x = A·sin(aτ + δ), y = B·sin(bτ) — watch the phase sweep
3
4
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1200
About: Lissajous figures are the paths traced by a point whose x and y coordinates are independent sinusoids. When the frequency ratio a:b is rational, the curve is closed — it has exactly a·b crossings. The phase δ continuously morphs the figure: at δ=0 and δ=π it degenerates to a line, while δ=π/2 produces the most "open" form. They appear in oscilloscope analysis, antenna polarization patterns, and mechanical harmonograph drawings. Jules Antoine Lissajous demonstrated them optically in 1857.