Worked example
Lissajous Curve: Parametric Art from Sine Waves
Before oscilloscopes had digital displays, physicists studied frequency ratios by feeding two sine waves into the horizontal and vertical plates of a cathode-ray tube. The glowing patterns they traced — Lissajous figures — reveal the ratio of the two frequencies at a glance. Here, x = sin(3t) oscillates three times for every two oscillations of y = cos(2t), weaving a closed, symmetric knot.
What makes this curve impossible as a regular y = f(x) graph is that it fails the vertical line test dramatically: a single x can correspond to many y values as the curve loops back on itself. Parametric equations sidestep that limitation by giving x and y their own formulas in a shared parameter t — the same idea animates everything from clock hands to planetary orbits.
Plotted expressions
- x = sin(3*t), y = cos(2*t)
The link preloads these exact expressions into the calculator — no typing needed.
What to notice
- The 3:2 frequency ratio determines the pattern: count the lobes touching each side of the bounding square.
- Change sin(3*t) to sin(4*t) for a 4:2 figure and compare the symmetry.
- Because t runs a full 2π, the curve closes perfectly — shorten the t-range and watch it become an open arc.