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Quadrifolium

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Quadrifolium

The quadrifolium is the 4-petalled rose curve having n=2. It has polar equation

r=asin(2theta),
(1)

where a>0, and Cartesian equation

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

The area of the quadrifolium is

A=1/2int_0^(2pi)[asin(2theta)]^2dtheta
(3)
=4int_0^(pi/4)[asin(2theta)]^2dtheta
(4)
=1/2pia^2.
(5)

Rather surprisingly, this means that the area inside the curve is equal to that of its complement within the disk bounded by the curve's circumcircle.

The arc length is

s=8aE(1/2sqrt(3))
(6)
=9.6884...a
(7)

(OEIS A138500), where E(k) is a complete elliptic integral of the second kind.

For the parametrization x(t)=asin(2t)cost, y(t)=asin(2t)sint, the arc length function measured from t=0, curvature, and tangential angle are

s(t)=aE(2t,1/2sqrt(3))
(8)
kappa(t)=(sqrt(2)[13+3cos(4t)])/(a[5+3cos(4t)]^(3/2))
(9)
phi(t)=1/2pi+t-tan^(-1)(cott-tant)+pi|_(2t)/pi_|,
(10)

where E(x,k) is an elliptic integral of the second kind and |_x_| is the floor function. The tangential angle at multiples of pi/2 is given by the continuous limits of the displayed expression.

See also

Bifoliate, Bifolium, Folium, Quadrifolium Catacaustic, Rose Curve, Trifolium

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References

Sloane, N. J. A. Sequence A138500 in "The On-Line Encyclopedia of Integer Sequences."Smith, D. E. History of Mathematics, Vol. 2: Special Topics of Elementary Mathematics. New York: Dover, p. 330, 1958.

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Quadrifolium

Cite this as:

Weisstein, Eric W. "Quadrifolium." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Quadrifolium.html

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