Lissajous curves

A Lissajous curve is what you get when two vibrations at right angles combine: x = sin(at + δ), y = sin(bt). The ratio a/b decides the number of lobes, and the phase δ twists the pattern. They are familiar from oscilloscopes and from the ABC logo.

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The Lissajous figure with a = 3, b = 4 and δ = π/2.

Equations to type

Ratio 3 : 4

(sin(3t + pi/2), sin(4t))

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With sliders for a, b and δ

(sin(a t + d), sin(b t)) a = 3 b = 2 d = 0.5

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Ratio 1 : 2 (a figure eight)

(sin(t), sin(2t))

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At a glance

Equations: x = A sin(at + δ), y = B sin(bt)
Closed curve: When a/b is rational
Edge touches: a times on the right-hand edge, b times on the top edge

Reading the frequency ratio

When a/b is a fraction in lowest terms, the curve closes. With x = sin(at + δ) and y = sin(bt), it touches the right-hand edge a times and the top edge b times (for a general phase), so you can read the ratio of the two frequencies straight off the picture, which is exactly how engineers used them to compare signals on an oscilloscope before frequency counters were common. If the ratio is irrational, the curve never closes and slowly fills a rectangle.

The phase δ

With a = b = 1 the phase alone decides the shape: δ = 0 gives a diagonal line, δ = π/2 a circle, and anything between an ellipse. For other ratios, changing δ makes the figure appear to rotate in three dimensions, which you can watch by pressing play on the δ slider.

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Ratio 1 : 1 with δ = 0.3, 1.2 and π/2: ellipses opening into a circle.

History

Nathaniel Bowditch studied these curves in 1815, and Jules Antoine Lissajous made them famous in 1857 by bouncing light off mirrors attached to two tuning forks vibrating at right angles. The patterns showed instantly whether the forks were in tune.