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Curtate Cycloid

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

The path traced out by a fixed point at a Radius $b<a$, where $a$ is the Radius of a rolling Circle, sometimes also called a Contracted Cycloid.

$\displaystyle x$ $\textstyle =$ $\displaystyle a\phi-b\sin\phi$ (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)

and $E(u)$ is a complete Elliptic Integral of the Second Kind and $\mathop{\rm sn}\nolimits u$ is a Jacobi Elliptic Function.

See also Cycloid, Prolate Cycloid


Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., 1989.

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

© 1996-9 Eric W. Weisstein