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

[画像: 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 = [画像:1/2a^2int_0^pi[(8costhetasin^2theta)/(3+cos(4theta))]^2dtheta]

(3)

= [画像:a^2int_0^(pi/2)[(8costhetasin^2theta)/(3+cos(4theta))]^2dtheta]

(4)

= [画像: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) = [画像:(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) = [画像: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."

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Cite this as:

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

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