Witch of Agnesi

Cartesian equation:
y(x2+a2)=a3y(x^{2} + a^{2}) = a^{3}
or parametrically:
x=at,y=a/(1+t2)x = at, y = a/(1 + t^{2})


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 Agn├ęsienne. 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 betweeny=0y = 0 and y=ay = a. It has points of inflection at y=3a/4y = 3a/4. The line y=0y = 0 is an asymptote to the curve.

The curve can be considered as the locus of a point PP defined as follows. Draw a circle CC with centre at (0,a/2)(0, a/2) through OO. Draw a line from OO cutting CC at LL and the liney=ay = a at MM. Then PP has the xx-coordinate of MM and the yy-coordinate of LL.

The tangent to the Witch of Agnesi at the point with parameter pp is
(p2+1)2y+2px=a(3p2+1)(p^{2}+1)^{2}y + 2px = a(3p^{2}+1).
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Xah Lee