Generalizing square wheels rolling on inverted catenaries 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.
   
 A: The link between Wheel and Ground is general : to any curve (W) in polar (rho, theta) is associated a ground (G) in cartesian orthonormal frame (x, y) and conversely. Gregory's transformation direct and inverse give parametric equations with one integration. 
James Gregory in "Geometriae pars universalis 1668" invented a direct transformation  GT equivalent :


*

*for a given wheel in polar coordinates (rho, theta)  it gives the ground (x,y) in orthonormal coordinates y=rho  and dx=Integral rho.d theta  Inverse transformation GT-1 defines, 

*for a given ground (x,y), the associated wheel : rho=y and theta=Integral dx/y if y<>0. 

*GT gives the ground if we know the wheel (rho, theta) and 

*GT-1 gives the wheel if we know the ground (y, x). 


In each case there is only one integration. Cesaro in NAM 1886 has given many examples and properties of these associated curves which have same arc length. The theory is linked with integration and area. The area of the wheel is half of the one under the ground. When the polar curve rolls on the ground (with initial conditions) the pole O runs along the x-axis (called base-line). When you fix the Wheel then the base line pass through the fixed pole if the ground rolls on the Wheel. The problem was much studied about 1845-1920 In NAM, Mathesis,JMPA, etc.
There is identity of arc length between the polar curve (rho, theta)and (x,y). 
A theorem of Steiner-Habich is important in the theory (pp 3-4 of the paper I Gregory's transformation). Apply the theory to special family of curves as wheels "sinusoidal spirals" for which pedals are in the same family gives examples: line-Catenary , Circle-double circle, parabola-parabola, Cardioid-Cycloid, Tractrix spiral-Tractrix, etc. 
You can view examples here http://christophe.masurel.free.fr/#s9 All papers are open-access. 
There are also many informations in "Nouvelles annales de mathematiques" (1842-1927) 
-but in french language- http://www.numdam.org/numdam-bin/feuilleter?j=nam 
or on Gallica.fr and also in Mathesis (Google books on line).
C. Masurel
A: 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.
A: Here's a Japanese webpage on related this subject:
http://www5d.biglobe.ne.jp/~the_imai/etymology/Musashikoganei.html
(Musashikoganei Square Wheel [= Peaucellier Linkage Wheel])
