Pretty Curve with Mathematics: Lissajous

Lissajous art by computer drawing

The other day while browsing some topics on waves, I came across some pretty pictures. These are nice, aren't they? Did you know that they are drawn by a computer? It used some simple mathematics that create the curves! These particular curves are called Lissajous curves in mathematics.

How did people discover the mathematics of these curves? Why? And what's their use?

Lissajous curves are named so after the French scientist Lissajous, who discovered them. He had built a machine to study sound waves. He would fix one mirror on the thing that was making sound. He would reflect a beam of light from it on to another mirror. This second mirror was attached to another vibrating object. When the mirrors vibrated, the light reflected from them fell on different points on a screen that he watched.

Apparatus that Lissajous built

But why two mirrors? Well, sound vibrates really fast - more than what our eye can catch. A fast moving light point appears like a straight line. Its like the blur of fan blades. To see a pattern, he had to move the beam in the perpendicular direction as well. When the second mirror vibrated in a direction perpendicular to the first one, the light beam appeared like a graph on the screen! There was another curious thing. Depending on how fast the second mirror was vibrating compared to the first one, the light would trace different interesting shapes. He then went about studying the mathematics behind it and discovered these curves.

You can also make a Lissajous curve by making a graph (plot) from two waves. Place two waves side by side. To mark a point on the graph, choose two values, one from each wave, and use them for the point coordinate. Lissajous curve is the result of two waves that are perpendicular to each other. You can use the demonstration below to play around with Lissajous curves (needs Java). Change the sliders and see how the waves and the curves change.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

There are some well known Lissajous shapes that are used when comparing waves. The ratio between "Wave 1" and "Wave 2" determines the shape. Pause the animation above (pause button on botton left corner) and try the steps below.

• Set sliders "Wave 1" and "Wave 2" to minimum. Change the "Shift Amount". Do you see some known figures like straight lines, circles and elipses?
• For each of the steps below, start with "Shift Amount" at zero. And then change it to see how the figure appears to rotate.
• Increase "Wave 2" by one step.
• Increase "Wave 1" by two steps.
• Increase "Wave 2" by two steps.
• Increase "Wave 1" by two steps.

Lissajous curves are used today in oscilloscopes, a kind of instrument that is used to study electrical waves. You usually find them in electronics and science labs. It is also used in space travel! Spacecrafts often follow an orbit that resembles a Lissajous curve. And have you watched laser shows? These are also used in laser shows to create beautiful patterns. And in computer graphics to create wonderful shapes and diagrams. Below is another demo for you to play around with. Change the sliders to get different effects. Take some time to explore the mathematical equations if you understand it and are interested!

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Photo credits:

• http://physics.kenyon.edu/EarlyApparatus/Oscillations_and_Waves/Lissajous_Figures/Lissajous_Figures.html
• http://www.bit-101.com/blog/