What is a Lissajous curve?

A Lissajous curve (also called a Bowditch curve) is the path traced by a point whose x and y coordinates each oscillate sinusoidally: x(t) = A·sin(a·t + δ), y(t) = B·sin(b·t). The shape depends on the frequency ratio a:b and the phase shift δ. When a:b is a rational number (ratio of two integers), the curve is closed and repeats after a finite time. When a:b is irrational, the curve never closes and eventually fills the entire rectangle — a Lissajous knot. They were first systematically studied by Nathaniel Bowditch in 1815 and later popularised by Jules Antoine Lissajous using tuning forks and optical mirrors in 1857.

What do the frequency ratios mean and which shapes do they produce?

The frequency ratio a:b directly determines the topology of the Lissajous figure. A 1:1 ratio produces an ellipse or straight line (depending on phase δ). A 1:2 ratio produces a figure-eight (infinity symbol) with δ = π/2. A 2:3 ratio gives a three-lobed figure resembling a figure-eight with an extra loop. In general, the curve crosses the horizontal axis 2b times and the vertical axis 2a times (for a:b in lowest terms). These ratios also correspond to musical intervals: 1:2 = octave, 2:3 = perfect fifth, 3:4 = perfect fourth, 4:5 = major third. An oscilloscope displaying the same ratios from audio signals produces these exact shapes.

What does the phase shift δ do to the shape?

The phase shift δ (delta) rotates and deforms the Lissajous figure without changing its fundamental topology. For 1:1 frequency ratio: δ=0 gives a diagonal line, δ=π/2 gives a circle, intermediate values give ellipses tilted at different angles. For 1:2: δ=0 gives a parabola, δ=π/2 gives the figure-eight. The 'Sweep Phase' button continuously varies δ from 0 to 2π and back, producing a mesmerising morphing animation. The phase shift is what electrical engineers measure when using a Lissajous figure on an oscilloscope to determine the phase difference between two sinusoidal signals.

Where are Lissajous figures used in real life?

Lissajous figures have both practical and scientific applications: (1) Electrical engineering — phase measurement between two sinusoidal signals displayed on an oscilloscope's X-Y mode; the shape immediately reveals the phase difference sin(δ) = (y-intercept)/(y-max). (2) Mechanics — the coupled pendulum (Blackburn pendulum) traces perfect Lissajous curves when the two suspension lengths give the required frequency ratio. (3) Seismology — two-component seismograph records trace Lissajous-like patterns revealing characteristics of seismic waves. (4) Optics — Thomas Young used interference patterns equivalent to parametric curves to study light. (5) Music — visualisers and synthesiser oscilloscope displays showing harmonic relationships. (6) Space missions — halo orbits near Lagrange points (e.g. SOHO spacecraft at L1) are three-dimensional Lissajous orbits.

Home▸Mathematics▸Lissajous Curves & Parametric Equations — Interactive Oscillation Simulator

🧪 Lissajous Curves & Parametric Equations — Interactive Oscillation Simulator

Interactive Lissajous figure simulator. Adjust frequency ratios a:b, phase shift δ, and amplitudes to trace beautiful parametric curves. Watch dual oscilloscope panels, enable rainbow trails, and explore the mathematics of coupled harmonic oscillators. Free browser-based tool.

Mathematics2DModerate60 FPS

⚙ Under the hood

This simulation allows users to explore Lissajous curves and parametric equations through an interactive oscillation simulator. By adjusting parameters like frequency and amplitude, they can visualize the resulting patterns and understand the mathematical relationships governing their movement.

Lissajous CurvesParametric EquationsOscillation

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install