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Roads and Wheels, Roulettes and Pedals

Fred Kuczmarski
The American Mathematical Monthly
Vol. 118, No. 6 (June‒July 2011), pp. 479-496
DOI: 10.4169/amer.math.monthly.118.06.479
Stable URL: http://www.jstor.org/stable/10.4169/amer.math.monthly.118.06.479
Page Count: 18
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Roads and Wheels, Roulettes and Pedals
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Abstract

Abstract 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.

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