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Interactive Lissajous Curve Explorer

Explore Mathematical Beauty

Lissajous curves are created by combining two perpendicular oscillations with different frequencies. These beautiful patterns appear in physics, engineering, and nature - from oscilloscope displays to planetary orbits!

Mathematical Equations:
x = A × sin(a×t + φ)
y = B × sin(b×t)
Where:
A, B = amplitudes | a, b = frequencies | φ = phase shift | t = time

Live Curve Generation

Frequency Ratios

2.0
3.0

Amplitudes

200
200

Animation

0.0
1.0
500

Presets

Instructions

  • • Adjust sliders for real-time changes
  • • Use 'Clear Trail' to see immediate effects
  • • Try different frequency ratios
  • • Use presets for classic patterns
  • • Watch how phase shift rotates patterns

Real-World Applications

  • Oscilloscopes: Display electrical waveforms
  • Astronomy: Planetary orbital patterns
  • Music: Harmonic frequency relationships
  • Engineering: Vibration analysis
  • Art: Mathematical beauty in design

Pattern Types

  • 1:1 Ratio: Circle or ellipse
  • 1:2 Ratio: Figure-eight pattern
  • 2:3 Ratio: Three-lobed pattern
  • Integer Ratios: Closed, repeating curves
  • Irrational Ratios: Never-repeating patterns