A square wheel rolling on a catenary road maintains the wheel center at a fixed height, a well-known construction previously discussed on MO (e.g., "Generalizing square wheels rolling on inverted catenaries"). I wondered if this fundamentally one-dimensional example could be generalized to two dimensions, in the following sense:

Is there a solid body $B$ and a non-flat surface $S$ which together have the property that, from some one, special fixed position of $B$ resting on $S$, $B$ can roll on $S$ in any horizontal direction $v$ so that some point in $B$ (its

center) remains at a fixed height?

Of course if $B$ is a sphere and $S$ is a plane, then the constant-height property holds. Note I am asking that this only hold for some special initial position of $B$, but demand that rolling in any direction of the full $360^\circ$ retains constant height along that ray.

My guess is that the requirement that this hold for *every* position of $B$ on $S$
forces a sphere on a plane.

The following is meant to be suggestive only!

What brought this to mind is the traditional Easter Egg Roll. :-)

**Addendum**. Here is another suggestive image, not metrically accurate, of a revolved square diamond that can roll on a revolved catenary, as per Anton's answer.