Lissajous Figures

What you are seeing: a parametric curve generated by two perpendicular harmonic oscillations, $x(t) = A \sin(a t + \delta)$ and $y(t) = B \sin(b t)$ . The shape depends only on the ratio $a / b$ and the phase $\delta$ . When $a / b$ is rational, the curve closes after time $T = 2\pi / \gcd(a, b)$ ; when irrational, it never closes and densely fills a bounding box.

The classical 1:1 case with $\delta = \pi/2$ gives a circle; with $\delta = 0$ it gives the line $y = x$ . The 1:2 ratio gives a figure-eight. As the ratio gets more complex (3:4, 3:5, 5:7), the pattern becomes a denser quasi-grid. Phase $\delta$ rotates and re-shapes the figure smoothly.

Figure 1. Lissajous figure. Method: closed-form parametric evaluation, animated pen with trailing curve.

a (x freq)3

b (y freq)5

phase d1.57

speed2

WHAT TO TRY

Watch the wave above and the wave on the left: their current values are the dot's x and y. The figure is just those two oscillations combined.
Tap a ratio in the gallery, or set $a$ and $b$ by hand. Simple ratios like 1:2 and 2:3 give clean loops; 5:7 fills the box with a dense mesh.
With $a = b$, slide the phase $\delta$ from 0 to $\pi/2$ and watch a diagonal line open into a circle.