Two related curves result when we include another parameter, L, which represents the ratio of pen length to the radius of the circle:
$x\left(\mathrm{\θ}\right)equals;r\cdot \left(\mathrm{theta;}-L\cdot \mathrm{sin}\left(\mathrm{theta;}\right)\right)$
$y\left(\mathrm{theta;}\right)equals;r\cdot \left(1-L\cdot \mathrm{cos}\left(\mathrm{theta;}\right)\right)$
When $L1$, the curve is called a curtate cycloid; when $Lgt;1$, the curve is called a prolate cycloid.