Roads and Wheels
Given a road, one can solve a differential equation to get a wheel that will roll smoothly along the road in the sense that the center of the wheel does not move up or down. This Demonstration presents eight examples, the first being the famous one of a square wheel rolling on a series of linked catenaries. This example derives from the pure catenary case. The triangular case is of interest because the wheel actually crashes into the road before reaching the cusp. The parabola, given by , is unusual in that the wheel is exactly congruent to the road. For a given road the wheel will not necessarily close up; this is illustrated by the three cosine examples. When the standard cosine function has the distance of the road's top from the axis as , then the wheel closes up over periods of the road. The cases of and are illustrated, as well as a cosine case where the wheel does not close up.
Given the road in the form , the wheel is given in polar coordinates by , where is the solution to the differential equation with . The details were first worked out by G. B. Robison, who discovered the relationship between the catenary and the square wheel, and also the parabolic case for which the road equals the wheel. Further details and examples can be found in the paper by Hall and Wagon. A photo of a working square-wheel tricycle can be found at the author's web site.
G. B. Robison, "Rockers and Rollers," Mathematics Magazine, 33, (1960) pp. 139–144.
L. Hall and S. Wagon, "Roads and Wheels," Mathematics Magazine, 65, (1992) pp. 283–301.
S. Wagon, http://stanwagon.com