## 3 Results and Discussion
### Prototype Development System
#### 3.1.2 Test for Virtual Measurement System Repeatability and Accuracy  
To test the stability of the measurement system, the measurements were repeated thrice with the unloading test. During the unloading test, the sample is not removed or repositioned. The tooth height and thickness of 16 teeth were measured, as shown in Tables 1 and 2. To easily compare the three sets of results, we subtracted the second and third measurement results from the first measurement results, to obtain the repeatability measurement error. The maximum repeatability measurement error is only 0.004 mm for the tooth height (Table 1), indicating that the measurements were accurate and repeatable. As seen in Table 2, the maximum repeatability error is \(-0.129\) mm for the repeatability measurement results for tooth thickness, and this, too, was an outlier. The tooth thickness error is the sum of the signal \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{6}{c}{**Measurement System Repeatability Study**} \\ \hline \multicolumn{2}{c}{**(Unit: mm)**} & \multicolumn{2}{c}{**Tooth Height**} & \multicolumn{2}{c}{**Repeatability Error**} \\ \hline
**No. of Tooth** & **Test 1** & **Test 2** & **Test 3** & **2 nd-1 st** & **3 rd-1 st** \\ \hline T1 & 3.059654 & 3.058098 & 3.056252 & \(-0.002\) & \(-0.003\) \\ T2 & 2.992897 & 2.99575 & 2.995895 & 0.003 & 0.003 \\ T3 & 3.238052 & 3.23745 & 3.241447 & \(-0.001\) & 0.003 \\ T4 & 3.220901 & 3.22065 & 3.219147 & 0.000 & \(-0.002\) \\ T5 & 3.212646 & 3.210396 & 3.210106 & \(-0.002\) & \(-0.003\) \\ T6 & 3.184547 & 3.185455 & 3.185547 & 0.001 & 0.001 \\ T7 & 3.165848 & 3.166351 & 3.165047 & 0.001 & \(-0.001\) \\ T8 & 3.200005 & 3.2015 & 3.201103 & 0.001 & 0.001 \\ T9 & 3.181755 & 3.181557 & 3.179451 & 0.000 & \(-0.002\) \\ T10 & 3.164795 & 3.163551 & 3.1604 & \(-0.001\) & \(-0.004\) \\ T11 & 3.150452 & 3.151344 & 3.150154 & 0.001 & 0.000 \\ T12 & 3.212845 & 3.216354 & 3.216148 & 0.004 & 0.003 \\ T13 & 3.133095 & 3.129196 & 3.131699 & \(-0.004\) & \(-0.001\) \\ T14 & 3.108749 & 3.108849 & 3.108704 & 0.000 & 0.000 \\ T15 & 3.142197 & 3.140297 & 3.14225 & \(-0.002\) & 0.000 \\ T16 & 3.154251 & 3.151299 & 3.152298 & \(-0.003\) & \(-0.002\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Measurement system repeatability test on tooth height (measurements were repeated thrice). Figure 8: (a) Calculation result of ring gauge eccentricity curve. (b) Eccentric correction results. noise error and sampling error. The signal noise error is the high-frequency noise in measurement, and the sampling error is determined from the sampling ratio. For the prototype system, the total number of CCD scan data points was 48,709, and thus, the angle of interval between each data point was 0.00739\({}^{\circ}\). The ring gauge diameter was 192.68 mm, and the sampling interval (0.012 mm sampling blind spot size) was obtained by using Equation (12). In Table 3, it can be seen that the percentage of errors of 0 to 0.01 mm in width are as high as 40.63%. Because the thickness of one tooth spans (pass) two tooth steps, in the worst case, the maximum error is twice the sampling interval: 0.024 mm (approximately 0.03 mm). Figure 9 shows the statistical values in each error range and the percentage distribution of the error area in Table 3. Regarding the error ranges, 87.5% of errors are within 0.05 mm, and 96.88% of errors are within 0.07 mm.