Roads and Wheels, Roulettes and Pedals

We revisit the idea of road-wheel pairs, first introduced 50 years ago by Gerson Robison and later popularized by Stan Wagon and his square-wheeled tricycle. We show how to generate such pairs geometrically: the road as a roulette curve and the wheel as a pedal curve. Along the way we gain geometric insight into two theorems proved by Jakob Steiner relating the area and arc length of a roulette to those of a corresponding pedal. Finally, we use our results to generate parabolas, ellipses, and sine curves as roulettes.