Trifolium curve

The trifolium curve (also three-leafed clover curve, 3-petaled rose curve, and paquerette de mélibée) is a type of quartic plane curve. The name comes from the Latin terms for 3-leaved, defining itself as a folium shape with 3 equally sized leaves.

It is described as

${\displaystyle x^{4}+2x^{2}y^{2}+y^{4}-x^{3}+3xy^{2}=0.\,}$

By solving for y, the curve can be described by the following function:

${\displaystyle y=\pm {\sqrt {\frac {-2x^{2}-3x\pm {\sqrt {16x^{3}+9x^{2}}}}{2}}},}$

Due to the separate ± symbols, it is possible to solve for 4 different answers at a given point.

It has a polar equation of

$${\displaystyle r=-a\cos 3\theta }$$

and a Cartesian equation of

$${\displaystyle (x^{2}+y^{2})[y^{2}+x(x+a)]=4axy^{2}}$$

The area of the trifolium shape is defined by the following equation:

${\displaystyle A={\frac {1}{2}}\int _{0}^{\pi }cos^{2}(3\theta )d\theta }$^[1]

And it has a length of

${\displaystyle 6a\int _{0}^{\tfrac {\pi }{2}}{\sqrt {1-{\frac {8}{9}}sin^{2}t}}*dt\thickapprox 6,7a}$^[2]

The trifolium was described by J. Lawrence as a form of Kepler's folium when

${\displaystyle b\in (0,4,a)}$^[3]

A more present definition is when ${\displaystyle a=b}$

The trifolium was described by Dana-Picard as

${\displaystyle (x^{2}+y^{2})^{3}-x(x^{2}-3y^{2})=0}$

He defines the trifolium as having three leaves and having a triple point at the origin made up of 4 arcs. The trifolium is a sextic curve meaning that any line through the origin will have it pass through the curve again and through its complex conjugate twice.^[4]

The trifolium is a type of rose curve when ${\displaystyle k=3}$^[5]

Gaston Albert Gohierre de Longchamps was the first to study the trifolium, and it was given the name Torpille because of its resemblance to fish.^[6]

The trifolium was later studied and given its name by Henry Cundy and Arthur Rollett.^[7]