# Effect of Wheel Path in Raster Grinding on Surface Accuracy of an Off-Axis Parabolic Mirror
## 4 Discussion
### The Correlation between Scallop Height and Surface Shape Error  
The unremoved area of the scallop height affects the accuracy of the grinding sur-face. The larger the scallop height, the larger the surface error. Figure 12 shows the spatial correspondence between the scallop height and the surface shape error, the yellow line represents the projection of the line connecting the scallop height vertex and the bottom point on the XOZ plane and the YOZ plane respectively, thus forming the angle \(\alpha\) and \(\beta\) with the coordinate axis, and the black dashed line represents the line parallel to the coordinate axis. The surface shape error is the difference between the actual profile and the theoretical profile in the vertical direction in a cross-section of the parallel stepping direction. The surface error values of each point on the entire off-axis parabolic surface are different. Therefore, in order to express the surface accuracy of the entire off-axis parabolic sur-face, the surface error values of each point are integrated to obtain the distribution law of the surface error of the off-axis parabolic surface. The solution process of the surface shape error stepping along the meridian direction:  
When stepping along the meridian direction, the feed direction is in the arc vector direction, and the same wheel path is on the arc with the same radius of curvature, so the scallop height is equal in the feed direction. After determining the coordinates of the wheel contact point (\(x_{c}\), \(y_{c}\), \(z_{c}\)), the radius of curvature \(R\) of the point in the stepping direction can  
Figure 12: Diagram of correlation between scallop height and surface shape error: (**a**) step along the meridian direction; (**b**) step along the arc vector direction. \[y_{t}=y_{d}+H\cos\beta \tag{27}\]  
The grinding wheel is continuously fed. Therefore, it is considered that the \(x\) coordinate of the scallop height's bottom point generated by the wheel path is continuous in the \(X\) direction of the surface, and its relationship with the \(x\) coordinate of the scallop height vertex is:  
\[x_{t}=x_{d}+h\cos\alpha \tag{28}\]  
The solution of the scallop height vertex \(z\) coordinate \(z_{t}\) is shown in Equation (24). The calculation of the scallop height vertex coordinates generated by the two paths is completed by Equations (19) to (28), and the obtained coordinates are substituted into the theoretical surface equation (Equation (2)). The theoretical \(z\) coordinates can be obtained from the following formula. \[z=F(x_{t},y_{t}) \tag{29}\]  
The solution of the surface shape error:  
\[\sigma=z_{t}-z \tag{30}\]  
Through the above formula, the simulation of the calculated distribution law of the off-axis parabolic surface error was realized and compared with the experimental measurement results. As shown in Figures 13 and 14, it is found that the distribution law of the surface error obtained by the simulation calculation is consistent with the distribution law of the surface error obtained in the experiment, which proves the correctness of the model. The scallop height distribution along the meridian direction is shown in Figure 15. The blue part represents the scallop height, and the red points represent the vertex and bottom point of the scallop height. The difference between the scallop height and the theoretical  
Figure 14: Diagram of the surface error distribution of the raster path stepping along the arc vector direction: (**a**) measurement; (**b**) simulation. Figure 13: Diagram of the surface error distribution of the raster path stepping along the meridian direction: (**a**) measurement; (**b**) simulation.