Harmonograph (2D)

A harmonograph pen is driven by two (here sums of) damped pendulum motions: x(t) = sum A_i sin(w_i t + p_i) e^(-d_i t) and y(t) = sum B_j cos(v_j t + q_j) e^(-e_j t). With undamped motion and an integer ratio w_x : w_y = a : b (coprime) the curve closes after T = 2 pi a / w_x; for a generic phase it has 2ab - a - b self-crossings. Damping shrinks every partial by its own exponential, so the figure spirals inward and, when partial decay rates differ, slowly changes shape. A small frequency detuning makes the loops rotate (beats).

Simplifications: the pendulum motions are given by their closed-form damped solutions (small-angle, linear drag), not integrated from a physical double-pendulum rig; the Lab default uses the same amplitudes, frequencies, phases and decay rates as the 3D harmonograph. Real machines also couple the pendulums and show a slight amplitude-dependent frequency shift, which is not modelled.