Eight Curve

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

A curve also known as the Gerono Lemniscate. It is given by Cartesian Coordinates

$\begin{displaymath} x^4=a^2(x^2-y^2), \end{displaymath}$

(1)

$\begin{displaymath} r^2=a^2\sec^4\theta\cos(2\theta), \end{displaymath}$

(2)

and parametric equations

$\displaystyle x$	$\textstyle =$	$\displaystyle a\sin t$	(3)
$\displaystyle y$	$\textstyle =$	$\displaystyle a\sin t\cos t.$	(4)

$\begin{figure}\begin{center}\BoxedEPSF{EightCurveInfo.epsf scaled 750}\end{center}\end{figure}$

$\displaystyle \kappa(t)$	$\textstyle =$	$\displaystyle -{3\sin t+\sin(3t)\over 2[\cos^2t+\cos^2(2t)]^{3/2}}$	(5)
$\displaystyle \phi(t)$	$\textstyle =$	$\displaystyle -\tan^{-1}[\cos t\sec(2t)].$	(6)

References

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 71, 1989.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 124-126, 1972.