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.

• If you're still interested in this, I have some old Mathematica 1.2 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. – J. M. is not a mathematician Sep 7 '10 at 12:47
• Here's a French webpage on this subject: mathcurve.com/courbes2d/engrenage/engrenage2.shtml – J. M. is not a mathematician Oct 19 '10 at 23:15
• @J.M.: I love those animated GIFs! :-) – Joseph O'Rourke Oct 19 '10 at 23:59

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.

• @abel: Thanks! I will study this. Here is a link to the paper: maa.org/pubs/sampMMA.pdf . – Joseph O'Rourke Jun 30 '10 at 10:17
• A beautiful paper! They derive or compute wheel-road pairs for an amazing variety of wheel shapes, including a hippopedal, a piroform, a rosette, a limacon, and a cuspitate rosette, among others. They do not seem to answer the question of what is the full class of curves that can serve as wheels--perhaps this remains unknown. Not all their wheels are centrally symmetric, but all illustrated are star-shaped. – Joseph O'Rourke Jun 30 '10 at 10:31
• Indeed they describe their wheels by polar functions $r(\theta)$, so all are star-shaped. Whether every polar function, or only polar functions, are possible wheels, remains unclear to me. – Joseph O'Rourke Jun 30 '10 at 11:33
• @Joseph O'Rourke - it is indeed a beautiful paper. Thanks for the pointer to it. It's very much like a spirograph children's toy straightened out to become the road. – sleepless in beantown Sep 8 '10 at 3:07

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

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])

• To me, the author Imai comes across as a total kook. His page and his English are basically incomprehensible. – Ben Crowell Feb 23 '13 at 4:49
• @Ben Crowell, you might want to rethink your comment above. It's possible that asifsound is the author Imai. – Joel Reyes Noche Feb 23 '13 at 9:03