Trifolium

Cartesian equation:
(x² + y²)(y² + x(x + a)) = 4axy²
Polar equation:
r = a cos (4sin² - 1)

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

(x² + y²)(y² + x(x + b)) = 4axy²

or, in polar coordinates

r = -b cos + 4a cos sin².

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 trifolium. There are separate entries for the simple folium and the double folium.

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