Lissajous Curves
Curves traced by a point whose x and y coordinates oscillate sinusoidally at different frequencies and phases.
The Equations
A Lissajous curve is defined parametrically:
x(t) = A sin(at + δ)
y(t) = B sin(bt)
The ratio a:b determines the shape; δ is a phase offset. When a/b is rational the curve is closed; when irrational it is dense in the rectangle.
The family portrait: a grid of Lissajous curves for integer frequency ratios a:b from 1:1 to 4:4, with phase δ = π/2. Each closed curve is a different Lissajous figure.
Frequency Ratio
The ratio a:b determines how many lobes the curve has:
1:1with δ=π/2 — a circle1:2— a figure-eight (parabola-like)2:3— three vertical lobes3:4— a more complex woven shape
The 3:4 ratio with δ = π/4 phase offset. The curve weaves through 12 crossings before closing. These curves appear in physics (two perpendicular oscillations), music (frequency ratios), and oscilloscope calibration — a Lissajous figure is the traditional way to verify that two signals are at an exact rational frequency ratio.