Lissajous Curve
Parametric curvesThe figure drawn by two perpendicular oscillations x = sin(at + δ), y = sin(bt) — the oscilloscope’s signature art.
Adjust parameters
Equation
Open the Creator with a ready-made animation brief for this curve — review the scene plan and Manim code before rendering.
What is the Lissajous Curve?
Feed one sine wave to the x-axis and another to the y-axis, and the point traces a Lissajous curve. Nathaniel Bowditch drew the first ones with a pendulum in 1815; Jules Antoine Lissajous produced them optically in 1857 by bouncing light off mirrors attached to two tuning forks, turning invisible frequency ratios into visible geometry.
The curve is a frequency detector you can read by eye: it closes only when a : b is rational, and counting the lobes along each edge reveals the ratio. The phase δ then rotates the figure’s character — with a = b it morphs a diagonal line through ellipses into a circle. This is why analog engineers used oscilloscope Lissajous figures to tune one signal against a reference.
Key properties
- Closed curve ⟺ the frequency ratio a : b is rational.
- Lobe counts along the two edges are in the ratio b : a.
- a = b: a line (δ = 0), ellipses, then a circle (δ = π/2, equal amplitudes).
- The curve always fits in the square [−1, 1] × [−1, 1], touching each edge.
- The 3 : 1 Lissajous figure is the logo of the Australian Broadcasting Corporation.
Where you'll see it
Oscilloscope phase measurement, laser light shows, harmonographs, and broadcast logos.