Article
Article
Cycloid
Article By:
Blumenthal, Leonard M. Formerly, Department of Mathematics, University of Missouri, Columbia, Missouri.
Last reviewed:June 2021
DOI:https://doi.org/10.1036/1097-8542.176200
A curve traced in the plane by a point on a circle that rolls, without slipping, on a line. If the line is the x axis of a rectangular coordinate system, at whose origin O the moving point P touches the axis, parametric equations of the cycloid are x = a(θ − sin θ), y = a(1 − cos θ), when a is the radius of the rolling circle, and the parameter θ is the variable angle through which the circle rolls (see illustration). One arch is obtained where θ assumes all values from 0 to 2π. The length of an arch is 8a, and the area bounded by an arch and the x axis is 3πa2 (three times the area of the rolling circle).
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