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 second is the curve with polar equation
(2)
which has the corresponding parametric equations
(3)
(3)
(Bourke, Fay 1989, Fay 1997, Kantel-Chaos-Team, Wassenaar; right figure).
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