Further Understanding for Lissajous Figures

At the undergraduate level, the composite motion of simple harmonic vibrations has always been the main content of physics as well as several other scientific disciplines. Many textbooks tell us clearly that when the frequency ratio ω1/ω2 of two perpendicular vibrations is simple integer ratio n1/n2 (n1 and n2 are integers), their combined motion has stable and closed trajectories. This trajectory graph is the famous Lissajous figure. The Lissajous figure has been widely studied since it was discovered more than 100 years ago. Especially in the last 30 years, the shape, evolution trend, periodicity, and symmetry of the Lissajous figure have been studied and a series of meaningful characteristics and laws have been summarized. However there has been no analysis of the Lissajous figure from the perspective of three-dimensional space curve so far.