# Effect of Wheel Path in Raster Grinding on Surface Accuracy of an Off-Axis Parabolic Mirror
## 4 Discussion
### Analysis of the Factors Affecting the Scallop Height of the Grinding Surface
#### 4.2.3 Effect of Workpiece Curvature Radius on Scallop Height  
The scallop height \(h\) is a function of the workpiece curvature \(R\), and the partial derivative of the workpiece curvature \(R\) occurs on both sides of Equation (6):  
\[\frac{\partial h}{\partial R}=\frac{l^{2}(R-r)}{2R^{2}}(1-\frac{R-r}{R}) \tag{13}\]  
According to Equation (13):  
\[\frac{\partial h}{\partial R}>0 \tag{14}\]  
The scallop height \(h\) is positively correlated with the workpiece curvature \(R\). The simulation calculation is shown in Figure 10. It can be seen that the greater the curvature \(R\) of the workpiece, the greater the scallop height. The dashed line is the asymptote. Therefore, in the grinding process, to obtain a lower scallop height, the processing step is selected in the direction of the small radius of curvature of the workpiece.  
The characteristics of the parabolic surface are that the curvature in the meridian direction is different from that in the arc vector direction. The radius \(R_{1}\) of curvature in the meridian direction can be described by the following equation on the parabolic generatrix. The radius \(R_{2}\) of curvature in the arc vector direction is the same as the off-axis amount of the point.  
Figure 10: Effect of workpiece curvature \(R\) on scallop height.  
Figure 9: Effect of step pitch \(l\) on scallop height.  
The parabolic generatrix (XOZ plane):  
\[z^{2}=\frac{2}{c}\cdot x \tag{15}\]  
The radius of curvature in the meridian direction \(R_{1}\):  
\[R_{1}=\left|\frac{\left[1+\left(z^{\prime}\right)^{2}\right]^{\frac{3}{2}}}{z^{ \prime\prime}}\right|=\sqrt{c^{4}z^{6}+3c^{2}z^{4}+3z^{2}+\frac{1}{c^{2}}} \tag{16}\]  
The radius of curvature in the arc vector direction \(R_{2}\):  
\[R_{2}=|z| \tag{17}\]  
Comparing the radius of curvature in the meridian direction and the arc vector direction of the off-axis parabolic surface yields the following:  
\[\frac{R_{1}}{R_{2}}=\sqrt{c^{4}z^{4}+3c^{2}z^{2}+3+\frac{1}{c^{2}z^{2}}}>1 \tag{18}\]  
The radius of curvature of the same point in the off-axis parabolic surface in the meridian direction is always greater than the radius of curvature in the arc vector direction.  
According to the scallop height model, the scallop height at the two center lines of the off-axis parabolic surface was calculated. The calculated results were compared with the experimental results, as shown in Figure 11. The measured scallop height along the meridian direction is about 0.0964 mm, and the theoretical scallop height is about 0.0928 mm. The measured scallop height along the arc vector direction is about 0.0933 mm, and the theoretical scallop height is about 0.0892 mm. The error between the simulation value and the actual value is less than 5%. The accuracy of the scallop height model is verified.  
Combined with Equation (18), the curvature radius of the off-axis parabola in the meridian direction is always greater than that in the arc vector direction, and combined with Equation (14), it can be seen that the larger the curvature radius of the workpiece, the larger the scallop height. Therefore, the scallop height of the raster path along the arc vector direction is smaller than that of the raster path along the meridian direction.  
Figure 11: Comparison of actual and theoretical scallop height values.