A Lissajous figure is the path traced when two perpendicular oscillations of different frequency are combined. In three dimensions we add a third axis, giving a harmonic knot woven through space.

Parametric equations

x = A·sin(a·t + δx)
y = B·sin(b·t + δy)
z = C·sin(c·t + δz),   t ∈ [0, 2π]

The point cloud is rotated in 3D and projected to the screen with a simple perspective camera; points farther from you are drawn dimmer to give depth.

Oscilloscope XY mode

Set c = 0 and the figure collapses to a flat 2D Lissajous curve — exactly what an oscilloscope shows when two signals drive its X and Y plates. The frequency ratio and phase determine the loops you see.

Closing into knots

When a : b : c are small coprime integers the curve closes after one period into a tidy lattice or knot. If they share a common factor the ratio reduces; irrational ratios never close and would fill a box densely.

Beats and phase

• Nearly equal frequencies produce slow beats — the figure appears to rotate and breathe.
• The phase offsets δ rotate and tilt the loops; a 1:1 ratio sweeps from a line to a circle to an ellipse as δ goes from 0 to π/2.
• This is the same maths behind the harmonograph, where swinging pendulums draw decaying Lissajous patterns.