# 2D Triangulation of Signals Source by Pole-Polar Geometric Models
## 2 The Proposed Geometric Models
### Convex Hull Centroid Model (CHC)  
Figure 4(a),b shows results closer to the solutions together with those shown in Figure 2(a),b. Nonetheless, the result shown in Figure 7 is better than that of Figures 2(c) and 4(c). This improvement in accuracy is related to the exclusion of non-symmetrical polar-points (called bad points) to the emitter actual location.  
It must be noticed that in Figure 7 the set of polar-points, adding a new receiver to the system, allows obtaining a region (ROI) more symmetrical to the emitter actual position. Accuracy in results improves by adding new receivers to the system. This filtered ROI is more symmetrical, both PPC and CHC (Algorithm 2) produce the same correct result.  
Figure 4: (**a**) Incoherence in data. The spatial position of a receiver is inside of range of another receiver. In this case, some polar-points are complex numbers (see the polar-points out of the line circles)—but their real parts belong to the region of interest (ROI). (**b**) Accurate data. Exact solution produced by PPC using redundant data (four receivers). For this case, is better applying the exact solution providing by Section 2.1. (**c**) Accurate data. A collinear arrangement among receivers (geometric arrangement not recommended) and the exact solution provided by PPC. The solution to this problem is impossible for Weighted Least Squares (WLSm), Least-Square (LSm), and Newton–Rapson (NRm).  
```
Data Input \(m>2\) is the number of receivers. \(C_{k}(x_{k},y_{k})\), \(k=1,2,\ldots,m\), is the planar position of each receiver. \(r_{k},k=1,2,\ldots,m\), is the signal range of each receiver to an emitter. Procedure
1: Execute the steps 1 until 4 of algorithm Polar Points Centroid Model.
2: Apply (d-8), find the convex hull polygon for all polar-points in \(S\). The obtained polygon is the minimal convex polygon that involves all interest points in \(S\). This polygon constitutes the ROI.
3: Exclude all polar-points on the boundary of this convex polygon, called bad polar-points, from \(S\).
4: Apply (d-9) in \(S\), compute the location estimation, \(E_{xy}\big{(}E_{x},E_{y}\big{)}\), of the emitter. Information Output
5: Emitter location estimation \(E_{xy}\big{(}E_{x},E_{y}\big{)}\).
```  
**Algorithm 2** CHC--Convex Hull Centroid Model Algorithm  
Figure 5: Results produced by Convex Hull Centroid Model (CHC). (**a,b**) highlight the possible presence of noise in acquired data. (**c**) represents the best case for acquired data. For this case, is better applying the exact solution provided by Section 2.1.  
Figure 6: Results produced by CHC using satisfactory redundant data (four receivers). For this case, is better applying the exact solution provided by Section 2.1.  
When the acquired data are coherent, the responses produced by the two models are similar (Figure 7b). If there are some discrepancies on the data, there may be divergences among the results (Figure 7a). Consequently, data errors promote differences in results. In this work, we do not evaluate data quality, but show consistency of results presented by the proposed models, whenever there is guarantee on the data quality.