Witch of Agnesi

In mathematics, the Witch of Agnesi (Italian pronunciation: [a.ˈɲe.zi]), sometimes called the "Witch of 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 to O. The line OA intersects the tangent of M at the point N. The line parallel to OM through N, and the line perpendicular to OM through A intersect at P. As the point A is varied, the path of P is the Witch of Agnesi.

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

The Witch of Agnesi gets its name from Italian mathematician Maria Gaetana Agnesi.

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

Note that if a = 1/2, then this equation becomes rather simple:

This is the derivative of the arctangent function.

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

Another parameterization, with θ being the angle between OA and the x-axis, increasing anti-clockwise is

The following properties can be derived from integral calculus.

The curve was studied by Pierre de Fermat in 1630. In 1703, Guido Grandi gave a construction for the curve. In 1718 Grandi suggested the name for the curve, the Latin term for sheet, the rope which turns the sail, and used the Italian word for it, versiera, a hint to sinus versus that appeared in his construction.

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