “spindle” and a “capsule”.

The hollow green point is the one that ensures smoothest – in the analythical sense – linking of the circular arcs.

The solid green point – which is the single position on the ellipse where the red one can be located – is such that, if R1, R2 are the radii of the circular arcs for it, then (R1/Ra)·(R2/Rb)=1, where Ra and Rb are the radii of curvature of the ellipse at its apeces (i.e., the deviations of the curvature of the pseudo-ellipse with respect to the curvature of the ellipse compensate each other at the apeces). The point is also special in other interesting ways.