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Concerning Roulettes

American Mathematical Monthly, Vol. 23 (1916),
237-241.

If one curve rolls on another, the curve traced by any point in the plane of the rolling curve is called a roulette. The rolling curve is called the moving centrode and the fixed curve is called the fixed centrode. In case both centrodes are circles, the traced curve is called an epicycloid or hypocycloid depending on whether the moving centrode is on the inside or outside of the fixed centrode.

This paper is about the locus of the centers of curvature of the points on the traced curve. Horton uses her theorems in the paper to give a construction for the center of curvature of the epicycloid or hypocycloid corresponding to any point. She also proves that the center of curvature of the element of a roulette described by any point in the plane of the rolling circle is given by a construction due to Felix Savary. Finally, she gives an application of one theorem to prove the well-known result that the evolute of any epicycloid is a similar epicycloid.