Lissajous Figures
A Lissajous figure is the graph of a system of parametric equations which describe complex harmonic motion. Invented by Jules Antoine Lissajous in 1857, these figures are generated by the intersection of two sinusoidal waves at right angles. Change the frequency and phase shift to discover intricate, periodic knots.
The Mechanics of Lissajous Curves
The curve is defined parametrically by the equations:
$$ x(t) = A \sin(at + \delta) $$
$$ y(t) = B \sin(bt) $$
Where $A$ and $B$ are amplitudes, $a$ and $b$ are the frequencies of the motion, and $\delta$ is the phase shift.
The appearance of the figure is highly sensitive to the ratio $\frac{a}{b}$. For a rational ratio, the curve is closed and periodic. When the ratio is irrational, the curve never closes, but instead eventually fills a rectangle defined by the amplitudes.