Quadrifolium

Rotated quadrifolium

Quadrifolium created with gears

The quadrifolium (also known as four-leaved clover^[1]) is a type of rose curve with n=2. It has the 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

The area inside the curve is $\tfrac 12 \pi$ , which is exactly half of the area of the circumcircle of the quadrifolium. The length of the curve is about 9.6884.^[2]

Notes

↑ C G Gibson, Elementary Geometry of Algebraic Curves, An Undergraduate Introduction, Cambridge University Press, Cambridge, 2001, ISBN 978-0-521-64641-3. Pages 92 and 93
↑ Quadrifolium - from Wolfram MathWorld

References

J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. p. 175. ISBN 0-486-60288-5.

External links

Interactive example with JSXGraph

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