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

angular frequency In physics, angular frequency (symbol ''ω''), also called angular speed and angular rate, is a scalar measure of the angle rate (the angle per unit time) or the temporal rate of change of the phase argument of a sinusoidal waveform or sine ...

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 cu ...

genus Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...

zero. The

dual curve In projective geometry, a dual curve of a given plane curve is a curve in the dual projective plane consisting of the set of lines tangent to . There is a map from a curve to its dual, sending each point to the point dual to its tangent line. I ...

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 the length of a closed boundary that encompasses, surrounds, or outlines either a two-dimensional shape or a one-dimensional line. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimet ...

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 In mathematics, the arithmetic–geometric mean (AGM or agM) of two positive real numbers and is the mutual limit of a sequence of arithmetic means and a sequence of geometric means. The arithmetic–geometric mean is used in fast algorithms f ...

and

'

denotes the

derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...

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