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.

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

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

and a Cartesian equation of

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

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 \,}$${\displaystyle =\pi a^{2}/4}$

And it has a length of

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

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

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

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.

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

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.

The trifolium was later studied and given its name by Henry Cundy and Arthur Rollett.

• Trefoil knot in topology
• Folium of Descartes
• Bifolium