# Merrill Lessley

In our interdisciplinary research and creative work, we create laser images in motion that represent specific mathematical curves (epicycloids, hypocycloids, roses, epitrochoids, hypotrochoids, and other special sine/cosine cases). We create these images by using a computer-controlled laser projection system that we have designed and built. Graphing such curves in multiple laser colors produces a wide variety of images that are really quite beautiful. Unlike drawing them on paper, however, projecting such curves with a laser, or several lasers, poses a particularly challenging problem: while a laser is often referred to as a kind of “pencil” in light, it can only be used to generate a complete picture by moving its projected “dot” rapidly and repeatedly over a reflective surface. The images we create must be scanned at rates between 15 and 2000 times per second. Our primary goal is to create computerized tools that can be utilized by laser artists throughout the world.

“Fourier's Epitrochoid Sawtooth” was created by applying mixtures of sawtooth signals to three lasers programmed to scan rapidly on “X” and “Y” axis lines. These images were extracted from a video recording of all lasers scanning a seventeen-petal epitrochoid curve, but sine and cosine waves were replaced with sawtooth forms. Being incommensurate, the laser images continuously rotate over each other. Creating the art required an approach similar to the graphing of any epitrochoid curve. But, since we use base and trace oscillators to form images, modified parametric equations were used to accommodate a “dynamic” scanning process. These equations consider both base and trace frequencies: x = (a+b) cos(ωt) - h cos(((a+b)/b) ωt); y = (a+b) sin(ωt) - h sin(((a+b)/b) ωt), where T=2π/ω is the time needed for the base oscillation to complete a full cycle of petals. Our sawtooth version is based on Fourier's theorem which allows us to generalize sine and cosine to any periodic shape.