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Cycloid Evolute

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

The Evolute of the Cycloid

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

is given by
$\displaystyle x(t)$ $\textstyle =$ $\displaystyle a(t+\sin t)$  
$\displaystyle y(t)$ $\textstyle =$ $\displaystyle a(\cos t-1).$  

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

© 1996-9 Eric W. Weisstein