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.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Trifolium_curve_chart.png/220px-Trifolium_curve_chart.png)
It is described as
By solving for y, the curve can be described by the following function:
Due to the separate ± symbols, it is possible to solve for 4 different answers at a given point.
It has a polar equation of
![](http://upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Three-leaved_clover_polar.svg/220px-Three-leaved_clover_polar.svg.png)
and a Cartesian equation of
The area of the trifolium shape is defined by the following equation:
And it has a length of
![](http://upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Trifolium_equations.png/220px-Trifolium_equations.png)
The trifolium was described by J. Lawrence as a form of Kepler's folium when
A more present definition is when
The trifolium was described by Dana-Picard as
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 [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]
References
edit- ^ Barile, Margherita; Weisstein, Eric. "Trifolium Curve Equations". mathworld.wolfram.com.
- ^ Ferreol, Robert (2017). "Regular Trifolium". mathcurve.com.
- ^ Lawrence, J. Dennis (1972). A catalog of special plane curves. New York: Dover Publications. ISBN 978-0486602882. Retrieved 19 July 2024.
- ^ 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.
- ^ Lee, Xah. "Rose Curve Information". xahlee.info.
- ^ de Longchamps, Gaston (1884). Cours De Mathématiques Spéciales: Geométrie Analytique À Deux Dimensions (French ed.). France: Nabu Press. ISBN 978-1145247291.
- ^ Cundy, Henry Martyn; Rollett, Arthur P. (2007). Mathematical models (3., repr ed.). St. Albans: Tarquin Publications. ISBN 978-0906212202.