Harmonograph

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Frequencies:
Phases:
Amplitudes:

After talking with John Baez and looking at some pictures of harmonographs (1, 2), it seems to me the two pendula attached to the pen can be approximated as two (probably orthogonal) vectors $v_1$ and $v_2$ that combine additively. The third pendulum adds an offset to the paper, so its effect is also additive.

So the position of the pen (relative to the paper) at time $t$ is:

$$v_1 + v_2 + v_3$$

where

$$v_i = A_i\sin(f_it + \Phi_i)$$

where:

• $f_i$ is the frequency of pendulum $i$,
• $\Phi_i$ is an adjustment to the phase,
• $A_i$ is the amplitude