Prolate Cycloid

$\begin{figure}\begin{center}\BoxedEPSF{cycloid_prolate.epsf}\end{center}\end{figure}$

$\begin{figure}\begin{center}\BoxedEPSF{ProlateCycloidMovie.epsf}\end{center}\end{figure}$

The path traced out by a fixed point at a Radius , where 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

$\begin{displaymath} s=2(a+b)E(u), \end{displaymath}$

(3)

where

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

(4)

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

(5)

See also Curtate Cycloid, Cycloid

References

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