# Merrill Lessley and Paul Beale

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.

“Laser Cornucopia” was constructed by applying mixtures of sine and cosine signals to a green laser programmed to scan rapidly and repeatedly in “X” and “Y” directions. Creating this art required a mathematical approach similar to the traditional graphing of hypotrochoid curves. However, since we use base and trace oscillators to form our images, traditional parametric equations are modified to accommodate the “dynamic” scanning process. Revised equations, therefore, consider the elements of 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). Also, ω = 2πf, where the base frequency f is the number of times per second that the base oscillator completes a cycle. With this particular image, the fundamental hypotrochoid scan was modulated by a third cosine signal, which formed a pseudo “Z” axis, thus generating the three-dimensional quality of the image.

“Laser Heart” was constructed by applying mixtures of sine and cosine signals to three lasers programmed to scan rapidly on “X” and “Y” axis lines moving rapidly and repeatedly in various directions. An image was extracted from a video and made transparent in Photoshop. Three copies were overlaid to gain the montage effect. Creating the art started with a mathematical approach similar to the graphing of any hypotrochoid curve. However, since we use base and trace oscillators to form images, traditional parametric equations were modified to accommodate the “dynamic” scanning process. Revised equations considered 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). Also, ω = 2πf, where the base frequency f is the number of times per second that the base oscillator completes a cycle. Since the Rose curve is a special case of the hypotrochoid function, a = (2n) h/(n+1), b = (n-1)/(n+1) h, where n is the number of petals.