Lissajous-like figures with triangular and square waves

Abstract

We show a generalization of the well-known Lissajous figures, changing the two orthogonal simple harmonic oscillations to triangular and square-wave-type oscillations. All figures cross the origin of axes when there is no phase difference between the oscillations. Aside from this common feature, square-wave-type figures show a very different behavior, with discontinuous phase changes and impossibility of using a simple formula to get the frequency ratio between the two component oscillations.

Medindo a velocidade do som utilizando figuras de Lissajous

“…Os primeiros estudos destas curvas foram desenvolvidos por Bowditch e Lissajous durante o Século XIX [7]. Atualmente elas são utilizadas em diversas aplicações, tanto de cunho didático quanto tecnológico, como por exemplo para obter a frequência de um sinal desconhecido, ou determinar a diferença de fase entre dois sinais [8].…”

Using Mathematica software to graph Lissajous figures

“…We can generate Lissajous figures in several ways, here we highlight the mechanical method [29], two-dimensional harmonic oscillations [22] and low-cost assembly [21].…”

“…There are also some generalizations of Lissajous figures, such as in triangular [24], or more commonly in three dimensions [7,26].…”