**Cartesian equation: **

*y*(*x*^{2} + *a*^{2}) = *a*^{3}

**or parametrically: **

*x* = *at*, *y* = *a*/(1 + *t*^{2})

**Click below to see one of the Associated curves.**

Click THIS LINK to experiment interactively with this curve and its associated curves.

This was studied and named

The curve had been studied earlier by Fermat and Guido Grandi in 1703.

The curve lies between *y* = 0 and *y* = *a*. It has points of inflection at *y* = 3*a*/4. The line *y* = 0 is an asymptote to the curve.

The curve can be considered as the locus of a point *P* defined as follows. Draw a circle *C* with centre at (0, *a*/2) through *O*. Draw a line from *O* cutting *C* at *L* and the line *y* = *a* at *M*. Then *P* has the *x*-coordinate of *M* and the *y*-coordinate of *L*.

The tangent to the Witch of Agnesi at the point with parameter *p* is

(*p*^{2}+1)^{2}*y* + 2*px* = *a*(3*p*^{2}+1).

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JOC/EFR/BS January 1997

The URL of this page is:

http://www-history.mcs.st-andrews.ac.uk/Curves/Witch.html