# Witch of Agnesi

Mathematica Notebook for This Page .

## History

Studied by
Maria Gaetana Agnesi (1718 to 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.

## Formula

Let the construction circle be centered at {0,1} with radius 1, then:

- Parametric: {2*t, 2/(1+t^2)}, 0 < t < ∞.
- Parametric: 2 {Tan[t], Cos[t]^2}, -π/2 < t < π/2.
- Cartesian: y*(x^2+4)==8

The parametric equation {2*t, 2/(1+t^2)} is derived easily by starting with a equation of circle x^2+(y-1)^2==1 and a line y==t*x and solve the equation of circle and lines, and simplify the result by the replacement t→1/t. (or, start with y==1/t*x instead). Elimating t and we find the Cartesian equation.

## Properties

## Graphics Gallery

Normals, and Evolute of witch of Agnesi.

## Related Web Sites

See: Websites on Plane Curves, Plane Curves Books .

Robert Yates: Curves and Their Properties .