## 3 Results  
Once the model for calculating the distances \(d_{k}\) and for estimating its associated uncertainties was defined, the measurement of the glass line scale was performed in a position approximately parallel to the \(X\) axis of the machine.  
The measurement was performed with the optical equipment Ultra-QV350, whose characteristics have already been defined in Section 2.1. The glass line scale images were obtained by using transmitted illumination at 20% of its nominal value with 10\(\times\) magnification. The glass line scale temperature during measurement remained in the range 20 \({}^{\circ}\)C \(\pm\) 0.1 \({}^{\circ}\)C. Thus, no temperature correction is applied to the distances \(d_{k}\).  
Table 3 shows the results obtained by measuring the glass line scale considering the Hypothesis 1. Figure 3a,b show the histograms of the distances \(d_{2}\) and \(d_{4}\). These results obtained by developing the Hypotheses 2 and 3 are shown in Table 4, Figure 4a,b and Table 5, Figure 5a,b, respectively. Analyzing the results of the above tables, it is observed that variations within a single estimate of the measurand \(d_{k}\) obtained in the various hypotheses are between 1 nm and 22 nm, values that can be considered negligible considering that the nominal size pixel in the measurement conditions used was 0.98 \(\upmu\)m.  
By performing the same analysis for the standard uncertainty of the distances \(d_{k}\), it was found that it varied between 26 nm and 74 nm. These variations were greater as distance \(d_{k}\) increased. When the results of Hypotheses 1 and 3 were compared, it was determined that its variation was between 1 nm and 30 nm. As in the previous case, these variations can be considered negligible.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Distances**} & \multirow{2}{*}{\begin{tabular}{c} **Estimate \(d_{k}\)** \\ **(mm)** \\ \end{tabular} } & \begin{tabular}{c} **Deviations to** \\ **the Reference** \\ **Value (nm)** \\ \end{tabular} & \begin{tabular}{c} **Standard** \\ **Uncertainty** \\ **_u(d\({}_{k}\))_ **(mm)** \\ \end{tabular} & \
\begin{tabular}{c} **Shortest 95\% Coverage Interval** \\ **Lower Limit** \\ **(mm)** \\ \end{tabular} &
\begin{tabular}{c} **Upper Limit** \\ **(mm)** \\ \end{tabular} \\ \hline \(d_{1}\) & 0.10002 & 7 & 0.00023 & 0.09960 & 0.10043 \\ \(d_{2}\) & 0.20006 & 4 & 0.00043 & 0.19921 & 0.20082 \\ \(d_{3}\) & 0.30009 & \(-\)75 & 0.00044 & 0.29935 & 0.30082 \\ \(d_{4}\) & 0.40004 & \(-\)49 & 0.00058 & 0.39903 & 0.40108 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Measurement results of the glass line scale in horizontal position. Hypothesis 2.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Distances**} & \multirow{2}{*}{\begin{tabular}{c} **Estimate \(d_{k}\)** \\ **(mm)** \\ \end{tabular} } & \begin{tabular}{c} **Deviations to** \\ **the Reference** \\ **Value (nm)** \\ \end{tabular} & \begin{tabular}{c} **Standard** \\ **Uncertainty** \\ **_u(d\({}_{k}\))_ **(mm)** \\ \end{tabular} & \
\begin{tabular}{c} **Shortest 95\% Coverage Interval** \\ **Lower Limit** \\ **(mm)** \\ \end{tabular} &
\begin{tabular}{c} **Upper Limit** \\ **(mm)** \\ \end{tabular} \\ \hline \(d_{1}\) & 0.10003 & 2 & 0.00020 & 0.09961 & 0.10047 \\ \(d_{2}\) & 0.20006 & \(-\)4 & 0.00038 & 0.19930 & 0.20094 \\ \(d_{3}\) & 0.30009 & \(-\)76 & 0.00037 & 0.29930 & 0.30083 \\ \(d_{4}\) & 0.40002 & \(-\)