Shaping a Road and Finding the Corresponding Wheel
For a given road shape, the corresponding wheel (given in polar form centered at the origin) might not close up. This Demonstration lets you vary the shape of the periodic road using locators. You can then raise or lower the road to see how the wheel can become closed, which is when the integral above the plot is an integer. Choosing to be 2, 3, or 4 in the setter bar causes the height of the road to be changed so that the wheel is automatically closed, with one revolution taking periods of the road.
The relationship between the road and the wheel that rolls smoothly on it is described in detail in the paper by Hall and Wagon. The main point is that if the road is given by the graph of , then the wheel is given in polar coordinates by , where is the solution to the differential equation , with . In this Demonstration the road is obtained by periodic interpolation on the locator points. The wheel is then found by numerically solving the numerical differential equation for . Raising or lowering the road causes the wheel function to be recomputed. The wheel will not always close up in a nice way. Whether it closes depends on whether is an integer, where is the half-period of the road, chosen here to be for consistency with the famous square-wheel case. Choosing 2, 3, or 4 for a closed wheel starts a search, via numerical integration and root-finding, for the height that will work for the desired wheel.
L. Hall and S. Wagon, "Roads and Wheels," Mathematics Magazine, 65(5), 1992 pp. 283–301.