Angular resolution and scanning density estimation based on a 3D to 2D Lissajous transition

Angular resolution and scanning density are important factors affecting the quality of Lissajous images. In this work, the transition characteristics from 3D to 2D Lissajous scanning curves and the relationship between 3D Lissajous scanning rotation and phase change were studied. The concept of the characteristic phase difference between 2D and 3D Lissajous scanning trajectories is introduced, and an extension method for multidimensional Lissajous scanning characteristic phase difference is established. On this basis, three key rules for the characteristic trajectories of 3D Lissajous scanning are defined. Additionally, the special patterns and phenomena of 2D Lissajous scanning from the perspective of 3D Lissajous scanning are explored, such as the occurrence of the "non-closed endpoints". On this basis, the quadrant patterns of 2D Lissajous scanning with the "non-closed endpoints" are derived. A method for calculating the scanning density for the number of 3D Lissajous scanning intersection points was proposed. Then, the double-junction circle calculation method for the angular resolution of the center region is deduced. The characteristics of the curves were analyzed for specific frequency ratios (9:7, 13:10, and 17:13). Finally, the scanning density and properties of 3D Lissajous scanning trajectories were experimentally verified. The results indicate trajectory densities of 110, 234, and 412 for frequency ratios 9:7, 13:10, and 17:13. The angular resolution under computed a frequency ratio of 17:13 was <i>ar </i>= 0,0089. The angular resolution was calculated with a computational error of <i>E </i>= 0.0001 and an accuracy of <i>ε</i>≤0.009, enabling high-resolution and high-density Lissajous scanning.