Witch of Agnesi

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In mathematics, the witch of Agnesi, sometimes called the witch of Maria Agnesi (named for Maria Agnesi) is the curve defined as follows.

Starting with a fixed circle, a point O on the circle is chosen. For any other point A on the circle, the secant line OA is drawn. The point M is diametrically opposite O. The line OA intersects the tangent at M at the point N. The line parallel to OM through N, and the line perpedicular to OM through A intersect at P. As the point A is varied, the path of P is the witch.

The curve is asymptotic to the line tangent to the fixed circle through the point O.

1 Equations
2 Examples
3 Properties
4 History
5 External link

[edit] Equations

Suppose the point O is the origin, and that M is on the positive y-axis. Suppose the radius of the circle is a.

Then the curve has cartesian equation $y = \frac{8a^3}{x^2+4a^2}$ .

Note that if a=1/2, then this equation becomes the very simple $y = \frac{1}{x^2+1}.$

Parametrically, if $θ$ is the angle between OM and OA, measured clockwise, then the curve is defined by the equations

$x = 2a \tan \theta, y = 2a \cos ^2 \theta.\,$

[edit] Examples

The figure to the right shows examples of this curve for a=1, a=2, a=4, and a=8.

[edit] Properties

The area between the Witch and its asymptote is four times the area of the fixed circle (i.e., $4π a 2$ ).

The volume of revolution of the Witch, about its asymptote, is $4π 2 a 3$ .

The centroid of the curve is at $(0,\frac{a}{2})$ .

[edit] History

The curve was studied by Fermat, Guido Grandi in 1701, and by Maria Agnesi in 1748.

In Italian, this curve is called la versiera di Agnesi which means "the curve of Agnesi". This was once interpreted by a Cambridge professor to mean "l'avversiera di Agnesi" where "avversiera" means "witch", and the mistranslation stuck as mentioned in "Fermat's Enigma" by Simon Singh.