Definition:Witch of Agnesi

Definition

 * WitchOfAgnesi.png

Let $OAM$ be a circle of radius $a$ whose center is at $\tuple {0, a}$.

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.

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


 * 1) It is defined for all $x$.
 * 2) $0 < y \le 2 a$.
 * 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 $x = 0$.

Also see

 * Cartesian Equation of Witch of Agnesi