Harmonograph: Damped Pendulum Trace

Controls

Live readouts

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Numerical check

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Performance graph

How to use

Four pendulums swing with slightly different frequencies; two move the pen along x and two along y. When the frequency ratio is close to a simple fraction the curve is a Lissajous figure; a small detuning makes it precess into rings. Raise damping to make the figure spiral into the centre, set it to 0 for a closed curve. Drag on the canvas to nudge the phase shift. Press R to restart the pen.

The science

Each pendulum contributes a damped sinusoid, the solution of the damped harmonic oscillator x'' + 2d x' + w0^2 x = 0 in the weakly damped limit:

x(t) = A1 sin(f1 t + p1) e^(-dt) + A2 sin(f2 t + p2) e^(-dt)
y(t) = A3 sin(f3 t + p3) e^(-dt) + A4 sin(f4 t + p4) e^(-dt)

With d = 0 and f1 : f3 = p : q (integers) the pen retraces the same Lissajous curve after a common period. With 1:1 and a 90 degree shift the curve is a circle, x^2 + y^2 = A^2. Damping multiplies every term by e^(-dt), so successive peaks shrink by the factor e^(-d 2 pi / f).

Honest simplifications: real harmonographs have frequency and damping that depend on the pendulum geometry, the frequency shifts slightly with amplitude, and the damping is not exactly viscous. This page uses ideal linear damped oscillators and a shared damping constant.

Keyboard, touch and accessibility

Space pauses or resumes the simulation.
R resets it to its initial state.
C re-runs the numerical check.
All sliders are native range inputs and work with arrow keys, Home and End; on touch devices drag on the canvas where the sim supports pointer input.
Rendering is capped at a device pixel ratio of 2 so phones stay smooth; the frame-time graph above shows the cost of the current settings (green line is 16.7 ms, i.e. 60 frames per second).

What is checked and why you can trust it

Every time the page loads, and whenever you press Re-run check, the simulation evaluates its own mathematics against an independent reference, such as a closed-form formula, a finite-difference derivative or an exhaustive count. Each line above reports PASS or FAIL together with the measured numbers, so you can see by how much it agrees. Results are also published on window.__msChecks for automated testing. The check never uses the same code path as the animation it validates, otherwise it would only confirm itself.

How to read a failure: a FAIL line means the measured discrepancy exceeded the stated tolerance for the current slider values. Try Reset, then move the sliders back toward their defaults; if a check still fails at the defaults it is a bug worth reporting. Tolerances are chosen from the numerical method itself, never tuned to make a test pass: round-off level (1e-9) for exact algebra, about 1e-3 for finite differences and Monte Carlo estimates with a few hundred thousand samples, and a documented margin for stochastic learning runs.

Reproducibility: the particle and pen simulations use the browser random generator for initial positions, so exact pixels differ between visits, but the checks depend only on mathematics and not on those positions. Fixed wave tables (curl noise) and fixed grammars (L-system) are deterministic.

This is an educational model. It shows the mechanism clearly, but it is deliberately simplified; the science panel lists what was left out.