# General Total Least Squares Theory for Geodetic Coordinate Transformations
## 5 Numerical Examples
The results indicate that the translation parameters differ significantly among the distinct transformation models, whereas the estimated elements within the transformation matrix change slightly, which matches the precision \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Point No.** & \(\mathbf{x_{s}}\) & \(\mathbf{y_{s}}\) & \(\mathbf{z_{s}}\) & \(\mathbf{x_{t}}\) & \(\mathbf{y_{t}}\) & \(\mathbf{z_{t}}\) \\ \hline
80,601 & 5,234,251.25 & 905,003.2011 & 3,518,869.674 & 5,233,991.482 & 905,003.106 & 3,519,305.459 \\
32,127 & 5,218,851.932 & 919,148.9749 & 3,537,928.348 & 5,218,595.021 & 919,152.324 & 3,538,363.627 \\
80,600 & 5,220,818.669 & 772,128.3613 & 3,569,828.606 & 5,220,565.466 & 772,130.563 & 3,570,253.01 \\
32,136 & 5,148,067.252 & 803,912.306 & 3,668,491.426 & 5,147,806.722 & 803,921.322 & 3,668,928.371 \\
80,598 & 5,081,676.23 & 771,786.812 & 3,765,023.787 & 5,081,410.788 & 771,799.426 & 3765460.689 \\
80,597 & 5,022,479.06 & 955,283.5487 & 3,801,754.143 & 5,022,218.176 & 955,297.254 & 3,802,185.975 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The common points of the 3D transformation. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**Parameter**} \\ \cline{2-5}  & **Affine** & **Orthogonal** & **Similarity** & **Rigid** \\ \hline \(\hat{\xi}_{11}\) & 0.99902905 & 0.99902817 & 0.99900748 & 0.99915487 \\ \(\hat{\xi}_{12}\) & 0.04111867 & 0.04109721 & 0.04109806 & 0.04110413 \\ \(\hat{\xi}_{21}\) & \(-0.04107747\) & \(-0.04109892\) & \(-0.04109806\) & \(-0.04110413\) \\ \(\hat{\xi}_{22}\) & 0.99898590 & 0.99898678 & 0.99900748 & 0.99915487 \\ \(\Delta\hat{\kappa}\) & \(-141.26879\) & \(-141.26546\) & \(-141.26279\) & \(-141.28363\) \\ \(\Delta\hat{\gamma}\) & \(-143.93120\) & \(-143.92843\) & \(-143.93164\) & \(-143.95288\) \\
**Objective** & 0.00061868 & 0.00063141 & 0.00064325 & 0.00124379 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Estimates of the 2D transformation parameters. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**Standard Deviation**} \\ \cline{2-5}  & **Affine** & **Orthogonal** & **Similarity** & **Rigid** \\ \hline \(\hat{\xi}_{11}\) & \(1.4969\times 10^{-4}\) & \(1.2342\times 10^{-4}\) & \(7.6328\times 10^{-5}\) & \(3.9027\times 10^{-6}\) \\ \(\hat{\xi}_{12}\) & \(1.4974\times 10^{-4}\) & \(8.7393\times 10^{-5}\) & \(7.6328\times 10^{-5}\) & \(9.4866\times 10^{-5}\) \\ \(\hat{\xi}_{21}\) & \(1.4969\times 10^{-4}\) & \(8.7397\times 10^{-5}\) & \(7.6328\times 10^{-5}\) & \(9.4866\times 10^{-5}\) \\ \(\hat{\xi}_{22}\) & \(1.4974\times 10^{-4}\) & \(1.2346\times 10^{-4}\) & \(7.6328\times 10^{-5}\) & \(3.9027\times 10^{-6}\) \\ \(\Delta\hat{\kappa}\) & \(3.2661\times 10^{-2}\) & \(2.3286\times 10^{-2}\) & \(1.7817\times 10^{-2}\) & \(1.7641\times 10^{-2}\) \\ \(\Delta\hat{\gamma}\) & \(3.2661\times 10^{-2}\) & \(2.4474\times 10^{-2}\) & \(1.7817\times 10^{-2}\) & \(1.7445\times 10^{-2}\) \\ \(\delta_{0}\) & \(0.017588\) & \(0.014508\) & \(0.012681\) & \(0.015772\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Precision evaluation of the 2D transformation parameters. analysis in Table 9. The other analysis results are almost the same as the 2D results in that the values of the objective function become larger with more constraints, the precision is higher with more constraints, and the variation of the estimates \(\hat{\sigma}_{0}\) has no rule. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{**Shift Parameters**} & \multicolumn{3}{c}{**The Transformation Matrix**} \\ \hline \(\Delta\hat{x}\) & \(-1956.3996\) & \(1.000224798\) & \(0.000041651\) & \(0.000137955\) \\ \(\Delta\hat{y}\) & \(168.5691\) & \(-0.000041663\) & \(0.999993142\) & \(0.000016147\) \\ \(\Delta\hat{z}\) & \(1495.9485\) & \(-0.000137998\) & \(-0.000016154\) & \(0.999907421\) \\ \hline \multicolumn{2}{c}{**Objective Function**} & \multicolumn{3}{c}{85.6586} \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Estimates of the 3D orthogonal transformation parameters.