witch of Agnesi

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

The Witch of Agnesi with labeled points

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 perpendicular 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.

Equations

An animation depicting the construction of the Witch of 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 the very simple

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

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

Properties

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

• The area between the Witch and its asymptote is four times the area of the fixed circle (i.e., 4πa²).

• The volume of revolution of the Witch, about its asymptote, is 4π²a³.

• The centroid of the curve is at .

History

The curve was studied by Pierre de Fermat in 1630, Guido Grandi in 1703, and by Maria Agnesi in 1748.^[1]

In Italian, the curve is called la versiera di Agnesi which means "the curve of Agnesi". Early on this was read by Cambridge professor John Colson as "l'avversiera di Agnesi", where "avversiera" meaning "woman contrary to God" was then identified as "witch," and the mistranslation stuck.^[2][3][4]

The Witch of Agnesi is also a fiction novel by Robert Spiller.

See also

• Cauchy distribution

References

Sources

Wikisource has the text of the 1911 Encyclopædia Britannica article Witch of Agnesi.

• Weisstein, Eric W., "Witch of Agnesi" from MathWorld.
• "Witch of Agnesi" at MacTutor's Famous Curves Index
• "Cubique d'Agnesi" at Encyclopédie des Formes Mathématiques Remarquables (in French)
• "MacTutor biography of Agnesi". http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Agnesi.html. 
• "The Witch of Agnesi". John H. Lienhard. The Engines of Our Ingenuity. NPR. KUHF-FM Houston. 2002. No. 1741. Transcript.

External links

• Witch of Agnesi by Chris Boucher based on work by Eric W. Weisstein, The Wolfram Demonstrations Project.
• The Witch of Agnesi - Mathforum.org Java applet