This answer to Is there a name for the great circle where latitude and longitude are equal? says (in part):
More generally, a clélie is the name given to any spherical curve where the longitude $\varphi$ and colatitude $\theta$ have the relationship $\varphi=c\theta,\quad c>0$, and the curve of Viviani corresponds to the locus of a geosynchronous orbit, $c=1$.
Is it really true that a polar geosynchronous orbit (displayed in a synodic or rotating frame) is described by Viviani's curve? Can this be demonstrated mathematically?
Answers to Is an Earth synchronous (?) polar orbit possible? don't mention this,
but this answer does show some similar-looking curves for geosynchronous orbits that are not polar.