This is a prototype of **JavaSketchpad**, a World-Wide-Web
component of *The
Geometer's Sketchpad.* Copyright ©1990-1997 by Key Curriculum Press, Inc. All
rights reserved. Portions of this work are being funded by the National Science Foundation
(awards DMI 9561674 & 9623018 ).

## The Witch of Agnesi

**Credits. **This page, and the Geometer's Sketchpad document on
which it is based, is from the WWW project A Visual
Dictionary of Special Plane Curves, Copyright 1995, 1996 by Xah Lee (xah@best.com). It
can be freely distributed provided this information is intact.

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### Sketchpad Notes

The Witch of Agnesi is defined as the curve traced by *X* as *Drag Me* moves
around the circle. You can drag *Drag Me* with the mouse, or click *Animate* to
move it around the circle. Click *Show Trace* to begin tracing point *X* as it
moves; click *Hide Trace *to stop. You can also drag any other red point, or while
the sketch is animating, press '>' or '<' keys to speed up and slow down the
animation. Click the red X in the lower-right corner to clear any visible traces.

Step by step description

- Let there be a circle of radius a with center at {0,a}.
- Let there be a horizontal line L passing through {0, 2 a}.
- From any point M on the circle, draw a secant passing the Origin and M. Let the
intersection of this secant and line L be N.
- Witch of Agnesi is the locus of intersections of a horizontal line passing through M and
a vertical line passing through N.

Formulas

- Parametric: 2 a {Tan[t], Cos[t]^2}, -Pi/2 < t < Pi/2.
- Cartesian: y (x^2 + 4 a^2) == 8 a^3

(a is the scaling factor. Geometrically, it's the radius of the circle on which the
Witch is constructed.)

History

Studied by Maria Gaetana Agnesi (1718-1799) in 1748. Also studied by Fermat (1666), and
Guido Grandi (1703). The name of this curve has a colorful history. "Versaria"
is the name given by Grandi, meaning "turning in every direction". In the course
of time the word versaria took on another meaning. The Latin words adversaria, and by
aphaeresis, versaria, signify a female that is contrary to God. Thus gradually the curve
versaria came to be known in English as "the witch."

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