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Prolate Cycloid



The path traced out by a fixed point at a Radius $b>a$, where $a$ is the Radius of a rolling Circle, also sometimes called an Extended Cycloid. The prolate cycloid contains loops, and has parametric equations

$\displaystyle x$ $\textstyle =$ $\displaystyle a\phi-b\sin\phi$ (1)
$\displaystyle y$ $\textstyle =$ $\displaystyle a-b\cos\phi.$ (2)

The Arc Length from $\phi=0$ is
\end{displaymath} (3)

\sin({\textstyle{1\over 2}}\phi)=\mathop{\rm sn}\nolimits u
\end{displaymath} (4)

k^2={4ab\over (a+c)^2}.
\end{displaymath} (5)

See also Curtate Cycloid, Cycloid


Wagon, S. Mathematica in Action. New York: W. H. Freeman, pp. 46-50, 1991.

© 1996-9 Eric W. Weisstein