Butterfly curve (transcendental)

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:

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

The sin term has been added for purely aesthetic reasons.[1]

In 2006, two mathematicians using Mathematica analyzed the function, and found variants where leafs, flowers or other insects became apparent.[3]

• Butterfly curve (algebraic)
• Oscar’s Butterfly Polar Equation

r= (cos50)^2 + sin30 + .3 A Polar equation discovered by Oscar Ramirez a UCLA student in the fall of 1991. https://books.google.com/books?id=laXgAAAAMAAJ&q=Oscar%27s+Butterfly+polar+math+equation+by+David+Cohen+inauthor:David+inauthor:Cohen&dq=Oscar%27s+Butterfly+polar+math+equation+by+David+Cohen+inauthor:David+inauthor:Cohen&hl=en&sa=X&ved=2ahUKEwjf1bPwqJ3uAhVKJzQIHTvUDfMQ6AEwAHoECAAQAg

• Butterfly Curve plotted in WolframAlpha