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Bifolium

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Bifolium

The bifolium is a folium with b=0. The bifolium is a quartic curve and is given by the implicit equation is

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

and the polar equation

r=4asin^2thetacostheta.
(2)

The bifolium has area

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

Its arc length is

s=4sqrt(2)aint_0^(pi/2)sqrt(3+3cos(2t)+2cos(4t))sintdt
(6)
=(35[E(phi,k)+E(k)]c_1+4sqrt(2)[(7c_1F(phi,k)+K(k))+2c_2^2])/(42)a
(7)
=7.1555...a
(8)

(OEIS A118307), where F(phi,k), K(k), E(phi,k), and E(k) are elliptic integrals with

c_1=sqrt(4sqrt(2)-5)
(9)
c_2=sqrt(4sqrt(2)+5)
(10)
phi=sin^(-1)((9-4sqrt(2))/7)
(11)
=sin^(-1)((4-c_1^2)/7)
(12)
k=sqrt(-(40sqrt(2)+57)/7)
(13)
=sqrt(-(c_2^4)/7).
(14)

The curvature is given by

kappa(t)=(sqrt(2)csctheta(3cos^3theta+sin^4theta))/([3+3cos(2theta)+2cos(4theta)]^(3/2))
(15)
=([3+3cos(2theta)+cos(4theta)]csctheta)/(asqrt(2)[3+3cos(2theta)+2cos(4theta)]^(3/2)).
(16)

The bifolium is the pedal curve of the deltoid where the pedal point is the midpoint of one of the three curved sides.

See also

Bifoliate, Folium, Kepler's Folium, Links Curve, Quadrifolium, Rose Curve, Trifolium

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 214, 1987.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 152-153, 1972.MacTutor History of Mathematics Archive. "Double Folium." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Double.html.Sloane, N. J. A. Sequence A118307 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

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

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