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

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 $$0 \le K \le \frac 1 a$$, and it achieves that maximum at $$x = 0$$.

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

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

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 $$\frac {y^2}{y \left({2a - y}\right)} = \frac {\left({2a}\right)^2}{x^2}$$.

Hence $$\frac y {2a - y} = \frac {4 a^2}{x^2}$$ from which $$y = \frac {8 a^3}{x^2 + 4 a^2}$$.

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

Note on the name
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". It has been suggested that the initial misnaming may have been mischievous.

When referred to in other languages, the term "witch" is not seen.