Main Concept

A cycloid is the path that is traced out by a point on the circumference of the circle as it rolls along a straight line (without slipping).

The cycloid can be defined by the following two parametric equations:

$x\left(\mathrm{\θ}\right) \= r\cdot \left(\mathrm{\θ}- \sin \left(\mathrm{\θ}\right)\right)$

$y\left(\mathrm{\θ}\right) \= r\cdot \left(1-\cos \left(\mathrm{\θ}\right)\right)$

where r is the radius of the circle that is rolled, and $\mathrm{\θ}$ is the angle through which the circle was rolled.

Curtate and prolate cycloids

Two related curves result when we include another parameter, L, which represents the ratio of pen length to the radius of the circle:   $x\left(\mathrm{\&theta;}\right) \&equals; r\cdot \left(\mathrm{\&theta;}- L\cdot \sin \left(\mathrm{\&theta;}\right)\right)$ $y \left(\mathrm{\&theta;}\right)\&equals; r\cdot \left(1-L\cdot \cos \left(\mathrm{\&theta;}\right)\right)$   When $L < 1$, the curve is called a curtate cycloid; when $L \&gt; 1$, the curve is called a prolate cycloid.

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