Quadrifolium

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

:$r\; =\; cos(2\; heta)\; ,$,

with corresponding algebraic equation

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

Rotated by 45°, this becomes

:$r\; =\; sin(2\; heta)\; ,$

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

:$(x^2-y^2)^4\; +\; 837(x^2+y^2)^2\; +\; 108x^2y^2\; =\; 16(x^2+7y^2)(y^2+7x^2)(x^2+y^2)+729(x^2+y^2)\; ,$.

References

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