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

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

This was studied and named versiera by Maria Agnesi in 1748 in her book Istituzioni Analitiche. It is also known as Cubique d'Agnesi or Agnésienne. There is a discussion on how it came to be called witch in Agnesi's biography.

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).
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Xah Lee

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JOC/EFR/BS January 1997

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