info prev up next book cdrom email home

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)

Polar Coordinates,
\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}

The Curvature and Tangential Angle are

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

See also Butterfly Curve, Dumbbell Curve, Eight Surface, Piriform


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.

Lee, X. ``Lemniscate of Gerono.'' http://www.best.com/~xah/SpecialPlaneCurves_dir/LemniscateOfGerono_dir/lemniscateOfGerono.html.

MacTutor History of Mathematics Archive. ``Eight Curve.'' http://www-groups.dcs.st-and.ac.uk/~history/Curves/Eight.html.




© 1996-9 Eric W. Weisstein
1999-05-25