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.
Elliptical wheels are interesting.
If the axle is at a focus, the road can be described in terms of the cosine function, and if the axle is at the center, the road involves the Jacobian elliptic functions. Here are some more pictures of roads and wheels. The rolling square and its catenary road. Wheels can be any shape and the axle can be anywhere.