this is an intresting curve ,,,,,,one of tthe most beautiful in mathematics
There are two curves known as the butterfly curve.
The first is the
sextic plane curve given by the implicit equation
 | (1) |
(Cundy and Rollett 1989, p. 72; left figure). The total area of both wings is then given by
(Sloane's
A118292). The
arc length is
 | (5) |
The second is the curve with polar equation
![r==e^(sintheta)-2cos(4theta)+sin^5[1/(24)(2theta-pi)],](http://mathworld.wolfram.com/images/equations/ButterflyCurve/equation3.gif) | (6) |
which has the corresponding
parametric equations 
do rate me for this >>>>>>plzzzzzzzzzz
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![sint[e^(cost)-2cos(4t)+sin^5(1/(12)t)]](http://mathworld.wolfram.com/images/equations/ButterflyCurve/inline12.gif){kind=small}
![cost[e^(cost)-2cos(4t)+sin^5(1/(12)t)],](http://mathworld.wolfram.com/images/equations/ButterflyCurve/inline15.gif){kind=small}
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