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Bifoliate

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Bifoliate

The bifoliate is the quartic curve given by the Cartesian equation

x^4+y^4=2axy^2
(1)

and the polar equation

r=(8costhetasin^2theta)/(3+cos(4theta))a
(2)

for theta in [0,pi].

It has a cusp at the origin (0,0).

BifoliateArea

The area of the bifoliate is given by

A=1/2a^2int_0^pi[(8costhetasin^2theta)/(3+cos(4theta))]^2dtheta
(3)
=a^2int_0^(pi/2)[(8costhetasin^2theta)/(3+cos(4theta))]^2dtheta
(4)
=pi/(2sqrt(2))a^2
(5)
=1.110720...a^2
(6)

(OEIS A093954).

Its perimeter is

s=6.4799119598464...
(7)

(OEIS A118289). Taking t=theta as the parameter, the bifoliate has curvature and tangential angle given by

kappa(t)=(8sqrt(2)[3+cos(4t)]^3[3+2cos(2t)+cos(4t)]csct)/(a[182+174cos(2t)+152cos(4t)+19cos(6t)-14cos(8t)-cos(10t)]^(3/2))
(8)
phi(t)=tan^(-1)[(4costsint[3+cos(4t)])/(6+25cos(2t)+2cos(4t)-cos(6t))].
(9)

See also

Bifolium, Folium, Kepler's Folium, Quadrifolium, Rose Curve, Trifolium

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References

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 72, 1989.Sloane, N. J. A. Sequences A093954 and A118289 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

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

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