Harmonograph Generator

How the Harmonograph Generator works

Harmonograph generator: getting a good result

The pen's position is the sum of two damped sine waves in each direction. Each wave has a frequency, a starting phase and an amplitude that shrinks over time by an exponential factor. The two pendulums are coupled so that each contributes to both x and y, which is what gives real harmonograph drawings their slightly off-axis, woven look rather than the clean mirror symmetry of a plain Lissajous figure.

Frequencies are whole-number ratios, as on the Lissajous page, and the ratio decides the basic figure: 1:2 gives a figure of eight, 2:3 a three-lobed knot, 3:4 a denser weave. On a physical machine the frequency is set by the pendulum's length; here it is a slider.

Detune adds a small offset to one pendulum in each pair. With detune at 0 the figure closes on itself and the pen retraces the same path as it shrinks, giving crisp nested lines. A little detune makes each pass land slightly beside the last, so the lines fan out into the shaded, rotating bands that make harmonograph drawings recognisable. Too much and the figure loses its shape.

Damping is how quickly the swing dies. Low damping keeps the pen near full amplitude for the whole run and fills the outer figure densely; high damping pulls it into the centre quickly and leaves most of the drawing as a tight spiral.

Run time is how long the pendulums are allowed to swing, in simulated minutes. Longer runs add more inner passes. A run of 10 with low damping draws hundreds of thousands of segments and takes a moment at 4K.

Seed and colour. The starting phases of the four waves are chosen from the seed, so Regenerate rotates and reshapes the figure without changing the frequencies. Rainbow shifts the hue over the run from outside to inside; Single uses one hue. Strokes are drawn semi-transparent so overlapping passes build up brightness.

Related tools