The quadrifolium (also known as four-leaved clover) is a type of rose curve with an

angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit ti ...

of 2. It has the polar equation: :

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

with corresponding algebraic equation :

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

Rotated counter-clockwise by 45°, this becomes :

r = a\sin(2\theta) \,

with corresponding algebraic equation :

(x^2+y^2)^3 = 4a^2x^2y^2. \,

In either form, it is a

plane algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...

genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial n ...

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). \,

The area inside the quadrifolium is

\tfrac 12 \pi a^2

, which is exactly half of the area of the circumcircle of the quadrifolium. The

perimeter A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several pr ...

of the quadrifolium is :

8a\operatorname\left(\frac\right)=4\pi a\left(\frac+\frac\right)

where

\operatorname(k)

is the complete elliptic integral of the second kind with modulus

k

\operatorname

is the arithmetic–geometric mean and

'

denotes the

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

with respect to the second variable.Quadrifolium - from Wolfram MathWorld
/ref>

Notes

References

* {{cite book , author=J. Dennis Lawrence , title=A catalog of special plane curves , publisher=Dover Publications , year=1972 , isbn=0-486-60288-5 , pag
175
, url-access=registration , url=https://archive.org/details/catalogofspecial00lawr/page/175

External links

Interactive example with JSXGraph
Sextic curves