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.
Best Answer
see the article roads and wheels by leon hall and stan wagon in the mathematics magazine, vol 65, no 5 pp. 283-301. they expand on a shorter artcle rockers and rollers by gerson b. robinson in the same magazine vol. 33(1960) 139-144.