Witch of Agnesi

Cartesian equation: y(x²+a²) = a³

or parametrically: x = at, y = a/(1+t²)

Click below to see one of the Associated curves.

Definitions of the Associated curves	Evolute
Involute 1	Involute 2
Inverse curve wrt origin	Inverse wrt another circle
Pedal curve wrt origin	Pedal wrt another point
Negative pedal curve wrt origin	Negative pedal wrt another point
Caustic wrt horizontal rays	Caustic curve wrt another point

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 = 3a/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

(p²+1)²y + 2px = a(3p²+1).

Other Web sites:

The Guardian, UK
Xah Lee
Eric Weisstein

The URL of this page is:
http://www-history.mcs.st-andrews.ac.uk/history/Curves/Witch.html