y(x2 + a2) = a3
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).
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