Lissajous Harmonograph (3D)

What the 3D scene shows

A Victorian-style harmonograph. One pendulum slides the paper table left and right, the other slides the pen carriage forwards and backwards. The pen is fixed to its carriage, so the ink on the paper is exactly the Lissajous curve (-x(t), y(t)) of the two displacements. Pendulum angles and link rods are derived from the same model as the 2D page.

The parametric curve

A Lissajous figure is the path of a point whose x and y coordinates oscillate harmonically: x = A sin(a t + δ), y = B sin(b t). When a/b is a ratio of integers p/q in lowest terms the motion repeats after t = 2π/gcd(a,b) and the curve is closed; otherwise it never closes and slowly fills the box.

Counting crossings

For a generic phase the closed curve has 2pq − p − q self-crossings (p, q coprime). The page counts them by brute force on a polyline and compares with this formula. Special phases (for example δ = 0) collapse the curve onto a retraced arc, so for those the count is made at a generic phase.

Ellipse area

For a = b the figure is an ellipse of area π A B |sin δ|: a line at δ = 0, a circle for A = B and δ = 90°. The shoelace sum over 4000 sample points reproduces this value.

Harmonograph

A real harmonograph uses damped pendulums, so the amplitudes decay as e−γt, and a slight detuning ε of one frequency makes the figure precess. This is the damped, detuned form simulated here; it ignores pendulum non-linearity (finite-angle period lengthening) and any coupling between the two pendulums, which are idealised as independent damped harmonic oscillators.

Honest simplifications

The displacements are prescribed sinusoids (damped by e^(-g t)); the pendulum angle is asin(x/1.4 m) for the picture only. A real pendulum has period 2π√(L/g), so the integer frequency ratios used here correspond to lengths in ratio 1/a² : 1/b².