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Trifolium curve - Wikipedia

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This image shows a graphical trifolium curve using its Cartesian Equation.

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

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}}\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]

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. 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]

  1. ^ Barile, Margherita; Weisstein, Eric. "Trifolium Curve Equations". mathworld.wolfram.com.
  2. ^ Ferreol, Robert (2017). "Regular Trifolium". mathcurve.com.
  3. ^ Lawrence, J. Dennis (1972). A catalog of special plane curves. New York: Dover Publications. ISBN 978-0486602882. Retrieved 19 July 2024.
  4. ^ Dana-Picard, Thierry (June 2019). "Automated Study of a Regular Trifolium". Mathematics in Computer Science. 13 (1–2): 57–67. doi:10.1007/s11786-018-0351-7. Retrieved 19 July 2024.
  5. ^ Lee, Xah. "Rose Curve Information". xahlee.info.
  6. ^ de Longchamps, Gaston (1884). Cours De Mathématiques Spéciales: Geométrie Analytique À Deux Dimensions (French ed.). France: Nabu Press. ISBN 978-1145247291.
  7. ^ Cundy, Henry Martyn; Rollett, Arthur P. (2007). Mathematical models (3., repr ed.). St. Albans: Tarquin Publications. ISBN 978-0906212202.