Lissajous curves are intricate patterns that emerge when two perpendicular harmonic motions are combined—typically one along the x-axis and another along the y-axis, each with its own frequency and phase difference. Depending on the ratio of frequencies, the curves can appear as elegant loops, figure-eights, or complex woven patterns. They were first studied in the 19th century by Jules Antoine Lissajous, who used optical methods with mirrors to visualize the motion. These curves are not only ma