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Epicycloid--1-Cusped

\begin{figure}\begin{center}\BoxedEPSF{epicycloid_1cusped.epsf scaled 800}\end{center}\end{figure}

A 1-cusped epicycloid has $b=a$, so $n=1$. The radius measured from the center of the large circle for a 1-cusped epicycloid is given by Epicycloid equation (9) with $n=1$ so

$\displaystyle r^2$ $\textstyle =$ $\displaystyle {a^2\over n^2}\left[{(n^2+2n+2)-2(n+1)\cos(n\phi)}\right]$  
  $\textstyle =$ $\displaystyle a^2[(1^2+2\cdot 1+2)-2(1+1)\cos(1\cdot\phi)]$  
  $\textstyle =$ $\displaystyle a^2(5-4\cos\phi)$ (1)


\begin{displaymath}
r=a\sqrt{5-4\cos\phi},
\end{displaymath} (2)

and
\begin{displaymath}
\tan\theta = {2\sin\phi-\sin(2\phi)\over2\cos\phi-\cos(2\phi)}.
\end{displaymath} (3)

The 1-cusped epicycloid is just an offset Cardioid.




© 1996-9 Eric W. Weisstein
1999-05-25