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 Cartesian equation is:
(x2 + y2)(y2 + x(x + a)) = 4axy2.
(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
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 created with Desmos.com
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