Not having any other place to note this down, I suppose this blog entry is as good as any.

From this m.SE answer, user Raskolnikov demonstrates a polar equation whose graph is a regular -gon (slightly reformulated):

Through similar considerations, one can derive parametric equations for Reuleaux polygons ( of course being odd):

Here’s a side-by-side comparison of some odd-sided regular and Reuleaux polygons:

I’m not sure if there are simpler expressions for the parametric equations of the Reuleaux polygon, though, and I would be interested if you, the reader, might come up with something a bit simpler.