Definition:Witch of Agnesi

Curve

 * WitchOfAgnesi.png

Let $OAM$ be a circle radius $a$ centered at $\left({0, a}\right)$ such that $OM$ is a diameter.

Let $OA$ be extended to cut the tangent to the circle through $M$ at $N$.

Generate $NP$ perpendicular to $MN$ and $AP$ parallel to $MN$.

As $A$ moves around the circle, the point $P$ traces the curve known as the Witch of Agnesi.

The equation of this curve is:
 * $\displaystyle y = \frac {8 a^3}{x^2 + 4 a^2}$

Properties
Various properties of the Witch of Agnesi are as follows.


 * 1) It is defined for all $x$.
 * 2) $0 < y \le 2a$.
 * 3) $y$ reaches its maximum at $x = 0$.
 * 4) The curvature $K$ of the curve is such that $0 \le K \le \dfrac 1 a$, and it achieves that maximum at $x = 0$.

Proof
Let $P = \left({x, y}\right)$ and $A = \left({d, y}\right)$.

We have that:
 * $\dfrac {OM} {MN} = \dfrac {2 a} x = \dfrac y d$.

Also, by Pythagoras's Theorem:
 * $\left({a - y}\right)^2 + d^2 = a^2 \implies y \left({2a - y}\right) = d^2$

Eliminating $d$ gives us:
 * $\dfrac {y^2}{y \left({2a - y}\right)} = \dfrac {\left({2a}\right)^2}{x^2}$

Hence:
 * $\dfrac y {2a - y} = \dfrac {4 a^2}{x^2}$

from which:
 * $y = \dfrac {8 a^3}{x^2 + 4 a^2}$

It had previously been written about by others, for example.

Linguistic Note
The word witch appears to be a mistranslation from the Italian vertere (to turn: the term comes from the rope used to turn a sail) as avversiera which means witch or she-devil (from the same root as the word adversary, an archaic soubriquet for Satan). It has been suggested that the initial misnaming may have been mischievous.

When referred to in other languages, the term witch is not seen, and the less colorful term curve of Agnesi is usually used instead.