There are two curves known as the butterfly curve.
The first is the sextic plane
curve given by the implicit equation
|
(1)
|
(Cundy and Rollett 1989, p. 72; left figure). The total area of both wings is then given by
(OEIS A118292). The arc
length is
|
(5)
|
(OEIS A118811).
The second is the curve with polar equation
|
(6)
|
which has the corresponding parametric equations
(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 Curve
Portions of this entry contributed by Margherita
Barile
Explore with Wolfram|Alpha
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.
Cite this as:
Barile, Margherita and Weisstein, Eric W. "Butterfly Curve." From MathWorld--A
Wolfram Web Resource. https://mathworld.wolfram.com/ButterflyCurve.html
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