Butterfly curve (transcendental)

Image:Animated construction of butterfly curve.gif (Click for enlarged version).]]

The curve is given by the following parametric equations:{{MathWorld|title=Butterfly Curve|urlname=ButterflyCurve}}

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

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

:$0\; \backslash le\; t\; \backslash le\; 12\backslash pi$

or by the following polar equation:

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

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The {{math|sin}} term has been added for purely aesthetic reasons, to make the butterfly appear fuller and more pleasing to the eye.

In 2006, two mathematicians using Mathematica analyzed the function, and found variants where leaves, flowers or other insects became apparent.{{cite journal|date=June 2008| title = On the analysis and construction of the butterfly curve using Mathematica| journal = International Journal of Mathematical Education in Science and Technology | volume = 39| issue = 5| pages = 670–678| doi = 10.1080/00207390801923240| last1 = Geum| first1 = Y.H.| last2 = Kim| first2 = Y.I.| s2cid = 122066238}} New developments regarding such curves are still under research by mathematicians.

• Butterfly curve (algebraic)

{{reflist}}

• [https://www.wolframalpha.com/input/?i=parametric+plot+%28sin%28t%29+%28+e+%5E+cos%28t%29+-+2+cos%284t%29+-+sin%5E5+%28t%2F12%29%29+%2C+y+%3D+cos%28t%29+*+%28e+%5E+cos%28t%29+-+2+cos%284t%29+-+sin%5E5+%28t%2F12%29%29+%29++from+t+%3D+0+to+t+%3D+12+pi Butterfly Curve plotted in WolframAlpha]

Category:Plane curves

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