Lissajous Figures - Harmonic Motion Synthesis

What are Lissajous Figures?

Lissajous figures are patterns formed by combining two perpendicular simple harmonic motions. When a point moves with harmonic motion in both X and Y directions, the resulting path creates intricate geometric patterns that depend on the frequency ratio and phase difference between the two motions.

Applications

Oscilloscopes: Used to compare two signal frequencies and measure phase differences. When the frequency ratio is simple (like 1:1, 1:2, 2:3), stable patterns form that make it easy to visualize the relationship.
Physics & Engineering: Analyzing vibrations, resonance, and wave interference. Common in acoustics, electronics, and mechanical systems.
Mathematics: Beautiful examples of parametric equations and how small parameter changes create dramatically different patterns.

Pattern Types

Line (ω₁:ω₂ = 1:1, δ = 0° or 180°): Both oscillations in phase or anti-phase, creating a straight line at ±45°.
Circle (ω₁:ω₂ = 1:1, δ = 90°): Equal frequencies with 90° phase shift creates a perfect circle.
Ellipse (ω₁:ω₂ = 1:1, other δ): General case creates ellipses of varying eccentricity.
Figure-8 (ω₁:ω₂ = 1:2): Y frequency twice X frequency with 90° phase creates a figure-8 pattern.
Complex Patterns: Higher frequency ratios (2:3, 3:4, 3:5, etc.) create increasingly complex closed curves with multiple lobes.

How to Use

Adjust the frequency sliders to change the ratio ω₁:ω₂ and observe how the pattern complexity increases. Modify the phase difference δ to see patterns morph from lines to circles to ellipses. Use preset buttons for classic Lissajous patterns, or experiment with custom combinations to discover new shapes.