Lissajous figures are closed curves traced by combining two perpendicular sinusoids at integer frequency ratios

A Lissajous figure is parametrically defined as x = sin(t * freqX + phi), y = sin(t * freqY). When freqX and freqY are in a simple integer ratio (2:3, 3:5, 4:7…) the curve closes after one full period and traces a recognisable knot. The shape depends only on the frequency ratio and the phase offset phi: 90 degrees at 1:1 gives a circle; 0 degrees at 1:1 gives a diagonal line. Modulating each axis by a second frequency (amplitude modulation) produces more complex star-burst variants. In generative design, connecting all sampled points with lines whose opacity fades by distance creates a dense, jewel-like pattern.

Examples

x = p.sin(angle * freqX + p.radians(phi)) * p.cos(angle * modFreqX); y = p.sin(angle * freqY) * p.cos(angle * modFreqY); — M_2_5_01 connects 1000 sampled points and draws lines between near neighbours.

Assessment

What Lissajous figure does freqX=3, freqY=2, phi=0 produce? How does changing phi from 0 to 90 change it? Draw both on paper or in code.