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.

Mathematics Physics Free A-Level Maths IB Maths AA GCSE Extension Beginner

Lissajous Curves & Parametric Equations

Trace the elegant figures produced by two coupled sinusoidal oscillations at right angles. Explore how frequency ratios encode musical intervals, watch the phase shift morph the shape continuously, and read the x and y motions from the oscilloscope side panels.

Ratio a:b

3:2

Period

2π

Phase δ

0.50π

Laps

Preset Ratios

Frequency A (x-axis)

Frequency B (y-axis)

Phase δ (0 → 2π)

0.50π

Speed

1.00x

Trail Fade

18%

Amplitude X

85%

Amplitude Y

85%

Rainbow Trail Persist Trail Oscilloscope X Oscilloscope Y

Space pause C clear S sweep 1–5 presets

What Are Lissajous Curves?

A Lissajous curve (pronounced lee-SAH-zhoo) is the graph of a system of parametric equations:

x(t) = A · sin(a·t + δ)
y(t) = B · sin(b·t)

where a and b are the angular frequencies along the x and y axes, δ is the phase offset, and A, B are amplitudes. They were first analytically studied by American mathematician Nathaniel Bowditch (1815) — hence the alternative name Bowditch curves — and later recreated optically by French physicist Jules Antoine Lissajous (1857) who used tuning forks and mirrors to project the figures in light.

Closed vs Open Curves

When a:b is a rational number (ratio of two integers), the curve is closed — it repeats exactly after a period T = 2π / gcd(a,b). When a:b is irrational, the curve is open and eventually fills the entire A×B rectangle, never exactly retracing its path.

Counting Lobes

For a ratio a:b in lowest terms, the Lissajous curve crosses the vertical bounding line 2b times and the horizontal bounding line 2a times. The number of interior lobes is (a−1)(b−1) + (a−1)/2 + (b−1)/2 for odd sums, giving a quick visual check.

Phase Shift δ

The phase shift rotates and deforms the figure. For 1:1 ratio, δ = 0 gives a diagonal line, δ = π/4 gives a tilted ellipse, δ = π/2 gives a circle. The Sweep Phase button animates this transition, used in physics labs to demonstrate SHM superposition.

Musical Intervals

Frequency ratios map to musical intervals. 1:2 = octave, 2:3 = perfect fifth, 3:4 = perfect fourth, 4:5 = major third. This connection between geometry and harmony fascinated 19th-century scientists and inspired Helmholtz's work on consonance.

Common Frequency Ratios

Ratio a:b	Phase δ = π/2	Musical Interval	Lobes
1:1	Circle or ellipse	Unison	1
1:2	Figure-8 (∞)	Octave	2
1:3	Triple loop	Octave + fifth	3
2:3	Two humps / three loops	Perfect fifth	6
3:4	Complex interlaced loops	Perfect fourth	12
4:5	Dense interlaced figure	Major third	20
5:6	Very dense figure	Minor third	30

The Oscilloscope Connection

Engineers use X-Y mode on an oscilloscope to compare two sinusoidal signals. If one signal drives the x-deflection plates and another the y-deflection plates, the resulting trace is exactly a Lissajous figure. The shape reveals the frequency ratio and phase difference between the signals:

A circle → same frequency, δ = π/2 (90° phase difference)
A diagonal line → same frequency, δ = 0 (in phase)
A figure-eight → 1:2 frequency ratio
Phase difference formula: sin(δ) = Y₀ / Y_max, where Y₀ is the y-intercept

The side oscilloscope panels in this simulator show the individual sinusoidal projections that combine to produce the Lissajous figure — exactly what you'd measure on two separate oscilloscope channels.

Physics: Coupled Harmonic Oscillators

A Blackburn pendulum consists of a weight suspended from two pivots at right angles. By setting the pendulum lengths to achieve integer frequency ratios, the bob traces a perfect Lissajous figure in sand. For a pendulum of length L, the natural frequency is f = (1/2π)√(g/L). To get a 2:3 ratio, the y-axis pendulum must be 9/4 times longer than the x-axis pendulum, since f ∝ 1/√L.

Period of a Closed Lissajous Curve

A closed Lissajous curve with frequency ratio a:b (in lowest terms) completes one full cycle in time:

T = 2π / gcd(a, b)

For a=3, b=2: T = 2π/1 = 2π. For a=4, b=6: first reduce to 2:3, T = 2π/gcd(2,3) = 2π. The HUD 'Period' readout shows this value in radians.

Applications in Science & Engineering

⚡ Electrical Engineering

Phase measurement, frequency ratio verification, signal analysis using oscilloscopes in X-Y mode. Standard technique in electronics labs and circuit testing.

🌍 Seismology

Two-component seismograph records produce Lissajous-like particle motion plots revealing the polarisation of seismic waves and source direction.

🚀 Space Mechanics

Spacecraft near Lagrange points (SOHO at L1, JWST at L2) follow three-dimensional Lissajous orbits — halo orbits with incommensurate frequencies in each plane.

🎵 Music Technology

Oscilloscope visualisers in synthesisers display Lissajous figures from stereo signals, letting musicians visualise stereo phase and harmonic relationships in real time.

📖 Deep Dive Article

Read the full educational article: Lissajous Curves & Parametric Equations: The Mathematics of Coupled Oscillations — covering history, Bowditch's original analysis, musical intervals, oscilloscope phase measurement, and Lissajous knots in 3D.