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Swallow Curve

A harmonograph is a mechanical apparatus that employs pendulums to create a geometric image. The drawings created typically are Lissajous curves, or related drawings of greater complexity.

A typical harmonograph has two pendulums that move in such a fashion, and a pen that is moved by two perpendicular rods connected to these pendulums. Therefore, the path of the harmonograph figure is described by the parametric equations

$\begin{align*} x(t) = A_1 \sin (f_1 t + p_1) e^{-d_1t} + A_2 \sin(f_2 t + p_2) e^{-d_2 t} \\ y(t) = A_3 \sin (f_3 t + p_3) e^{-d_3t} + A_4 \sin(f_4 t + p_4) e^{-d_4 t} \end{align*}$

The swallow curve has parameters

$\begin{align*} &A= (7, 5, 2, 5) \\ &f = (2, 1, 9, 6) \\ &p = \\ &d = (0.02, 0.002, 0.02, 0.002)\\ \end{align*}$

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