Quadrifolium - Alchetron, The Free Social Encyclopedia

The quadrifolium (also known as four-leaved clover) is a type of rose curve with n=2. It has the polar equation:

r = cos ⁡ ( 2 θ ) ,

with corresponding algebraic equation

( x 2 + y 2 ) 3 = ( x 2 − y 2 ) 2 .

Rotated by 45°, this becomes

r = sin ⁡ ( 2 θ )

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 ) .

The area inside the curve is 1 2 π , which is exactly half of the area of the circumcircle of the quadrifolium. The length of the curve is about 9.6884.

References

Quadrifolium Wikipedia

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