## Witch of Agnesi

Parallels of Witch of Agnesi

## 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 is understood in English as the Witch.

## Description

Witch of Agnesi (Versiera) is defined as follows.
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}.
- Draw a line passing the Origin and any point M on the circle. 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.

Tracing Witch of Agnesi (20 k)

## 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. It is the radius of the circle the Witch is constructed on)

## Properties

## Graphics Gallery

Normals, and Evolute of witch of Agnesi.

Osculating circles of the Witch.

Conchoids of witch of Agnesi.

Conhoids wrt a moving point (57 k)

Parallels of the Witch.

Inversion curves of the Witch {Tan[t], Cos[t]^2} with respect to points {{0,-1}, {0, -.8},...,{0,1.6}} and radius of inversion 1, corresponding to curves with light to dark shades.

Pedal curves of the Witch {Tan[t], Cos[t]^2} with respect to points {{0,-1}, {0, -.8},...,{0,1.6}}, corresponding to curves with light to dark shades.

A java applete (JavaSketchpad) that generates the curve. http://www.keypress.com/sketchpad/java_gsp/witch.html
© copyright 1995-97 by Xah Lee.

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