Quadrifolium

Rotated Quadrifolium
This article is about the geometric shape. For the plant, see Four-leaf clover. For the symmetrical shape framework, see Quatrefoil.

The quadrifolium (also known as four-leaved clover[1]) 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

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 ca. 9.6884.[2]

Notes

  1. ↑ 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
  2. ↑ Quadrifolium - from Wolfram MathWorld

References

External links