Double Folium

Cartesian equation:
(x2 + y2)2 = 4axy2
Polar equation:
r = 4a cosθ sin2θ

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.

The general form of the folium is given by the formula
(x2+y2)(y2+x(x+b)) = 4axy2
or, in polar coordinates
r = -b cosθ + 4a cosθ sin2θ.
The word folium means leaf-shaped.

There are three special forms of the folium, the simple folium, the double folium and the trifolium. These correspond to the cases

b = 4a, b = 0, b = a
respectively in the formula for the general form.

The graph plotted above is the double folium. There are separate entries for the simple folium and the trifolium.

The double folium is the pedal curve of the tricuspoid where the pedal point is mid-point of one of the three curved sides.

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JOC/EFR/BS January 1997

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