## 3 Results and Discussion
The average error after semi-finishing machining was 15.0 \(\upmu\)m with a standard deviation of 6.0 \(\upmu\)m. After error compensation was performed, the average error was 0.7 \(\upmu\)m, and the standard deviation was 2.0 \(\upmu\)m. The average error was improved by 14.3 \(\upmu\)m, and the standard deviation was improved by 4.0 \(\upmu\)m, which demonstrated that the proposed error compensation method could significantly improve the surface variation. Similar results were obtained for the other three replicas of Surface one. Similar to the experiments on Surface one, we also experimented with Surface two and Surface three. Figures 11 and 12 show the error distributions for one of the replicas before and after compensation on Surfaces two and three, respectively. Figure 11: (**a**) The error distribution of replica one before (blue circles) and after (red plus signs) tool radius compensation for Surface two; (**b**) the error distribution of replica one before (blue circles) and after (red plus signs) mirror compensation for Surface two. Figure 10: (**a**) The error distribution of replica one before (blue circles) and after (red plus signs) tool radius compensation for Surface one; (**b**) the error distribution of replica one before (blue circles) and after (red plus signs) mirror compensation for Surface one. The experimental results are summarized in Table 3. As mentioned previously, Surfaces one, two, and three had 70, 49, and 68 measurement points for each replica, respectively, and each surface design had four replicas, so the number of errors for Surface one, two, and three was 280, 196, and 272, respectively. We used the errors to calculate the mean and standard deviation of the errors for each surface design, and the results are listed in Table 3. For all the three different surfaces, it can be seen that both compensation methods can greatly improve the mean errors. The mean errors can be reduced to be within 2.0 \(\upmu\)m or less for both methods. Regarding the standard deviations for Surfaces one, two, and three, the standard deviations after using mirror compensation improved by 3.8 \(\upmu\)m, 2.3 \(\upmu\)m, and 1.0 \(\upmu\)m, which was equivalent to 61%, 61%, and 32%. The standard deviation can only improve by 0.1 \(\upmu\)m by using the tool radius compensation for all three surfaces. It is evident that to improve the standard deviation of the error, the mirror compensation method proposed in the paper is more effective compared to the tool radius compensation method. Regarding the different types of surface, Surface one is the least complicated surface, and Surface three is the most complicated one, as discussed previously. Intuitively, we expected Surface three to have the largest standard deviation of the errors and Surface one the smallest before error compensation. However, the actual standard deviations of errors were not as expected. Surface one had the largest variation. The reason for this phenomenon is unknown. We also used the surface profile per ISO standard 1101 as a measure for evaluating the two compensation methods, and the results are summarized in Table 4. For Surfaces one, two, and three, the surface profile error after using tool radius compensation could be improved by 24.8 \(\upmu\)m, 32.3 \(\upmu\)m, and 30.5 \(\upmu\)m, which were equivalent to 45%, 62%, and 56%, respectively. However, using mirror compensation could improve the surface profile error by 44.8 \(\upmu\)m, 44.4 \(\upmu\)m, and 41.2 \(\upmu\)m, which were equivalent to 78%, 84%, and 77%, respectively. For Surfaces one, two, and three, the mirror compensation method outperformed the tool radius compensation method by 33%, 22%, and 21%, respectively. It is obvious that for improving the surface profile error, the mirror compensation method proposed in the paper is also more effective than the tool radius compensation method.