# Curves

### Witch of Agnesi

Cartesian equation:
$y(x^{2} + a^{2}) = a^{3}$
or parametrically:
$x = at, y = a/(1 + t^{2})$

### Description

This was studied and named versiera by Maria Agnesi in 1748 in her book Istituzioni Analitiche. It is also known as Cubique d'Agnesi or Agnésienne. There is a discussion on how it came to be called witch in Agnesi's biography.

The curve had been studied earlier by Fermat and Guido Grandi in 1703.

The curve lies between$y = 0$ and $y = a$. It has points of inflection at $y = 3a/4$. The line $y = 0$ is an asymptote to the curve.

The curve can be considered as the locus of a point $P$ defined as follows. Draw a circle $C$ with centre at $(0, a/2)$ through $O$. Draw a line from $O$ cutting $C$ at $L$ and the line$y = a$ at $M$. Then $P$ has the $x$-coordinate of $M$ and the $y$-coordinate of $L$.

The tangent to the Witch of Agnesi at the point with parameter $p$ is
$(p^{2}+1)^{2}y + 2px = a(3p^{2}+1)$.
Other Web site:

Xah Lee

### Associated Curves

Definitions of the Associated curves