## 3 Test and Verification
### Calculation of Test Results
#### 3.2.2 Calculation of Thermal Condition Magnification  
The temperature gradient required by the indicator is 0.2 \({}^{\circ}\)C in the row direction and 0.5 \({}^{\circ}\)C in the column direction. According to the uniform distribution of the actual temperature in the 6 \(\times\) 3 matrix columns, the temperature gradient matrix in the row direction is a matrix of 6 \(\times\) 2, as shown in Equation (20). The temperature gradient matrix in the column direction is a matrix of 5 \(\times\) 6, as shown in Equation (21). \[\Delta T_{row}= \begin{bmatrix}a_{11}&a_{12}\\ a_{21}&a_{22}\\ \vdots&\vdots\\ a_{12,1}&a_{12,2}\\ \end{bmatrix} = \begin{bmatrix}0.1&0.1\\ 0.1&0.1\\ \vdots&\vdots\\ 0.1&0.1\\ \end{bmatrix}  \tag{20}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Displacement Direction** & \(\mathbf{X_{1}}\) & \(\mathbf{Y_{1}}\) \\ \hline Average displacement value (mm) & 1.25 & 0.83 \\ \hline Average drift error (mm) & 0.003778 & \(-\)0.01033 \\ \hline Absolute displacement value (mm) & 1.24622 & 0.8403 \\ \hline Average temperature change value \(\Delta T\) (\({}^{\circ}\)C) & 10.4633 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Calculation of spot displacement and temperature rise. Figure 23: Coordinate System Conversion. \[\Delta T_{col}= \begin{bmatrix}0.1&0.1&0.1&0.1&0.1&0.1\\ 0.1&0.1&0.1&0.1&0.1\\ 0.1&0.1&0.1&0.1&0.1\\ 0.1&0.1&0.1&0.1&0.1\\ 0.1&0.1&0.1&0.1&0.1\\ 0.1&0.1&0.1&0.1&0.1\\ \end{bmatrix}  \tag{21}\]  
To calculate the temperature gradient comparison matrix in the row direction, we merge the two temperature matrices in Figure 19 into a temperature matrix with 12 rows and 3 columns:  
\[T^{\prime}_{row}= \begin{bmatrix}T^{\prime}_{front}\\ \hline T^{\prime}_{back}\\ \end{bmatrix} = \begin{bmatrix}31.46&30.17&32.32\\ 30.72&31.53&30.06\\ 30.72&30.06&30.21\\ 30.39&30.28&30.85\\ 31.93&30.21&30.08\\ 31.36&30.45&31.25\\ \end{bmatrix} \]  
We calculate the temperature gradient matrix \(\Delta T_{col}^{\prime}\) in the row direction:  
\[\Delta T^{\prime}_{row}= \begin{bmatrix}\Delta T^{\prime}_{row,front}\\ \hline\Delta T^{\prime}_{row,back}\\ \end{bmatrix} = \begin{bmatrix}1.29&2.15\\ 0.81&1.47\\ 0.21&0.15\\ 1.72&0.13\\ 0.91&0.8\\ 0.27&0.01\\ 0.05\\ 0.12&0.06\\ 0.6&0.51\\ 0.06&0.09\\ \end{bmatrix} \]  
Similarly, in order to obtain the temperature gradient comparison matrix in the column direction, the two temperature matrices in Figure 16 are merged into a temperature matrix of 6 rows and 6 columns:  
\[T^{\prime}_{col}= \begin{bmatrix}T^{\prime}_{front}\\ \hline T^{\prime}_{back}\\ \end{bmatrix} = \begin{bmatrix}31.46&30.17&32.32&29.97&30.87&30.06\\ 30.72&31.53&30.06&30.48&30.21&30.22\\ 30.72&30.06&30.21&30.23&30.23&30.28\\ 30.39&30.28&30.85&30.17&30.05&29.99\\ 31.93&30.21&30.08&30.14&30.74&30.23\\ 31.36&30.45&31.25&30.06&30&30.09\\ \end{bmatrix} \]  
We calculate the temperature gradient matrix \(\Delta T_{col}^{\prime}\) in the column direction:  
\[\Delta T_{col}^{\prime}= \begin{bmatrix}\Delta T^{\prime}_{col,front}\\ \hline\Delta T^{\prime}_{col,back}\\ \end{bmatrix} = \begin{bmatrix}0.74&1.36&2.26&0.51&0.66&0.16\\ 0.45&1.47&0.15&0.25&0.02&0.06\\ 0.12&0.22&0.64&0.06&0.18&0.29\\ 1.54&0.07&0.77&0.03&0.69&0.24\\ 0.57&0.24&1.17&0.08&0.74&0.14\\ \end{bmatrix} \]By introducing Equations (11) and (12), the multiplication matrices for row and column directions are obtained as follows:  
\[N_{row}=\left[ \begin{array}{ccc}12.9&21.5\\ 8.1&14.7\\ 2.1&1.5\\ 9&8.1\\ 2.7&0.1\\ 0&0.5\\ \hline 1.1&5.7\\ 17.2&1.3\\ 9.1&8\\ 1.2&0.6\\ 6&5.1\\ 0.6&0.9\\ \end{array} \right] \tag{22}\]  
\[N_{col}=\left[ \begin{array}{ccccc}7.4&13.6&22.6\\ 4.5&14.7&1.5\\ 1.2&2.2&6.4\\ 15.4&0.7&7.7\\ 5.7&2.4&11.7\\ \end{array}