# 2D Triangulation of Signals Source by Pole-Polar Geometric Models
## 3 Experimental Cases
### Results and Analyses
The lower this value, the less sensitive is the method to the errors in the data and therefore has the ability to produce more accurate results. A simple analysis of the data in Table 3 shows that the proposed geometric models are more robust when they operate on data with errors than the traditional WLSm, NRm, and LSm methods. Among the geometric methods, the TLI has greater capacity to process data with errors. In order to analyze these results in a particular context, Figure 15 prove that the fourth experiment is the worst case of acquired signal data, since it promotes bigger errors, both for the numerical \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Methods**} & \multicolumn{8}{c}{**Magnitudes Errors (in Meters)**} \\ \cline{2-9}  & \multicolumn{3}{c}{**Minimum Error**} & \multicolumn{3}{c}{**Maximum Error**} & \multicolumn{3}{c}{**Mean Error**} \\ \hline  & **x-axis** & **y-axis** & **Distance** & **x-axis** & **y-axis** & **Distance** & **x-axis** & **y-axis** & **Distance** \\ \hline PPC * & 0.4 & 0.2 & 2.7 & 15.5 & 12.5 & 15.7 & 4.2 & 4.4 & 6.9 \\ CHC * & 0.1 & 0.1 & 0.6 & 11.9 & 6.0 & 12.8 & 2.5 & 2.4 & 3.7 \\ PLI * & 0.1 & 0.1 & 0.7 & 12.8 & 15.9 & 16.3 & 2.4 & 3.8 & 5.0 \\ MAI * & 1.6 & 0.3 & 1.7 & 7.9 & 5.3 & 8.9 & 4.2 & 1.5 & 4.7 \\ TLI * & 2.3 & 0.3 & 2.3 & 8.3 & 3.9 & 9.2 & 4.1 & 1.3 & 4.4 \\ NRm & 0.1 & 1.1 & 1.3 & 14.7 & 12.2 & 15.2 & 2.8 & 5.1 & 6.5 \\ LSm & 0.3 & 1.2 & 1.8 & 14.1 & 22.5 & 23.7 & 3.6 & 6.4 & 7.9 \\ WLSm & 0.1 & 0.7 & 1.3 & 13.6 & 15.3 & 17.7 & 3.5 & 4.9 & 6.5 \\ \hline \hline \end{tabular}
* Geometric Models proposed in this work.  
\end{table} 
Table 1: Magnitude of errors in the experimental cases. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Methods**} & \multicolumn{2}{c}{**Standard Deviation of the Errors**} & \multicolumn{2}{c}{**Effective Variabilityof the Errors**} \\ \cline{2-5}  & \(x\)-axis & **y-axis** & **Distance** & **Effective Variabilityof the Errors** \\ \hline PPC * & 4.1 & 3.8 & 4.5 & 4.1 \\ CHC * & 3.1 & 2.1 & 3.4 & 2.9 \\ PLI * & 3.3 & 4.1 & 4.9 & 4.1 \\ MAI * & 2.5 & 1.6 & 2.6 & 2.2 \\ TLI * & 2.3 & 1.0 & 2.3 & 1.9 \\ NEM & 4.0 & 4.0 & 4.9 & 4.3 \\ LSm & 4.3 & 6.7 & 7.4 & 6.1 \\ WLSm & 4.2 & 4.3 & 5.5 & 4.7 \\ \hline \hline \end{tabular}
* Geometric Models proposed in this work.  
\end{table} 
Table 3: Standard deviation of the errors. methods and for the geometric models (Figure 17a). An intermediate result is achieved by the eleventh (Figure 17b) experiment and the best result is obtained in tenth (Figure 17c). Figure 17c shows that when satisfactory data make all methods and models works with relative precision, whereas data incoherencies (dotted circle lines in Figure 17) make results divert of the expected solution, and Figure 17a-c confirms that the proposed geometric models point to a location closer to the exact position of the signal source than the numerical methods used and can better solve the existing problem of data inaccuracy. The worst case, Figure 17a, deals with the case of inconsistency of acquired data. Even, if the use of such data does not show the precise location of the emitter, the results produced by the geometric models are consistent with the geometric arrangement among data. The solution is found on the region where most of signals range (circles) intersects. The solution is inaccurate because of the serious  
Figure 17: Some particular experimental results. (**a**) The worst case. (**b**) An intermediate case. (**c**) The best case. The blue-dotted circle lines show the receivers/emitters that suffered severe interferences and promoted the acquisition of data with errors. deviations in the acquired data. If data inconsistency is moderate (Figure 17b), the proposed geometric models generate higher quality results than those produced by numerical methods. The Appendix A presents six other experimental results.