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 (except the triangle), 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.

**Addendum1** (*1July10*).
The delightful Hall-Wagon paper that user abel found (below) answers many of my questions, and may be
the last word (or the most recent work) on the topic. However, it does not seem to address the broader question I posed: For which class of wheel shape curves is a such a wheel-road construction
possible? I'll update further if anything comes to light.

**Addendum2** (*8June11*).
A paper just appeared in the *Amer. Math. Monthly* (Vol.118, No.6, 2011),
"Roads and Wheels, Roulettes and Pedals," by Fred Kuczmarski, which seems to
establish that a wheel-road construction is possible for every

continuously differentiable plane curve such that the angle of rotation of its tangent lines, as measured relative to some initial position, is a strictly monotonic function of arc length. We call such curves

rollable. The monotonic condition implies that rollable curves have no inﬂection points, while the strictness of the monotonicity precludes rollable curves from containing line segments.

Certainly this is not the full class (as he mentions), but he has a nice theorem that constructs a road for any rollable-curve wheel.

**Added** from mathcurve.com, as
cited by @J.M.isntamathematician: Ellipses on sinusoid.
Animation by Alain Esculier.

Mathematica1.2 code sent to me by Leon Hall a long time ago. It needs to be updated though to work with current versions, but you can at least see the formulae/algorithms used. I can send it if you're interested. $\endgroup$ – J. M. isn't a mathematician Sep 7 '10 at 12:47