Folium

Cartesian equation:
(x2 + y2)(y2 + x(x + b)) = 4axy2
Polar equation:
r = -b cosθ + 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


If your browser can handle JAVA code, click HERE to experiment interactively with this curve and its associated curves.

The general form of the folium is given by the formula above. 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 above.

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

The simple folium is the pedal curve of the tricuspoid where the pedal point is one of the cusps.


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JOC/EFR/BS January 1997

The URL of this page is:
http://www-history.mcs.st-andrews.ac.uk/Curves/Folium.html