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Trifolium -- from Wolfram MathWorld

  • ️Weisstein, Eric W.
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Trifolium

"The" trifolium is the three-lobed folium with b=a, 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 b in (0,4a), but this more general definition is not so commonly used.

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

r=-acos(3theta)

(1)

and Cartesian equation

(x^2+y^2)[y^2+x(x+a)]=4axy^2.

(2)

The Cartesian equation can also be written in the alternative form

(x^2+y^2)^2=a(x^3-3xy^2).

(3)

TrifoliumPair

The two mirror images of the trifolium together have Cartesian equation

(x^2+y^2)^4=a^2(x^3-3xy^2)^2.

(4)

TrifoliumArea

The area of the trifolium is given by

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

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

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

where E(x,k) is an incomplete elliptic integral of the second kind and |_x_| is the floor function.

The trifolium is the deltoid radial curve.

See also

Bifolium, Daisy, Folium, Kepler's Folium, Quadrifolium, Rose Curve, Trefoil Curve

Portions of this entry contributed by Margherita Barile

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References

Apéry, F. Models of the Real Projective Plane. Braunschweig, Germany: Vieweg, p. 85, 1987.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 152-153, 1972.MacTutor History of Mathematics Archive. "Trifolium." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Trifolium.html.Sloane, N. J. A. Sequence A093728 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

Barile, Margherita and Weisstein, Eric W. "Trifolium." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Trifolium.html

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