Lissajous figures

Combine two perpendicular oscillations and the moving point traces a Lissajous figure. Nathaniel Bowditch studied these curves with a compound pendulum in 1815. In 1857 Jules Antoine Lissajous made them visible by reflecting a beam of light off mirrors fixed to two tuning forks vibrating at right angles, and used the figures to compare the forks’ frequencies. When the two frequencies are in a ratio of small integers, the curve closes and its shape tells you the ratio at a glance.

What you see

The point $(x, y)$ with a trail of its last 2000 steps, colored in a rainbow that slowly cycles, older parts dimmer and the current position as a white dot. With axis projections on, faint lines run through the dot, and two small markers on the center lines show $x (t)$ and $y (t)$ on their own. The ratio $ω_{x} : ω_{y}$ is shown at the bottom left.

The presets set only the ratio and keep the current phase and decay. At the default phase δ = π/2:

1:1 gives a circle.

2:1 collapses into a parabola, $x = 1 - 2 y^{2}$ , traced back and forth.

3:2 is a closed curve with crossings that touches the left and right edges three times each and the top and bottom twice each.

4:3, 5:4, 5:3, 3:1, 4:1, 5:2: denser figures following the same rule. Those with an even $ω_{x}$ and an odd $ω_{y}$ (4:3 and 4:1) collapse into an open arc traced back and forth, like 2:1; most other phases open them into closed curves.

Governing equation

The simulation integrates two independent damped oscillators,

\ddot{x} = - ω_{x}^{2} x - 2 γ \dot{x}, \ddot{y} = - ω_{y}^{2} y - 2 γ \dot{y},

started so that without damping

x (t) = \sin (ω_{x} t + δ), y (t) = \sin ω_{y} t .

The slider values for $ω_{x}$ and $ω_{y}$ are integers that multiply a base frequency of 1.2 rad/s, so every figure closes after $2 π / 1.2 \approx 5.2$ s when the two integers have no common factor. For coprime integers the closed curve touches each side edge $ω_{x}$ times and the top and bottom edges $ω_{y}$ times. With damping the amplitude falls as $e^{- γ t}$ and the curve winds into the center. The default integrator is fourth-order Runge-Kutta.

Parameters

ωₓ, ωᵧ: integers from 1 to 9, defaults 3 and 2.

phase δ (×π): the phase of the horizontal oscillation in units of π, 0 to 2, default 0.5.

decay γ (s⁻¹): the amplitude decay rate, 0 to 0.5, default 0.

timestep dt: 0.005 to 0.1, default 0.02. The simulation takes one step per frame, so dt sets how fast the figure is drawn and how much of it the trail covers.