Quadrifolium

{{Short description|Rose curve with angular frequency 2}}

File:Quadrifolium.svg

File:Rose Curve animation with Gears n2 d1.gif

{{about|the geometric shape|the plant|Four-leaf clover|the symmetrical shape framework|Quatrefoil}}

The quadrifolium (also known as four-leaved cloverC 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) is a type of rose curve with an angular frequency of 2. It has the polar equation:

:$r\; =\; a\backslash cos(2\backslash theta),\; \backslash ,$

with corresponding algebraic equation

:$(x^2+y^2)^3\; =\; a^2(x^2-y^2)^2.\; \backslash ,$

Rotated counter-clockwise by 45°, this becomes

:$r\; =\; a\backslash sin(2\backslash theta)\; \backslash ,$

with corresponding algebraic equation

:$(x^2+y^2)^3\; =\; 4a^2x^2y^2.\; \backslash ,$

In either form, it is a plane algebraic curve of genus zero.

The dual curve to the quadrifolium is

:$(x^2-y^2)^4\; +\; 837(x^2+y^2)^2\; +\; 108x^2y^2\; =\; 16(x^2+7y^2)(y^2+7x^2)(x^2+y^2)+729(x^2+y^2).\; \backslash ,$

File:Dualrose.png

The area inside the quadrifolium is $\backslash tfrac\; 12\; \backslash pi\; a^2$, which is exactly half of the area of the circumcircle of the quadrifolium. The perimeter of the quadrifolium is

:$8a\backslash operatorname\{E\}\backslash left(\backslash frac\{\backslash sqrt\{3\}\}\{2\}\backslash right)=4\backslash pi\; a\backslash left(\backslash frac\{(52\backslash sqrt\{3\}-90)\backslash operatorname\{M\}\text{'}(1,7-4\backslash sqrt\{3\})\}\{\backslash operatorname\{M\}^2(1,7-4\backslash sqrt\{3\})\}+\backslash frac\{7-4\backslash sqrt\{3\}\}\{\backslash operatorname\{M\}(1,7-4\backslash sqrt\{3\})\}\backslash right)$

where $\backslash operatorname\{E\}(k)$ is the complete elliptic integral of the second kind with modulus $k$, $\backslash operatorname\{M\}$ is the arithmetic–geometric mean and $\text{'}$ denotes the derivative with respect to the second variable.[http://mathworld.wolfram.com/Quadrifolium.html Quadrifolium - from Wolfram MathWorld]