Butterfly curve (transcendental)

An animated construction gives an idea of the complexity of the curve (Click for enlarged version).

The butterfly curve.

The butterfly curve is a transcendental plane curve discovered by Temple H. Fay. The curve is given by the following parametric equations:

$x=\sin(t)\left(e^{\cos(t)}-2\cos(4t)-\sin ^{5}\left({t \over 12}\right)\right)$

$y=\cos(t)\left(e^{\cos(t)}-2\cos(4t)-\sin ^{5}\left({t \over 12}\right)\right)$

or by the following polar equation:

$r=e^{\sin \theta }-2\cos(4\theta )+\sin ^{5}\left({\frac {2\theta -\pi }{24}}\right)$

See also

• Butterfly curve (algebraic)

References

• Fay, Temple H. (1989). "The Butterfly Curve". Amer. Math. Monthly. 96 (5): 442–443. doi:10.2307/2325155. {{cite journal}}: Unknown parameter |month= ignored (help)
• Weisstein, Eric W. [ButterflyCurve "Butterfly Curve"]. MathWorld. {{cite web}}: Check |url= value (help)

External links

• An animation based on the butterfly curve: video. The script to reproduce it with gnuplot : script