Quadrifolium
Quadrifolium
The quadrifolium is the 4-petalled rose curve having n=2. It has polar equation
| r=asin(2theta) |
(1)
|
| (x^2+y^2)^3=4a^2x^2y^2. |
(2)
|
QuadrifoliumArea
The area of the quadrifolium is
A = [画像:1/2int_0^(2pi)[asin(2theta)]^2dtheta]
![画像:1/2int_0^(2pi)[asin(2theta)]^2dtheta](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/Quadrifolium/Inline4.svg){kind=link}
(3)
![画像:4int_0^(pi/4)[asin(2theta)]^2dtheta](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/Quadrifolium/Inline7.svg){kind=link}
= 1/2pia^2.
(5)
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
s = 8aE(1/2sqrt(3))
(6)
= 9.6884...a
(7)
(OEIS A138500), where E(k) is a complete elliptic integral of the second kind.
The arc length function, curvature, and tangential angle are
s(t) = aE(2t,1/4sqrt(3))
(8)
kappa(t) = [画像:(sqrt(2)[13+3cos(4theta)])/(a[5+3cos(4theta)]^(3/2))]
![画像:(sqrt(2)[13+3cos(4theta)])/(a[5+3cos(4theta)]^(3/2))](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/Quadrifolium/Inline23.svg){kind=link}
(9)
phi(t) = [画像: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.
See also
Bifoliate, Bifolium, Folium, Quadrifolium Catacaustic, Rose Curve, TrifoliumExplore with Wolfram|Alpha
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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.Referenced on Wolfram|Alpha
QuadrifoliumCite this as:
Weisstein, Eric W. "Quadrifolium." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Quadrifolium.html