Lissajous Figures

Parametric curves x = A·sin(aθ + δ), y = B·sin(bθ). Rational frequency ratios a/b produce closed curves; irrational ratios fill the plane densely.

Frequency a 3

Frequency b 2

Phase δ (rad) 0.00

Trail length 600

Speed 1.0

x=sin(3θ+δ), y=sin(2θ)

Crossings: a closed Lissajous with ratio a:b has (a+b−2) interior crossings when gcd(a,b)=1.

Ratio a/b rational → periodic closed curve.
Ratio irrational → dense, never closes.

Closed curve (gcd=1)