Harmonograph · 羊·實驗

羊·實驗中文

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Parametric equations

A harmonograph has the damped form

x = A sin (a t + δ) e^{- d t}, y = B sin (b t) e^{- d t}

A and B are the amplitudes on x and y. This figure uses the same amplitude for both.
a and b are the frequency parameters.
δ is the phase difference. It sets where the orbit starts and how symmetric it is.
d is the damping. It sets how fast the curve decays inward.

Interaction

Frequency a: switch the integer frequency. The orbit updates at once and the trace grows in again
Frequency b: switch the integer frequency. The ratio a:b is these two controls
Phase δ: move the starting phase and watch the orbit twist
Damping d: a larger d pulls the orbit into the center faster

What to notice

The factor $e^{- d t}$ shrinks the amplitude as t grows when d > 0, so the orbit spirals inward.
Different pairs of a:b and $δ$ draw patterns that do not repeat one another.
After a long trace, the places where the orbit packs together are the dense parts of the pattern.

Further reading

Harmonograph (Wikipedia)

Comments

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