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There are two curves known as the butterfly curve.
The first is the sextic plane curve given by the implicit equation
(1)
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(Cundy and Rollett 1989, p. 72; left figure). The total area of both wings is then given by
(2)
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(3)
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(4)
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(Sloane's A118292). The arc length is
(5)
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(Sloane's A118811).
The second is the curve with polar equation
which has the corresponding parametric equation s
(7)
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(8)
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(Bourke, Fay 1989, Fay 1997, Kantel-Chaos-Team, Wassenaar; right figure).
SEE ALSO: Bean Curve, Butterfly Catastrophe, Butterfly Effect, Butterfly Function, Butterfly Graph, Butterfly Lemma, Butterfly Polyiamond, Butterfly Theorem, Dumbbell Curve, Eight Curve, Piriform
Portions of this entry contributed by Margherita Barile
REFERENCES:
Bourke, P. "Butterfly Curve." http://astronomy.swin.edu.au/~pbourke/curves/butterfly/.
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 72, 1989.
Fay, T. H. "The Butterfly Curve." Amer. Math. Monthly 96, pp. 442-443, 1989.
Fay, T. H. "A Study in Step Size." Math. Mag. 70, pp. 116-117, 1997.
Kantel-Chaos-Team "Die Butterfly-Kurve." http://www.schockwellenreiter.de/pythonmania/pybutt.html.
Sloane, N. J. A. Sequences A118292 and A118811 in "The On-Line Encyclopedia of Integer Sequences."
Wassenaar, J. "2D Curves." http://www.2dcurves.com/exponential/exponentialb.html.