# Effect of Wheel Path in Raster Grinding on Surface Accuracy of an Off-Axis Parabolic Mirror
## 4 Discussion
### Modeling Scallop Height of Off-Axis Parabolic Surface  
The influences of the selection of wheel path and of process parameters such as grinding wheel radius, processing step pitch, and workpiece curvature on the scallop height are also greater. Therefore, it was of great significance to build a scallop height model and analyze the influence of wheel path and process parameters on the scallop height in off-axis paraboloid grinding.  
When stepping along the arc vector direction, the radius of curvature at each point of the workpiece on the stepping arc is equal, so the radius of curvature of the cutter contacts points and the radius of curvature of the small arc segment between the adjacent cutter contacts are both R along the wheel path direction.  
When stepping along the meridian direction, it is assumed that the coordinates of the cutter contact points corresponding to the adjacent grinding wheel paths are \(P_{1}(x_{1},y_{1})\) and \(P_{2}(x_{2},y_{2})\), respectively. Because the radius of curvature of each point on the parabola is not equal, the radius of curvature of the two cutter contacts points and the small arc between the two points on the parabola are not equal. However, in the grinding process, the step pitch between the adjacent grinding wheel path is usually very small. In the step direction, the curvature radius of the bottom point of the scallop height is close to that of the wheel contact point (\(R_{1}\approx R\)). To simplify the calculation, the radius of curvature \(R\) of the current wheel contact is used to replace the radius of curvature of the bottom point of the scallop height. On this basis, a generalized model of the scallop height of the two wheel paths was established.  
Figure 7 shows the calculation of the scallop height of the two adjacent wheel paths in the aspheric grinding. \(P_{1}\) and \(P_{2}\) are the two adjacent grinding points on the work surface, \(R\) is the radius of curvature of the bottom point of the scallop height on the workpiece, \(O\) is the center of the curvature radius of the workpiece, \(r\) is the wheel radius, \(D_{1}\) is the  
Figure 6: Comparison between the measured surface and the theoretical surface: (**a**) step along the meridian direction; (**b**) step along the arc vector direction.  
wheel center, \(D_{2}\) is the center of the wheel after the workpiece has gone around once, and \(l\) is the adjacent wheel center distance. The blue solid circle is the position of the wheel during current grinding, and the blue dotted line is the position of the wheel in the previous grinding path.  
The expression of the scallop height \(h\) is calculated as follows:  
\[h=R-\overline{OE}-\overline{EF} \tag{3}\]  
According to the Pythagorean theorem,  
\[\overline{OE}=(R-r)\cos(l/2R) \tag{4}\]  
\[\overline{EF}=\sqrt{r^{2}-\left(R-r\right)^{2}\sin^{2}(l/2R)} \tag{5}\]  
To bring Equations (4) and (5) into Equation (3), the following equation can be used:  
\[h=R-\sqrt{r^{2}-\left(R-r\right)^{2}\sin^{2}(l/2R)}-\left(R-r\right)\cos(l/2R) \tag{6}\]  
Based on the scallop height value calculated by the scallop height model and combined with the wheel attitude (fixed angle \(\alpha\) in the feed direction of the wheel), the value \(H\) of the scallop height in the vertical direction when the wheel is tilted can be obtained.  
\[H=h\sin\alpha=\left[R-\sqrt{r^{2}-\left(R-r\right)^{2}\sin^{2}(l/2R)}-\left( R-r\right)\cos(l/2R)\right]\sin\alpha \tag{7}\]  
According to the established scallop height model, it can be seen that the workpiece curvature, grinding wheel radius, and step pitch all affect the scallop height. Therefore, combined with the scallop height model, they provide a theoretical basis for the selection of wheel path and the determination of process parameters.