Cycloid Involute

$\begin{figure}\begin{center}\BoxedEPSF{CycloidInvolute.epsf scaled 700}\end{center}\end{figure}$

$\displaystyle x(t)$	$\textstyle =$	$\displaystyle a(t-\sin t)$
$\displaystyle y(t)$	$\textstyle =$	$\displaystyle a(1-\cos t)$

is given by

$\displaystyle x(t)$	$\textstyle =$	$\displaystyle a(t+\sin t)$
$\displaystyle y(t)$	$\textstyle =$	$\displaystyle a(3+\cos t).$

As can be seen in the above figure, the Involute is simply a shifted copy of the original Cycloid, so the Cycloid is its own Involute!