## 3 Results and Discussion  
This section presents the results of the automatic error compensation method proposed in this study. The number of measurement points of each replica for the three surface designs was different. Surfaces one, two, and three had 70, 49, and 68 measurement points for each replica, respectively. We used two different error compensation methods, tool radius modification and the mirror method as discussed previously, to carry out the error compensation and find their results. Because we machined four replicas for each surface design, as shown in Figures 2-4, we would like to examine the repeatability of the four replicas first. Table 2 shows the standard deviations of the errors of the four replicas for each surface design. The values without the parentheses represent the standard deviations for the replicas before and after error compensation, and the values within the parentheses represent the range (the difference between the lowest and highest values) for the standard deviation of the errors for the four replicas. As listed in Table 2, most of the ranges are less than 0.5 \(\upmu\)m, and the maximum range is 0.8 \(\upmu\)m, which comes from the measurement results of Surface three after mirror compensation. Except for this case, all the other results show consistent standard deviations among the four replicas for the three surfaces before and after compensation, regardless which compensation method we used. Because the repeatability of the four replicas is reasonably good in general, we will show the error distribution of one of the replicas (replica one) before and after error compensation by using either the tool radius compensation method or the mirror compensation method to illustrate the error distribution for all four replicas. The reason we do not show the distribution for all four replicas is that we would like to focus on the comparison between the two compensation methods without using too much space of the paper. After performing the semi-finishing machining on the four replicas of Surface one, we obtained the error distribution of the free-form surface of replica one, as the circles shown in Figure 10a. The average error after semi-finishing machining was 12.3 \(\upmu\)m with a standard deviation of 6.0 \(\upmu\)m. After the tool radius compensation was performed, the error distribution became the plus signs in Figure 10a. The average error was reduced to 3.1 \(\upmu\)m, but the standard deviation became 6.2 \(\upmu\)m, which was \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Tool Radius Compensation**} & \multicolumn{2}{c}{**Mirror Compensation**} \\ \hline
**Surface Design** & **Before** & **After** & **Before** & **After** \\ \hline Surface one & 6.0, 6.1, 6.3, 6.1 (0.3) & 6.2, 6.4, 5.8, 6.2 (0.6) & 6.0, 6.1, 6.3, 6.3 (0.3) & 2.0, 1.8, 2.0, 1.9 (0.2) \\ Surface two & 3.8, 3.7, 3.3, 3.8 (0.5) & 3.4, 3.4, 3.7, 3.3 (0.4) & 3.8, 3.4, 3.8, 3.9 (0.5) & 1.1, 1.5, 1.3, 1.2 (0.4) \\ Surface three & 2.9, 2.9, 3.0, 3.0 (0.1) & 3.2, 3.2, 3.2, 3.0 (0.2) & 3.0, 3.0, 2.9, 3.0 (0.1) & 1.4, 2.2, 2.0, 2.0 (0.8) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Standard deviations of the errors of the four replicas for the three surfaces by using tool radius compensation or mirror compensation. Each of the values within the parentheses represents the range of the four values above it. Unit: \(\upmu\)m.  
almost unchanged. It is evident that the average error was much improved, but with no significant improvement in the variation. On the other hand, we repeated the previous experiment by using the mirror compensation method proposed in this study. The error distribution of replica one before and after the mirror compensation is shown in Figure 10b. Because the error compensation was already performed on the free-form surfaces used in the previous experiment, we had to machine new surfaces for this experiment.