A **Trifolium curve** (or *3-petalled rose curve*) is a special case of the folium curve (where *b *= *a*), with three smooth parts that intersect each other transversally [1]. The word *folium *means “leaf-shaped” [2].

The Cartesian equation is:

(x^{2} + y^{2})(y^{2} + x(x + a)) = 4axy^{2}.

Alternatively,

(x^{2} + y^{2})^{2} = a(x^{3} – 3xy^{2}).

The variable *a* affects the **size of the trifolium**, not its shape. The **shape of the trifolium** is determined by the ratio between the upper and lower wheels that construct the curve (shown in the above image).

Two other **special cases** of the folium are the *simple folium*, and the *double folium*. They correspond to b = 4a, b = 0.

The curve was first studied in 1609 by Kepler, which gives it it’s alternate name, *Kepler’s Folium*. It is also called *paquerette de mélibée* [3] where “paquerette” is French for “wild daisy.” The curve was also studied by Longchamps in 1885, and Brocard and d’Ocagne in 1887 [4].

## Trifolium Curve: Alternate Definitions

Dana-Picard and Kovacs [5] define a regular trifolium (or trefoil) as a plane curve with the implicit equation

(x^{2} + y^{2})^{2} = ax(x^{2} – 3y^{2}),

where a is a positive real parameter.

The curve can also be defined by the polar equation r(t) = a | cos(3 θ) |.

The polar equation r = -a cos (3 θ) describes a trifolium with a lobe along the negative axis.

Lawrence [6] defines a trifolium as a folium curve with b ∈ (0, 4a), but this definition is not commonly used [7].

## Area of a Trifolium Curve

The area of a trifolium curve is given by the integral

## Folium Curve

folium of Descartes (x^{3}+ y

^{3}= 3xy) which is a single “leaf” with one node and two asymptotes at the ends.

## References

Folium created with Desmos.com

[1] Dragovic, V. & Radnovic, M. (2011). Poncelet Porisms and Beyond: Integrable Billiards, Hyperelliptic Jacobians and Pencils of Quadrics. Springer Basel.

[2] MacTutor. Curves : Trofolium. Retrieved February 2, 2022 from: https://mathshistory.st-andrews.ac.uk/Curves/Trifolium/

[3] Apéry, F. Models of the Real Projective Plane. Braunschweig, Germany: Vieweg, p. 85, 1987.

[4] Tan, S. (2020). Handbook of Famous Plane Curves Using Mathematica. Lulu.com.

[5] Dana-Picard, T. & Kovacs, Z. (2021). Oﬀsets of a regular trifolium. Retrieved February 22, 2022 from: https://www.academia.edu/64611330/Offsets_of_a_regular_trifolium

[6] Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 152-153, 1972.

[7] Giedre, S. Mathematics Curves. Retrieved February 22, 2022 from: https://www.academia.edu/23149113/Mathematics_Curves

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