# 2D Triangulation of Signals Source by Pole-Polar Geometric Models
## 3 Experimental Cases
### Results and Analyses  
Three experimental cases are chosen to show the results produced by the application of the proposed geometric models: (1) one that produces the worst result; (2) one that produces result with intermediate precision; and (3) one that generates more accurate result. The accuracy achieved in these cases is due solely to the quality of the data acquired. For more result, see Appendix A.  
Figure 15: Acquired data quality (graphical units in meter). (**a,b**) worst quality data. (**c**) Satisfactory data quality. These analyses consider three metrics: (1) the error in distance measurement (Euclidean distance between the estimated location of the emitter \(E_{xy}\big{(}E_{x},E_{y}\big{)}\) and the real emitter position \(E_{p}\big{(}x_{p},y_{p}\big{)}\)); (2) the error along \(x\)-axis given by \(\big{|}E_{x}-x_{p}\big{|}\); and (3) the error along \(y\)-axis given by \(\big{|}E_{y}-y_{p}\big{|}\). The nomenclature used on the graphics legend box is:  
* PPC--Polar Points Centroid Model (proposed). * CHC--Convex Hull Centroid Model (proposed). * PLI--Polar Lines Intersections Model (proposed). * TLI--Tangent Lines Intersections Model (proposed). * MAI--Tangent Lines with Minimal Angles Model (proposed). * NRm--Newton-Rapson Method (for comparison). * LSm--Least Square Method (for comparison). * WLSm--Weighted Least Square Method (for comparison). For accomplishing the analyses, twelve experimental cases are generated (Figure 14) and the proposed geometric models (PPC, CHC, PLI, TLI, and MAI) are utilized in each acquired data set. We also use WLSm, NRm, and LSm methods to demonstrate the superior quality of the proposed geometric models. The metrics of analyses (distance error, \(x\)-axis error, and \(y\)-axis error) are computed for each result obtained from the geometric models and numerical methods. Figure 16 shows the magnitude of these errors for each experimental case. In these graphs, the 13 th case represents the arithmetic mean of the magnitude errors for the respective metric and model/method. Table 1 presents a global evaluation of the error values considering all experiments for each geometric model and numerical method. These values explain that some geometric model produce results with errors equivalent to those produced by numerical methods. MAI and TLI models promote larger errors on the \(x\)-axis, but smaller on the \(y\)-axis and distance (dist). They also promote smaller errors and produce smaller maximum errors, which show that MAI and TLI achieve better results when there is inconsistency in the data. CHC and PLI miss the least on the \(x\)-axis and get most significant result on the \(y\)-axis and distance. Errors committed by PPC are equivalent to those committed by numerical methods (WLSm, NRm, and LSm). We can observe that errors may be larger or smaller depending on the quality of the acquired data, but in the general context, geometric models present better results than numerical methods. Figure 16: Magnitude error analyses. Table 2 displays a global mean error committed by the geometric models against numerical methods. That is to say, these values are the result of minimizing the errors reached by solving the problem of signal location. Errors are not committed by methods, but they are present in data, as Figure 15 displays. Table 2 shows that the proposed geometric models can achieve better results than those achieved by numerical methods. Table 3 displays the variability of the errors committed by each method. The column Effective Variability of the Errors is the arithmetic mean of the standard deviation of the errors committed on the \(x\)-axis, \(y\)-axis and in distance by each method. The Effective Variability of the Errors (Table 3) allows analyzing how a method is sensitive to the variation of data quality. In other words, this value is a metric to define how much a method can approach the exact solution when it operates on data with errors.