## Merrill Lessley and Paul Beale |

This is a high-resolution video done in MOV
format of a multi-laser projection in motion.
The video flat panel display will be
between 20” and 24” wide and 16” high.
The final version of the
high-resolution video was completed in February of 2009.

In this multi-laser projection of a hypotrochoid curve, each laser projects the same curve but in a different phase. In laser projections, where a small dot of light moves rapidly through a complete pattern, the math differs from traditional spirograph or roulette equations. Traditional equations are modified to accommodate the “dynamic” nature of having base and trace circles rotate simultaneously. The usual parametric equation for graphing a hypotrochoid curve is:

Hence, where base and trace oscillators form images, and the base frequency is the number of times per second that the base oscillator completes a cycle, the revised equation is:

A Quick
Time video of this image can be viewed at

Merrill Lessley, Professor of Theatre
and Paul Beale, Professor of Physics.

Merrill Lessley,Theatre Department, University of Colorado, Boulder.Colorado, USA.

Paul Beale, Physics Department, University of Colorado, Colorado, USA..

"We create laser images in motion that represent specific mathematical
curves (epicycloids, hypocycloids, roses, epitrochoids, hypotrochoids,
and other special sine/cosine cases). These image "sequences" are
created via a computer-controlled laser projection system designed by
Professor Merrill Lessley.
Graphing such curves in multiple colors produces a wide variety of
appealing images. Unlike drawing them with a pencil, however, projecting
such curves with lasers presents a particularly challenging problem:
while the laser is often referred to as a kind of "pencil" in light, it
can only generate a seemingly complete picture by moving its projected
"dot" rapidly and repeatedly over a reflective surface. The images we
create are scanned at rates between 15 and 600 times per second.
Scanning is accomplished by reflecting laser beams off of very small and
rapidly moving first-surface mirrors via precision galvanometers whose
movements are controlled through a combination of math, software, and
custom designed hardware."