Lissajous Figures: When Two Oscillations Draw

Make a point oscillate left and right while it also oscillates up and down, and it draws a Lissajous figure. Depending on the two frequencies and the phase between them, the trace is a line, an ellipse, a figure eight or an intricate woven knot. These curves have been used to compare tuning forks and test electronics, and they shape the orbits of some spacecraft. The waveform on our Oscillayne: The Harmonic Pulse tee is the one-dimensional version of the same harmonic motion.

What is the equation of a Lissajous curve?

A Lissajous curve is the graph of the parametric equations Lissajous curve, Wikipedia^[1]

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

where x and y are the displacements (in m, or in volts on an oscilloscope screen), A and B are the amplitudes in the same units, a and b are the angular frequencies in rad/s, t is time in s and δ is the phase offset in radians. The figure stays inside a rectangle of width 2A and height 2B.

What decides the shape?

The frequency ratio

The appearance of the figure is sensitive to the ratio a/b Lissajous curve, Wikipedia^[1]. The pattern closes only if a/b is rational, a ratio of whole numbers Lissajous curve, Wikipedia^[1]. If the ratio is irrational, the curve never repeats. Given enough time, it passes arbitrarily close to every point of its rectangle.

The phase

• For a/b = 1 the figure is an ellipse. Special cases are a circle (A = B, δ = π/2) and a straight line (δ a multiple of π) Lissajous curve, Wikipedia^[1].
• For b/a = 2 and δ = π/4 the curve is a parabola Lissajous curve, Wikipedia^[1].
• For a = 1 and b = N, with δ = (N − 1)/N · π/2, the curves are the Chebyshev polynomials of the first kind of degree N Lissajous curve, Wikipedia^[1]. With cosines this is easy to see. Put x = cos t and y = cos(Nt), and the identity T_N(cos θ) = cos(Nθ) Chebyshev polynomials, Wikipedia^[2] gives y = T_N(x).

Reading the ratio off the picture

In one full period, y(t) = B·sin(b·t) reaches its maximum b times and x(t) reaches its maximum a times. So, for a generic phase, the curve touches the top edge of its rectangle b times and a side edge a times (our derivation). Counting touches gives the frequency ratio. A figure that touches the top once and a side three times has a/b = 3.

Who discovered Lissajous figures?

The family of curves was investigated by Nathaniel Bowditch in 1815, and later in more detail in 1857 by Jules Antoine Lissajous, after whom it is named Lissajous curve, Wikipedia^[1]. That is why it is also called the Bowditch curve Lissajous curve, Wikipedia^[1]. Lissajous (1822–1880) was a French physicist Jules Antoine Lissajous, Wikipedia^[3]. His apparatus bounced a beam of light off a mirror on a vibrating tuning fork, then off a second mirror on a perpendicular tuning fork, usually of a different pitch, and onto a wall Jules Antoine Lissajous, Wikipedia^[3]. A musical interval became a visible shape.

The pendulum version

Before electronics, Lissajous curves could be drawn mechanically with a harmonograph Lissajous curve, Wikipedia^[1]. One simple version is the Blackburn pendulum, a Y-shaped string pendulum named after Hugh Blackburn, who described it in 1844. It was first discussed by James Dean in 1815 and analysed mathematically by Bowditch the same year Harmonograph, Wikipedia^[4]. Harmonographs began to appear in the mid-19th century and were most popular in the 1890s Harmonograph, Wikipedia^[4].

In a Blackburn pendulum the bob swings with an effective length L_x in one direction and L_y in the other. For small swings each motion has angular frequency √(g/L), so the frequency ratio is

ω_x / ω_y = √(L_y / L_x)

where ω is angular frequency in rad/s, L is effective pendulum length in m and g is the gravitational acceleration in m/s² (our derivation). For a 2:1 figure, one effective length must be four times the other. Friction makes a real harmonograph's swings shrink, so its drawings spiral inward. For more pendulum physics, see our post on the Foucault pendulum.

How are Lissajous figures used on an oscilloscope?

Feed one signal to an oscilloscope's horizontal input and another to its vertical input, and the screen draws their Lissajous figure. This was used to show that two sine waves had a simple frequency ratio, and to show the phase difference between two waves of the same frequency Oscilloscope, Wikipedia^[5]. In broadcast engineering, the same method is used to plot the left and right stereo channels Oscilloscope, Wikipedia^[5].

For equal frequencies the figure is an ellipse, and the phase can be read from it:

|sin δ| = y₀ / B

where y₀ is the height at which the ellipse crosses the vertical axis (x = 0) and B is its maximum height, in the same units (our derivation from the equations above). A line means 0 or π, and a circle with equal amplitudes means π/2. A circle is also what a charged particle traces in a magnetic field, two perpendicular oscillations a quarter cycle apart; see our post on cyclotron motion.

Where else do Lissajous curves appear?

• Spacecraft orbits. Near the Sun–Earth Lagrange points, spacecraft can fly Lissajous orbits. SOHO, ACE and DSCOVR have used them at L1, and WMAP, Herschel, Planck and Gaia at L2 Lissajous orbit, Wikipedia^[6]. The points themselves are the subject of our Libratrix: Lagrange Points sticker.
• Logos. The Australian Broadcasting Corporation logo uses a = 1, b = 3, δ = π/2, and MIT's Lincoln Laboratory uses a = 3, b = 4, δ = π/2 Lissajous curve, Wikipedia^[1].
• Film. John Whitney's title sequence for Alfred Hitchcock's 1958 film Vertigo is based on Lissajous figures Lissajous curve, Wikipedia^[1].

Find more oscillation physics in our Waves, Chaos & Matter collection and Waves & Optics collection.

Quick FAQ

Why does a near-1:1 figure slowly rotate?

If the frequencies differ slightly, the ellipse fails to close and appears to precess Lissajous curve, Wikipedia^[1]. The phase drifts at the frequency difference, so the shape cycles once per beat period, 2π/|a − b| (our derivation).

Are Lissajous figures chaotic?

No. They are completely regular. Even the dense, never-closing pattern of an irrational ratio is predictable at every moment.

Is a Lissajous figure the same as a harmonograph drawing?

A harmonograph drawing is usually a Lissajous curve or a more complex relative, often damped Harmonograph, Wikipedia^[4].

References

1. Wikipedia, “Lissajous curve”. https://en.wikipedia.org/wiki/Lissajous_curve
2. Wikipedia, “Chebyshev polynomials”. https://en.wikipedia.org/wiki/Chebyshev_polynomials
3. Wikipedia, “Jules Antoine Lissajous”. https://en.wikipedia.org/wiki/Jules_Antoine_Lissajous
4. Wikipedia, “Harmonograph”. https://en.wikipedia.org/wiki/Harmonograph
5. Wikipedia, “Oscilloscope”. https://en.wikipedia.org/wiki/Oscilloscope
6. Wikipedia, “Lissajous orbit”. https://en.wikipedia.org/wiki/Lissajous_orbit

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.