# 2D Triangulation of Signals Source by Pole-Polar Geometric Models
## 2 The Proposed Geometric Models
### Polar-Points Centroid Model (PPC)  
The Figures 3 and 4 show a set of possible cases of signal data and the respective application of PPC model (Algorithm 1). Figure 4a,c presents an intrinsic example of data incoherence, when the spatial position of a receiver is inside the signal range of another receiver, due to the presence of noise or excessive gain applied to the antennas. Even so, the results produced are consistent with data and no singularity was generated. If this case is possible, then some polar-points are complex numbers in the form \(a+bi\). However, the proposed models have their origins in the polar geometry projection, which is also used to represent complex numbers. This means that the real part, \(real(a+bi)=a\), of a complex polar-point can be used as an approximate value to represent an actual polar-point. Another way is to exclude all generated complex numbers from partial results. In this work, those complex numbers are not excluded. This option shows that the proposed geometric models are capable to solve these data incoherence producing a consistent result (Figure 4a,c, Figures 8b and 13).  
Figure 3c displays the optimal case for real application. The emitter location should be in the common intersection point among the three receiver range circles. Although the result was close to this point, there is no perfect match, because there are polar-points that do not have the pattern responsible for symmetrically enveloping the emitter location. These polar-points are bad points. Using redundant data, by applying \(m>3\) receivers to the system, these non-symmetrical points will have low influences on results (Figure 4b).  
Figure 4c present an atypical geometric arrangement for a system used to solve signal triangulation problems. All receivers are arranged in a straight line (they are collinear). It is not all triangulation  
Figure 3: Results produced by Polar-Points Centroid Model (PPC). (**a,b**) show the possible presence of noise in acquired data. (**c**) Represents the best case for acquired data because the signal data are _quasi_-exact. The red dotted rectangle mark all (four) polar points that belong to all circle lines (signals range). The exact solution is one of these points (see Section 2.1).  
methods that can solve the problem with this arrangement. The PPC resolves and, if the data is accurate, the solution can be exact.  
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Data Input \(m\geq 2\) is the number of receivers. \(C_{k}(x_{k},y_{k})\), \(k=1,2,\ldots,m\), is the planar position of each receiver. \(r_{b},k=1,2,\ldots,m\), 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:store the points \(P_{kj}\) and \(Q_{kj}\) in the set \(\big{(}S\gets S\cup\big{\{}P_{kj},Q_{kj}\big{\}}\big{)}\).
4:endfor
5: Apply (d-9) in the set \(S\), compute the location estimation, \(E_{xy}\big{(}E_{x},E_{y}\big{)}\), of the emitter. Information Output
```  
**Algorithm 1** PPC--Polar-Points Centroid Model