x3 + y3 = 3axy
x = 3at/(1 + t3), y = 3at2/(1 + t3)
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 problem to determine the tangent to the curve was proposed to Roberval who also wrongly believed the curve had the form of a jasmine flower. His name of fleur de jasminwas later changed.
The curve is sometimes known as the noeud de ruban.
The folium has an asymptote x + y + a = 0.
The equation of the tangent at the point with t = p is
p(p3 - 2)x + (1 - 2p3)y + 3ap2 = 0.
The curve passes through the origin at t = 0 and approaches the origin a second time as t goes to infinity.
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