mathworld.wolfram.com

Quadrifolium -- from Wolfram MathWorld

  • ️Weisstein, Eric W.
Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology Alphabetical Index New in MathWorld

Quadrifolium

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

r=asin(2theta)

(1)

and Cartesian equation

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

(2)

QuadrifoliumArea

The area of the quadrifolium is

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

The arc length is

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

The arc length function, curvature, and tangential angle are

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

See also

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

Explore with Wolfram|Alpha

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.

Cite this as:

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

Subject classifications