Projecting Mathematical Curves with Laser Light: A Tribute to Ptolemy
Merrill Lessley and Paul Beale · 2008
This paper describes the math and technology required to project a variety of mathematical curves with a computerized laser light control system. The focus is upon creating “large-scale” animated laser projections of roulette or spirograph shapes, such as those found in the epitrochoid, hypotrochoid, epicycloid, and hypocycloid families. The projection process described utilizes a geometric approach that was first presented by the Greek or Egyptian mathematician and astronomer Ptolemy, that of “epicycles.” Examples of these laser images can be viewed at http://spot.colorado.edu/~lessley/. Introduction Many mathematical curves in the epitrochoid, hypotrochoid, epicycloid, and hypocycloid families are beautiful to view. These curves are usually graphed by incorporating such devices as a plotter, printer, or video device. The most common mechanical method for graphing such curves involves a spirograph tool in which a small trace wheel is rotated within a larger stationary wheel. While these techniques produce interesting images, the images are normally rather small and not animated. Creating very large-scale animated images with high-intensity lasers such as those encountered in laser light shows, concerts, or art installations require some unusual graphing and projection strategies. The Problem: Moving from the Spirograph to the Laser Forming a projected pattern of any mathematical curve with a moving laser “dot” is done by “scanning” that dot rapidly with X and Y axes galvanometers through an image path at least sixteen times a second. At this rate, our “persistence of vision” makes the image appear solid. Scanning a laser dot rapidly in a circular pattern creates the appearance of a solid circle in light. Furthermore, in moving the laser dot to draw a series of smaller circles that follow the path of a larger circle (a roulette shape), we see that--unlike the spirograph tool with its static base circle--the trace and base circles rotate simultaneously. When scanning this way, both circles usually possess individual frequency and diameter factors. In terms of geometric patterns, this is similar to what the ancient Greek or Egyptian astronomer/mathematician Ptolemy expressed in his attempt to explain the visual motion of the planets by developing the idea of “epicycles.” Of critical importance in our project was how Ptolemy’s approach could help us create roulette patterns in a slightly different manner than we would normally do with a conventional spirograph tool. His method has the trace circle rotating “on the circumference” of the base circle (not inside or outside of it). Both circles can also maintain differing rotational speeds (frequencies) and directions. In terms of translating the math to electromechanical scanning devices (galvanometers), through summing amplifiers and digital-to-analog converter circuits, this approach is simple and very flexible. Epitrochoid and hypotrochoid formulas can be modified to include special-case curves like the rose and ellipse.