Trifolium

"The" trifolium is the three-lobed folium with , i.e., the 3-petalled rose curve. It is also known as the paquerette de mélibée (Apéry 1987, p. 85), with paquerette being the French word for "wild daisy."

Lawrence (1972) defines a trifolium as a Kepler's folium with , but this more general definition is not so commonly used.

The trifolium with lobe along the negative -axis has polar equation

(1)

and Cartesian equation

(2)

The Cartesian equation can also be written in the alternative form

(3)

The two mirror images of the trifolium together have Cartesian equation

(4)

The area of the trifolium is given by

			(5)
			(6)
			(7)

Rather surprisingly, this means that the area of the trifolium (left figure) is exactly one quarter of the area of the circumscribed circle, and even more surprisingly that the combined area of two mirror image trifoliums (middle figure) is identical to the area of the circle lying outside the curve (right figure).

The arc length of the trifolium is

			(8)
			(9)

(OEIS A093728), where is a complete elliptic integral of the second kind.

The arc length function, curvature, and tangential angle of the trifolium are

			(10)
			(11)
			(12)

where is an incomplete elliptic integral of the second kind and is the floor function.

The trifolium is the deltoid radial curve.