Butterfly curve (transcendental)

[
curve (click for larger image).]

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

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

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

or by the following polar equation:

:$r=e^\{sin\; heta\}\; -\; 2\; cos\; (4\; heta\; )\; +\; sin^5left(frac\{2\; heta\; -\; pi\}\{24\}\; ight)$

ee also

* Butterfly curve (algebraic)

References

*cite journal
first = Temple H.
last = Fay
year = 1989
month = May
title = The Butterfly Curve
journal = Amer. Math. Monthly
volume = 96
issue = 5
pages = 442–443
url = http://links.jstor.org/sici?sici=0002-9890%28198905%2996%3A5%3C442%3ATBC%3E2.0.CO%3B2-T
doi = 10.2307/2325155

External links

* [http://mathworld.wolfram.com/ButterflyCurve.html Butterfly Curve from MathWorld]
* An animation based on the butterfly curve: [http://www.freaknet.org/alpt/misc/ganim_video.avi video] . The script to reproduce it with gnuplot: [http://www.freaknet.org/alpt/src/misc/ganim.sh script]