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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

<center> <img src="http://latex.codecogs.com/gif.latex?\begin{align*}&space;x(t)&space;=&space;A_1&space;\sin&space;(f_1&space;t&space;&plus;&space;p_1)&space;e^{-d_1t}&space;&plus;&space;A_2&space;\sin(f_2&space;t&space;&plus;&space;p_2)&space;e^{-d_2&space;t}&space;\\&space;y(t)&space;=&space;A_3&space;\sin&space;(f_3&space;t&space;&plus;&space;p_3)&space;e^{-d_3t}&space;&plus;&space;A_4&space;\sin(f_4&space;t&space;&plus;&space;p_4)&space;e^{-d_4&space;t}&space;\end{align*}" title="\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*}" /> </center>

The swallow curve has parameters 

<center> <img src="http://latex.codecogs.com/gif.latex?\begin{align*}&space;&A=&space;(7,&space;5,&space;2,&space;5)&space;\\&space;&f&space;=&space;(2,&space;1,&space;9,&space;6)&space;\\&space;&p&space;=&space;(0,&space;\pi,&space;\pi/2,&space;\pi/2)&space;\\&space;&d&space;=&space;(0.02,&space;0.002,&space;0.02,&space;0.002)\\&space;\end{align*}" title="\begin{align*} &A= (7, 5, 2, 5) \\ &f = (2, 1, 9, 6) \\ &p = (0, \pi, \pi/2, \pi/2) \\ &d = (0.02, 0.002, 0.02, 0.002)\\ \end{align*}" /></center>

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