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Butterfly Curve

Butterfly curve.

The parametric equations of the butterfly curve can be given by

x=sin⁡t(ecos⁡t−2cos⁡4t−sin5⁡t12),y=cos⁡t(ecos⁡t−2cos⁡4t−sin5⁡t12)x = \sin t \left(\text{e}^{\cos t}-2 \cos 4t-\sin^{5} \frac{t}{12} \right), \, y = \cos t \left(\text{e}^{\cos t}-2 \cos 4t-\sin^{5} \frac{t}{12} \right)

In polar coordinates the equation of a butterfly curve is given by

r=esin⁡θ−2cos⁡(4θ)+sin5⁡[124(2θ−π)]r=\text{e}^{\sin \theta}-2 \cos \left(4 \theta \right)+ \sin^5\left[\frac{1}{24} \left(2 \theta – \pi \right) \right]

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