Quadrifolium

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Rotated Quadrifolium
Rotated Quadrifolium

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

r = \cos(2\theta) \,,

with corresponding algebraic equation

(x^2+y^2)^3 = (x^2-y^2)^2 \,.

Rotated by 45°, this becomes

r = \sin(2\theta) \,

with corresponding algebraic equation

(x^2+y^2)^3 = 4x^2y^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 + 108x^2y^2 = 16(x^2+7y^2)(y^2+7x^2)(x^2+y^2)+729(x^2+y^2) \,.
Dual Quadrifolium
Dual Quadrifolium
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