# 2D Triangulation of Signals Source by Pole-Polar Geometric Models
## 2 The Proposed Geometric Models
### Polar Lines Intersections Model (PLI)  
The polar-line \(p\) is a line that passes through two corresponding polar-points \(P(x_{1},y_{1})^{*}\) and \(Q(x_{2},y_{2})^{*}\) that belongs to a line of a conic section (circle in this case). These polar-points are obtained by a given pole-point \(E(x_{e},y_{e})\) (Figure 1b). The PLI model (Algorithm 3) consists finding all combinations of pole-polar lines using as pole-points the position of each receiver at a time. After, we must compute the points of intersection among all these lines.  
Figure 8a represents cases of data underestimation provide by sensors. Figure 8b shows a good case for real application. The emitter location should be on the common intersection point among the receiver range circles. Figure 8c shows the exact result. Reliable data quality improves accuracy.  
Figure 8: Results produced by Polar Lines Intersections Model (PLI). (**a**,**b**) show 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 7: Different cases and respective responses of the PPC and CHC models. (**a**) Five receivers system. The data has some discrepancies. (**b**) Four receivers system. The data has small inconsistencies. For both cases, is better applying the solution provided by Section 2.1.  
A lot of pole-polar lines pairs, in the geometry arrangement with \(m\) receivers, are parallel lines (Figures 8 and 9); this minimizes the number of intersection points. For \(m>3\) receivers, several bad intersections points are generated. The convex hull among polar-points establishes a ROI that ensures better symmetry among the points of interest. For large values of \(m\), many intersections will occur very far from the ROI, breaking then the desired symmetry among these points (Figure 9). The Pole-Polar Lines Intersections model achieves accurate values for emitter location because it generates significant symmetrical points of interest.  
```
Data Input \(m\geq 3\) 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:for each \(k=1,2,3,\ldots,m\)
2: \(\forall k,j:1,2,\ldots,m/k\
eq j\), by (d-6) and (d-7), for each receiver position \(C_{k}\), used as pole-points, compute all combinations of polar-points \(P_{kj}\big{(}x_{pkj},y_{pkj}\big{)}\) and \(Q_{kj}\big{(}x_{qkj},y_{qkj}\big{)}\) with the respective receiver at position \(C_{j}\) and signal range \(r_{j}\).
3: Stores the corresponding points \(P_{kj}\) and \(Q_{kj}\) in the set \(R\) \(\big{(}R\gets R\cup\big{\{}P_{kj},Q_{kj}\big{\}}\big{)}\).
4: For each corresponding \(P_{kj}\) and \(Q_{kj}\) points, by (d-6), compute the \(p_{kj}\) polar-line equation.
5:endfor
6: For all \(p_{kj}\) polar-lines, by (d-3), compute the intersections points, \((x_{+},y_{+})\), among all others polar-lines. Stores these intersections points in \(S\).
7: Apply (d-8), find the convex hull polygon for all polar-points in \(R\). The obtained polygon is the minimal convex polygon that involves all interest points in \(R\). This polygon constitutes our ROI.
8: Exclude from \(S\) all intersections points among all polar-lines on the boundary, or out, of the ROI, called bad intersections points.
9: Apply (d-9), compute the location estimation, \(E_{xy}\big{(}E_{x},E_{y}\big{)}\), of the emitter. Information Output
10: Emitter location estimation \(E_{xy}\big{(}E_{x},E_{y}\big{)}\).
```  
**Algorithm 3**. PLI--Polar Lines Intersections Model Algorithm  
Figure 9: Result produced by PLI four receivers system. The non-symmetrical points are removed from the ROI by the use of Convex Hull. For this case is better applying the exact solution provided by Section 2.1.