A Lissajous figure is the curve traced by a point whose coordinates are the superposition of two perpendicular simple harmonic motions.
If the horizontal motion is
\[
x(t)=A\cos(\omega_x t+\phi_x)
\]
and the vertical motion is
\[
y(t)=B\cos(\omega_y t+\phi_y),
\]
then the resulting path in the \(xy\)-plane is a parametric curve. These figures are famously seen on oscilloscopes when two sinusoidal signals
drive the horizontal and vertical deflections.
Role of the frequency ratio. The key geometric behavior is controlled by the ratio \(\omega_x/\omega_y\). If the ratio is a rational number,
\[
\frac{\omega_x}{\omega_y}=\frac{p}{q}\in\mathbb{Q},
\]
then the two oscillations repeat with a common period and the Lissajous curve is closed. In that case, the figure typically shows
\(p\) “lobes” in one direction and \(q\) in the other (exact counts depend on phase). If the ratio is irrational, the curve does not repeat
exactly and the point gradually fills a region rather than closing into a single loop.
Role of phase difference. The phase difference
\[
\Delta\phi=\phi_x-\phi_y
\]
changes the shape dramatically. For example, if \(\omega_x=\omega_y\):
- \(\Delta\phi=0\): the curve is a straight line with slope \(B/A\).
- \(\Delta\phi=\pm\frac{\pi}{2}\): the curve is an ellipse (a circle if \(A=B\)).
- \(\Delta\phi=\pi\): the straight line flips orientation.
When \(\omega_x\ne\omega_y\), varying \(\Delta\phi\) rotates and reshapes the lobes, producing different “knot-like” patterns.
Sampling and plotting. Because the curve is parametric, one common way to visualize it is to sample time values
\[
t_i=\frac{i}{N}t_{\max},\quad i=0,1,\dots,N
\]
and compute the corresponding points \((x(t_i),y(t_i))\). A larger \(N\) yields a smoother curve. Choosing \(t_{\max}\) matters:
to see a full closed figure when \(\omega_x/\omega_y=p/q\), a good choice is a time interval long enough to cover a common repeat cycle.
For many demos, setting \(t_{\max}=2\pi\) already shows the structure clearly.
Lissajous figures are used to compare frequencies, measure phase shifts, and create decorative patterns. In advanced settings,
you can extend the idea to three dimensions (3D Lissajous curves) or replace the cosine waves with other periodic signals to explore
more general parametric motion.
