# Trifolium Curve (Three Leaf Clover)

Feel like "cheating" at Calculus? Check out our Practically Cheating Calculus Handbook, which gives you hundreds of easy-to-follow answers in a convenient e-book.

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 . The word folium means “leaf-shaped” . The trifolium is complete when θ = π is reached (one-half revolution of the lighter gear).

The Cartesian equation is:
(x2 + y2)(y2 + x(x + a)) = 4axy2.
Alternatively,
(x2 + y2)2 = a(x3 – 3xy2).

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  where “paquerette” is French for “wild daisy.” The curve was also studied by Longchamps in 1885, and Brocard and d’Ocagne in 1887 .

## Trifolium Curve: Alternate Definitions

Dana-Picard and Kovacs  define a regular trifolium (or trefoil) as a plane curve with the implicit equation
(x2 + y2)2 = ax(x2 – 3y2),
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  defines a trifolium as a folium curve with b ∈ (0, 4a), but this definition is not commonly used .

## Area of a Trifolium Curve

The area of a trifolium curve is given by the integral ## Folium Curve

folium of Descartes (x3 + y3 = 3xy) which is a single “leaf” with one node and two asymptotes at the ends. ## References

Folium created with Desmos.com
 Dragovic, V. & Radnovic, M. (2011). Poncelet Porisms and Beyond: Integrable Billiards, Hyperelliptic Jacobians and Pencils of Quadrics. Springer Basel.
 MacTutor. Curves : Trofolium. Retrieved February 2, 2022 from: https://mathshistory.st-andrews.ac.uk/Curves/Trifolium/
 Apéry, F. Models of the Real Projective Plane. Braunschweig, Germany: Vieweg, p. 85, 1987.
 Tan, S. (2020). Handbook of Famous Plane Curves Using Mathematica. Lulu.com.
 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
 Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 152-153, 1972.
 Giedre, S. Mathematics Curves. Retrieved February 22, 2022 from: https://www.academia.edu/23149113/Mathematics_Curves

CITE THIS AS:
Stephanie Glen. "Trifolium Curve (Three Leaf Clover)" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/trifolium-curve-three-leaf-clover/
---------------------------------------------------------------------------  Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!