Witch of Agnesi

In mathematics, the Witch of Agnesi (Italian pronunciation: ), sometimes called the "Witch of Maria Agnesi" is the curve defined as follows.

The Witch of Agnesi with labeled points
An animation showing the construction of the Witch of Agnesi

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. [1]

Equations

The Witch of Agnesi with parameters a = 1, a = 2, a = 4, and a = 8

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

${\displaystyle y={\frac {8a^{3}}{x^{2}+4a^{2}}}.}$

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

${\displaystyle y={\frac {1}{x^{2}+1}}.}$

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

${\displaystyle x=2a\tan \theta ;\quad y=2a\cos ^{2}\theta =a\operatorname {vercosin} (2\theta ).\,}$

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

${\displaystyle x=2a\cot \theta ;\quad y=2a\sin ^{2}\theta =a\operatorname {versin} (2\theta ).}$
Other Languages
Afrikaans: Heks van Agnesi
български: Версиера
bosanski: Versiera
español: Curva de Agnesi
italiano: Versiera
norsk nynorsk: Agnesis heks
Piemontèis: Masca d'Agnesi
polski: Lok Agnesi
português: Curva de Agnesi
slovenščina: Agnesin koder
svenska: Agnesis häxa
українська: Локон Аньєзі