About This Tool
Lissajous curves are the paths traced by a point whose x and y coordinates each follow a sine wave, but at different frequencies. Named after French physicist Jules Antoine Lissajous who studied them in 1857, these figures appear in physics, engineering, and music — oscilloscopes display them to compare signal frequencies, and they arise naturally in any system with two independent oscillations. The shape of a Lissajous curve depends entirely on the ratio a:b of the two frequencies and the phase offset δ between them. As the phase δ slowly advances, the figure morphs continuously through the full family of shapes for that ratio.
How to Use
The animation starts automatically, cycling through the phase offset to reveal the full family of shapes.
Select a frequency ratio preset (e.g. 1:2, 3:4) using the Ratio buttons to change the type of curve.
Use the Speed buttons (Slow / Normal / Fast) to control how quickly the phase advances.
Press Pause / Play to freeze the curve at any point for closer inspection.
Frequently Asked Questions
What does the a:b ratio control?
The ratio of x-frequency to y-frequency. A 1:1 ratio produces ellipses and circles. A 1:2 ratio produces a figure-eight (parabolic) shape. Higher ratios produce more complex curves with more lobes — the number of horizontal lobes equals b and the number of vertical lobes equals a.
Why does the shape keep changing?
The phase offset δ between the two sine waves is continuously incremented. At δ = 0° the curve closes into its simplest form; at δ = 90° it forms a different canonical shape. Animating δ shows the entire continuous family of curves for a given a:b ratio.
Where are Lissajous curves used in real life?
Oscilloscopes use them to measure the frequency and phase relationship between two electrical signals. They also appear in laser light shows, harmonograph drawings, and the analysis of mechanical vibrations.
When does a Lissajous curve close into a stable shape?
When the ratio a:b is a ratio of small integers (1:2, 3:4, etc.) the curve eventually closes and repeats. When the ratio is irrational the curve never closes and will eventually fill a rectangular region densely.