Lissajous Figures and Harmonographs

The Lissajous script draws the curve known as a Lissajous curve.

You may be familiar with this curve from the pattern displayed on an oscilloscope when a sine wave (from an oscillator) is connected to the horizontal trace, and another sine wave connected to the vertical trace. A variety of pleasing patterns can be generated by adjusting the frequencies and phase offsets of the two signals. Lissajous figures have been used as special effects in science fiction films of the 60's and 70's including 2001: A Space Odyssey and Silent Running.

This curve can be described mathematically as two parametric equations:

 y(t) = H * sin(A*t + P)
 x(t) = W * sin(B*t)

H and W define the height and width of the curve while A, B and P determine the overall shape. For a more mathematical treatise, refer to the references at the end of this article.

The following dialog box is used to configure the curve parameters as well as the drawing tool, brush and color.

The width and height of the curve are calculated automatically from the size of the canvas, but reduced by the value of the Margin setting.

• A multiplier - sets the value of A.
• B multiplier - sets the value of B
• Phase offset - sets the value of P

The curve is drawn by varying the value of t from 0 to 360 (the number of degrees in a circle) in increments of the value set by Step Size.

The next two parameters, Decay Percentage and Cycles are described later in the Harmonograph section.

Tool - Choose the pencil, brush or airbrush for drawing the curve.

Brush - displays the GIMP brush selection dialog box.

Color - choose the color for the brush.

Setting both A and B multipliers to 1 and varying the phase value creates lines, ellipses and circles. Phase values of 0 and 180 give diagonal lines.

A value of 90 produces a circle.

And other values produce ellipses:

As the values of the multipliers are increase, more interesting shapes emerge. the following shapes were drawn with A set to 2 and B set to 1 with phase values of 0, 45, 90 and 135.

Even higher values give denser curves. Here are two curves with A and B set to 11 and 19, and A and B set to 23 and 21.

Even more interesting patterns can be created if the width and height of the curve (the amplitude) are decayed.

The Decay Percentage is used to reduce the amplitude by the specified amount in one cycle. The script draws the points for the curve going through one cycle.

Increasing the value of Cycles will cause the script to continue drawing additional cycles, applying the decay percentage. You will need to adjust this value in conjunction wiht the previous parameter. For example, if the decay percentage was 10% you would need to set Cycles to 10 to draw the completely deayed curve.

Here is a curve with A multiplier of 5, B multiplied of 3, decay percentage of 20, and 5 cycles. Smoother results can be obtained by drawing on a larger canvas than scaling.

And here are some more harmonographs:

Download the filter here.

See more of my GIMP filters and scripts here

Weisstein, Eric W. "Lissajous Curve." From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/LissajousCurve.html

Weisstein, Eric W. "Harmonograph." From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/Harmonograph.html

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