Quadrifolium

Quadrifolium

The quadrifolium is a type of rose curve with n=2. It has polar equation:

r = cos(2θ),

with corresponding algebraic equation

(x 2 + y 2) 3 = (x 2 - y 2) 2 .

Rotated by 45°, this becomes

r = sin(2θ)

with corresponding algebraic equation

(x 2 + y 2) 3 = 4 x 2 y 2 .

In either form, it is a plane algebraic curve of genus zero.

The dual curve to the quadrifolium is

(x 2 - y 2) 4 + 837(x 2 + y 2) 2 + 108 x 2 y 2 = 16(x 2 + 7 y 2)(y 2 + 7 x 2)(x 2 + y 2) + 729(x 2 + y 2).

Dual Quadrifolium