The bifolium is a folium with 
. The bifolium is a quartic curve and is given by the implicit equation is
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(1)
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and the polar equation
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(2)
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The bifolium has area
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(3)
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(4)
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(5)
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Its arc length is
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![1/(42)a{35[E(phi,k)+E(k)]x+4sqrt(2)[(7F(phi,k)+K(k))x+2x^'^2]}](http://mathworld.wolfram.com/images/equations/Bifolium/Inline13.gif) |
(6)
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(7)
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(Sloane's A118307), where 
, 
, 
, and 
are elliptic integrals with
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![sin^(-1)[1/7(9-4sqrt(2))]](http://mathworld.wolfram.com/images/equations/Bifolium/Inline23.gif) |
(8)
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(9)
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The curvature is given by
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![(sqrt(2)csctheta(3cos^3theta+sin^4theta))/([3+3cos(2theta)+2cos(4theta)]^(3/2))](http://mathworld.wolfram.com/images/equations/Bifolium/Inline29.gif) |
(10)
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![([3+3cos(2theta)+cos(4theta)]csctheta)/(asqrt(2)[3+3cos(2theta)+2cos(4theta)]^(3/2)).](http://mathworld.wolfram.com/images/equations/Bifolium/Inline32.gif) |
(11)
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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, Links Curve, Quadrifolium, Rose, Trifolium
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."
