Interactive Lissajous Curves

Published: June 27, 2026

This interactive playground demonstrates the math behind analog oscilloscope visualization. A Lissajous curve is traced by compounding two perpendicular sine waves: one driving the horizontal position (x), and another driving the vertical position (y).

x(t) = A · sin(a · t + δ)

y(t) = B · sin(b · t)

Horizontal Frequency (a): 5

Vertical Frequency (b): 7

Phase Shift (δ): 90°

How to Read the Interactive Curve

Frequency Ratio (a:b): The ratio of horizontal to vertical frequencies dictates the pattern complexity.
- If you set a = 5 and b = 7, you get the exact 5:7 pattern from the history article.
- Count the top/bottom loop peaks (lobes) to verify a = 5.
- Count the left/right loop peaks to verify b = 7.
Phase Shift (δ): Changing the phase rotates the pattern in a simulated 3D space. When δ = 90° (or π/2 radians) with a = b, you get a perfect circle. When δ = 0°, you get a diagonal line.
Phase Animation (Drift): In real-world analog oscilloscopes, two separate signal generators are rarely perfectly synchronized. A tiny difference in frequency causes the phase to slowly drift over time. This makes the curve continuously spin, providing that classic 3D rotating illusion.
Sweep Animation (Time): Enabling "Animate Sweep" simulates the progressive sweep of the oscilloscope's electron beam. The beam paints the curve over the time parameter t, creating a white-hot leading edge and a green phosphor decay trail behind it. Disabling it draws the entire static closed curve instantly.