Butterfly curve (transcendental)

The butterfly curve is a transcendental plane curve discovered by Temple Fay.[1]

The butterfly curve.

Equation

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

The curve is given by the following parametric equations:[2]

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

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

${\displaystyle 0\leq t\leq 12\pi }$

or by the following polar equation:

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

See also

• Butterfly curve (algebraic)

References

1. Fay, Temple H. (May 1989). "The Butterfly Curve". Amer. Math. Monthly. 96 (5): 442–443. doi:10.2307/2325155. JSTOR 2325155.
2. Weisstein, Eric W. "Butterfly Curve". MathWorld.