A square wheel with its axle at the center will roll smoothly (that is, the axle will stay level) on a road made up of pieces of an inverted catenary. A catenary is the curve formed by a hanging chain, and a famous structure based on an inverted catenary is the Gateway Arch in St. Louis. The catenaries forming the road for a rolling square meet at right angles. These corners can be smoothed out if the corners of the square are appropriately rounded off at the same time. The image above shows such approximations to the square and the road. The approximations were obtained using Fourier series. Details can be found in:
The rolling square and its catenary road.
Wheels can be any shape and the axle can be anywhere.
The road for this ellipse, with axle at the focus, is y = cos x - sqrt(2).
Leon Hall / lmhall@umr.edu / January 30, 1997