Definition:Witch of Agnesi

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