The name "butterfly curve" is used for either of two curves .
The first is the sextic plane
curve given by the implicit equation
(1)
(Cundy and Rollett 1989, p. 72; left figure). For real numbers curve has symmetric
about both coordinate axes . The total area
of both wings is then given by
(OEIS A118292 ), where gamma function .
The total arc length is
(5)
(OEIS A118811 ).
The second is the curve with polar
equation
(6)
which has the corresponding parametric equations
(Fay 1989, Fay 1997, Bourke, Kantel-Chaos-Team, Wassenaar; right figure). For real parameters , these
parametric equations correspond to the polar equation under the substitution curve
over any closed interval of parameter
values of length
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. Fay,
T. H. "The Butterfly Curve." Amer. Math. Monthly 96 ,
442-443, 1989. https://doi.org/10.1080/00029890.1989.11972217 . Fay,
T. H. "A Study in Step Size." Math. Mag. 70 , 116-117,
1997. https://doi.org/10.1080/0025570X.1997.11996513 . 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." https://www.2dcurves.com/exponential/exponentialb.html . Referenced
on Wolfram|Alpha Butterfly Curve
Cite this as:
Weisstein, Eric W. , with contributions by Margherita Barile . "Butterfly Curve." From MathWorld https://mathworld.wolfram.com/ButterflyCurve.html
Subject classifications More... Less...
and 
,
the curve has 
and is symmetric
about both coordinate axes. The total area
of both wings is then given by
is the gamma function.
The total arc length is
![r=e^(sintheta)-2cos(4theta)+sin^5[1/(24)(2theta-pi)],](/images/equations/ButterflyCurve/NumberedEquation3.svg)
![sint[e^(cost)-2cos(4t)+sin^5(1/(12)t)]](/images/equations/ButterflyCurve/Inline16.svg)
![cost[e^(cost)-2cos(4t)+sin^5(1/(12)t)],](/images/equations/ButterflyCurve/Inline19.svg)
. Each form traces the complete curve
over any closed interval of parameter
values of length 
.
