Abstract: Starting from the classical Lissajous figures formed by the simple harmonic vibrations in the transverse and longitudinal directions under the classical Cartesian coordinate system, we study the Lissajous figures formed by the vibrations in the radial and angular directions under the polar coordinates. The features of the Lissajous figures are analyzed when the equilibrium point is located at the origin of polar coordinates and the equilibrium point moves in a circular motion. These results are further extended to the periodic orbits formed by the bound motion of particles in the background of black holes. This further enriches and expands our knowledge and understanding of the Lissajous figures.

Key words: Lissajous figure, periodic orbit, black hole