The quadrifolium (also known as four-leaved clover) is a type of

rose curve A rose is either a woody perennial flowering plant of the genus ''Rosa'' (), in the family Rosaceae (), or the flower it bears. There are over three hundred species and tens of thousands of cultivars. They form a group of plants that can be ...

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 tim ...

of 2. It has the

polar equation In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...

: :

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 c ...

genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...

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 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 pract ...

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 In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising i ...

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. F ...

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