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MathVista 3D

Butterfly curve

r = e^{sin θ} − 2 cos 4θ + sin⁵((2θ−π)/24) x = r cos θ, y = r sin θ wings: rotate ±α sin ωt about the body

Speed (t)

3D Temple–Fay butterfly curve

3D particle system

The Temple–Fay butterfly is a famous polar curve: e^{sin θ} softens the body, cos 4θ cuts four lobes, and the fifth-power sine adds fine scallops.

Halves x ≥ 0 and x < 0 rotate about the body by ±α sin ωt, so the figure beats like a flying butterfly.