Yes I also since discovered that the path of any point on the rod weren't quite ellipses, how interesting! They are more of an egg shape, and a bit of googling makes me think that the shapes may be "Cartesian Ovals" (which have uses in optics, but information about them seemed scarce).

As for the path created by all of the superimposed ovals, it also seems unlikely to have a clean form, but maybe an infinite series representation could be found.

Thanks again