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Witch of Agnesi

\begin{figure}\begin{center}\BoxedEPSF{witch.epsf scaled 900}\end{center}\end{figure}

A curve studied and named ``versiera'' (Italian for ``she-devil'' or ``witch'') by Maria Agnesi in 1748 in her book Istituzioni Analitiche (MacTutor Archive). It is also known as Cubique d'Agnesi or Agnésienne. Some suggest that Agnesi confused an old Italian word meaning ``free to move'' with another meaning ``witch.'' The curve had been studied earlier by Fermat and Guido Grandi in 1703.

It is the curve obtained by drawing a line from the origin through the Circle of radius $a$ ($OB$), then picking the point with the $y$ coordinate of the intersection with the circle and the $x$ coordinate of the intersection of the extension of line $OB$ with the line $y=2a$. The curve has Inflection Points at $y = 3a/2$. The line $y =
0$ is an Asymptote to the curve.

In parametric form,

$\displaystyle x$ $\textstyle =$ $\displaystyle 2a\cot\theta$ (1)
$\displaystyle y$ $\textstyle =$ $\displaystyle a[1-\cos(2\theta)],$ (2)

$\displaystyle x$ $\textstyle =$ $\displaystyle 2at$ (3)
$\displaystyle y$ $\textstyle =$ $\displaystyle {2a\over 1+t^2}.$ (4)

In rectangular coordinates,
y={8a^3\over x^2+4a^2}.
\end{displaymath} (5)

See also Lamé Curve


Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 90-93, 1972.

Lee, X. ``Witch of Agnesi.''

MacTutor History of Mathematics Archive. ``Witch of Agnesi.''

Yates, R. C. ``Witch of Agnesi.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 237-238, 1952.

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© 1996-9 Eric W. Weisstein