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Eight Curve

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

\end{displaymath} (1)

Polar Coordinates,
\end{displaymath} (2)

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

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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


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.''

MacTutor History of Mathematics Archive. ``Eight Curve.''

© 1996-9 Eric W. Weisstein