Epicycloid Evolute

$\begin{figure}\begin{center}\BoxedEPSF{EpicycloidEvolute.epsf scaled 800}\end{center}\end{figure}$

$\displaystyle x$	$\textstyle =$	$\displaystyle (a+b)\cos t-b\cos\left[{\left({a+b\over b}\right)t}\right]$
$\displaystyle y$	$\textstyle =$	$\displaystyle (a+b)\sin t-b\sin\left[{\left({a+b\over b}\right)t}\right]$

is another Epicycloid given by

$\displaystyle x$	$\textstyle =$	$\displaystyle {a\over a+2b}\left\{{(a+b)\cos t+b\cos\left[{\left({a+b\over b}\right)t}\right]}\right\}$
$\displaystyle y$	$\textstyle =$	$\displaystyle {a\over a+2b}\left\{{(a+b)\sin t+b\cos\left[{\left({a+b\over b}\right)t}\right]}\right\}.$