Witch of Agnesi

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Curve

WitchOfAgnesi.png

Let $OAM$ be a circle of radius $a$ whose center is at $\left({0, a}\right)$.

Let $M$ be the point such that $OM$ is a diameter of $OAM$.

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 $OAM$, the point $P$ traces the curve known as the Witch of Agnesi.


The equation of this curve is:

$y = \dfrac {8 a^3} {x^2 + 4 a^2}$


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:

$y = \dfrac {8 a^3}{x^2 + 4 a^2}$



Source of Name

This entry was named for Maria Gaëtana Agnesi.


Historical Note

While the Witch of Agnesi is named for Maria Gaëtana Agnesi, its study does not actually originate from her.

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


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.


Note the name Agnesi is Italian: its pronunciation is something like an-ye-zi, and never in the apparently obvious way ag-nee-zee.


Sources