Trifolium curve

The trifolium curve (also three-leafed clover curve, 3-petaled rose curve, and pâquerette^{nl:madeliefje} 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.^[1]

This image shows a trifolium curve using its Cartesian equation.

It is described as

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

By solving for y by substituting y² and its square, the curve can be described by the following function(s):

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

Due to the separate ± symbols, it is possible to solve for 4 different answers at a given (positive) x-coordinate; 2 y-values per negative x-coordinate. One sees 2 resp. 1 pair(s) of solutions, mirroring points on the curve.

It has a polar equation of

This image shows the trifolium curve using its polar equation. Its area is equivalent to one quarter the area of the inscribed circle.

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

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:

${\displaystyle A={\frac {1}{2}}a^{2}\int _{0}^{\pi }\cos ^{2}(3\theta )\,d\theta \,}$^[2] ${\displaystyle ={\frac {\pi a^{2}}{4}}}$

And it has a length of

${\displaystyle 6a\int _{0}^{\tfrac {\pi }{2}}{\sqrt {1-{\frac {8}{9}}\sin ^{2}t}}\,dt\thickapprox 6.7a}$^[3]

This image shows two equations for the trifolium defined as ${\displaystyle x^{4}+2x^{2}y^{2}+y^{4}-x^{3}+3xy^{2}=0}$ (blue) and ${\displaystyle (x^{2}+y^{2})^{3}-x(x^{2}-3y^{2})=0}$ (red).

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

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

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

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.^[5]

The trifolium is a type of rose curve when ${\displaystyle k=3}$^[6]

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.^[7]

The trifolium was later studied and given its name by Henry Cundy and Arthur Rollett.^[8]

See also

• Trefoil knot in topology
• Folium of Descartes
• Bifolium