Hypocycloid Evolute

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

For ,

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

, then

$\displaystyle x$	$\textstyle =$	$\displaystyle {1\over n-2}[(n-1)\cos\phi-\cos[(n-1)\phi]a$
$\displaystyle y$	$\textstyle =$	$\displaystyle {1\over n-2}[(n-1)\sin\phi+\sin[(n-1)\phi]a.$

This is just the original Hypocycloid scaled by the factor

and rotated by

of a turn.