It is not uncommon to see in a science museum a bicycle with square wheels that rides smoothly over a washboard-like surface made from inverted catenary curves (e.g., at the Münich museum). The square wheel may be generalized to any regular polygon, which rolls on a similar curve without slippage. Here, for example, is a nice Mathematica demo.

My question is: For which wheel shapes does there exist a matching road shape that permits the wheel to roll over it without slippage so that: (a) the wheel center remains horizontal throughout its motion, (b) the wheel can turn at constant angular velocity, and (c) if possible, the wheel center also moves at constant horizontal velocity?

The square satisfies (a) and (b), but only regular hexagons and beyond satisfy (c). If you've experienced a square-wheel bicycle ride, you can feel it jerk because (c) fails to hold. It would be interesting to know the class of closed wheel curves that satisfy (a) and (b), and also those that in addition satisfy (c). For example, must all (a,b) curves be star-shaped from the wheel center $x$? (star-shaped: every point of the curve is visible from $x$).

This is probably all known, so an appropriate reference may suffice. I don't quite know how to tag this; please retag if you know better.