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Rolling Sinusoidal Spirals

Fred Kuczmarski
The American Mathematical Monthly
Vol. 119, No. 6 (June‒July 2012), pp. 451-467
DOI: 10.4169/amer.math.monthly.119.06.451
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Page Count: 17
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Rolling Sinusoidal Spirals
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Abstract A theorem of Apostol and Mnatsakanian states that as a circle rolls on a line, the area of the cycloidal sector traced by a point on the circle is always three times the area of the corresponding segment cut from the rolling circle. We generalize this result by showing that sinusoidal and logarithmic spirals rolling on lines have similar area ratio properties. We then extend our ideas to include one curve rolling on another. Such pairs of curves are a natural generalization of road-wheel pairs.

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