THC Science

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)

0\leq t\leq 12\pi

or by the following polar equation:

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

References

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

Polar: An animation constructing the butterfly curve (polar) from start to end : Online Polar Function Grapher.

Parametric: An animation constructing the butterfly curve (parametric) from start to end: Online Parametric Grapher.

@@ Line 5: / Line 5: @@
 :<math>x = \sin(t) \left(e^{\cos(t)} - 2\cos(4t) - \sin^5\left({t \over 12}\right)\right)</math>
 :<math>y = \cos(t) \left(e^{\cos(t)} - 2\cos(4t) - \sin^5\left({t \over 12}\right)\right)</math>
 :<math>0\le t \le 12\pi</math>
 or by the following [[polar equation]]:
@@ Line 32: / Line 32: @@
 *{{MathWorld|title=Butterfly Curve|urlname=ButterflyCurve}}
+==External links==
+* '''Polar''': An animation constructing the butterfly curve (polar) from start to end : [http://gcalcd.com/calculator/graphing/function/polar/ Online Polar Function Grapher].
+* '''Parametric''': An animation constructing the butterfly curve (parametric) from start to end: [http://gcalcd.com/calculator/graphing/parametric/ Online Parametric Grapher].
 [[Category:Curves]]