Lissajous Curve
The figure drawn by two perpendicular oscillations x = sin(at + δ), y = sin(bt) — the oscilloscope’s signature art.
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Formulas and key values
- Parametric equations
- Period · whole-number a, b
- Closed curve
- Bounding box
a and b are the two frequencies, δ is the phase shift, and t runs over one period.
Explainer video
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Start the video01 · Definition
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.
02 · Properties
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.
03 · Questions
Lissajous Curve: common questions
- What is the equation of the Lissajous Curve?
- Parametric equations: x = sin(at + δ), y = sin(bt). a and b are the two frequencies, δ is the phase shift, and t runs over one period.
04 · Wonders
See it in a math wonder
05 · Keep exploring