Lissajous curves are graceful patterns formed by combining two perpendicular oscillations, typically one horizontal and one vertical, each varying sinusoidally in time. The exact shape depends on the ratio of their frequencies and the phase difference between them. When the frequencies are equal, the curve may appear as a straight line, an ellipse, or a circle. When the frequencies differ, the path becomes more intricate, producing looping, woven figures that repeat periodically when the ratio i