Definition:Witch of Agnesi

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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.


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 known as

The witch of Agnesi is also known as the versiera.

Also see

  • Results about Witch of Agnesi can be found here.

Source of Name

This entry was named for Maria Gaëtana Agnesi.

Historical Note

The witch of Agnesi was studied in detail by Maria Gaëtana Agnesi in the $18$th century, but does not actually originate from her.

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

Linguistic Note

The witch of Agnesi was originally named the versorio by Luigi Guido Grandi, from the Italian vertere (to turn: the term comes from the rope used to turn a sail).

Maria Gaëtana Agnesi confused the word with versiera, from avversiera, which means witch or she-devil (from the same root as the word adversary, an archaic soubriquet for Satan).

I 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.