## 2 Analysis and Optimization of TIF Shape
### Modeling and Optimization of TIF  
The single-rotation small tool polishing device, as shown in Figure 1, is the earliest polishing tool applied in the computer-controlled optical surfacing technology. It has the advantages of a simple structural form, easy implementation, and a large polishing contact area. It is still commonly used in optical mirror polishing nowadays. However, the TIF of the single-rotation small tool polishing device is annular, deviating from the ideal Gaussian shape. After polishing, it is prone to leaving medium-frequency errors, resulting in lower polishing accuracy. According to the Preston equation, the velocity distribution and pressure distribution of the polishing tool are the main factors that determine the form of the TIF. The distribution of the contact pressure is related to the way the polishing pressure is applied. For a single-rotating small tool polishing device, its polishing pressure is applied to the polishing disk at a single point through the pressure output shaft. Meanwhile, the pressure output shaft is also the driving shaft for the rotation movement of the polishing disk, meaning that the pressure application point coincides with the center of the rotation movement. According to the eccentric compression theory [19], the contact pressure distributions generated by different positions of the pressure application point are different. On this basis, the contact pressure distribution can be changed by altering the position of the polishing pressure application point. However, the size of the object discussed in the eccentric compression theory is much larger than that of the polishing disk and is not very precise. To more accurately describe the influence of the position of the pressure application point on the contact pressure distribution, this paper opts to establish a contact pressure analysis model based on the finite element method. We utilized the ABAQUS software version 2020 to model the contact pressure. Subsequently, the Matlab software version 2020b was employed to post-process the calculation results from the ABAQUS software. With Matlab, we completed the modeling and analysis of the removal function. The modeling parameters are shown in Table 1. The settings of the boundary conditions in the model are as follows. We fixed all six degrees of freedom of the workpiece and established a contact relationship between its upper surface and the lower surface of the asphalt layer. A coupling constraint was established between the upper surface of the Pitch layer and the lower surface of the Metal disks. We applied a pressure of 20N on the pressure output shaft and established a contact relationship between the lower endpoint of the pressure output shaft and the upper surface of the Metal disks. The established finite element analysis model is shown in Figure 2.  
Figure 1: The single-rotation small tool polishing device and its TIF. (**a**) Schematic diagram of the single-rotation small tool polishing device. (**b**) Actual processing situation of the single-rotation small tool polishing device. (**c**) TIF of the single-rotation small tool polishing device. As shown in Figure 2, in order to explore the influence of the pressure application point on the pressure distribution, we first established a finite element analysis model in which the pressure application point coincides with the selected center of the polishing disk. On this basis, we moved the pressure output shaft to deviate from the center of the polishing disk (the center of the polishing rotation movement), defined the deviation distance as E, and defined the diameter of polishing disk as D. We analyzed the contact pressure distribution when the pressure application point coincides with the center of the polishing disk (E = 0 mm), and also analyzed the contact pressure distribution when the pressure application point is separated from the center of the polishing disk (E \(\
eq\) 0 mm).