# 2D Triangulation of Signals Source by Pole-Polar Geometric Models
## 2 The Proposed Geometric Models
### Tangent Lines Intersections Model (TLI)  
Using each center circle point \(C_{k}\) as pole-point, one by one, all polar-points \(P_{kj}\Big{(}x_{pkj},y_{pkj}\Big{)}\) and \(Q_{kj}\Big{(}x_{qkj},y_{qkj}\Big{)}\) can be computed for each other circle \(j\
eq k\) (Figures 1b, 3 and 10). It is possible to draw two lines for a particular \(C_{k}\) point (tangent lines to a circle \(j\
eq k\)): one passing through by \(P_{kj}\) and another by \(Q_{kj}\) (Figures 10 and 11a). If \(m\) is the number of receivers in the system (there are \(m\) circles), so, \(2(m-1)\) tangent lines (Figures 10 and 11a) pass through each circle and generate a large set of intersections points among these lines, but most of these points are bad points that are eliminated by the convex hull ROI. 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 the desired symmetry among these points.  
Figure 10: Results produced by Tangent Lines Intersections (TLI) model. (**a,b**) show the possible presence of noise in acquired data. (**c,d**) represents the best case for acquired data. (**d**) Use of four receivers (the legend box is omitted for best view). For cases (**c,d**) is better applying the exact solution provided by Section 2.1.  
There are intersections of tangent lines in each circle center because they pass by each circle center (receiver position) \(C_{k}\). Whether these center points must be used in the estimation of emitter location depends on the geometric arrangement among receivers. While the results produced by TLI model (Algorithm 4) are consistent (Figure 10), \(m\geq 4\) receiver usage is recommended (Figure 10d) for best accuracy.  
```
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. \(r_{k},k=1,2,\ldots,m\), 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 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 \(Q_{kj}\) polar-points, by (d-6), computes the respective two tangent lines equation \(t_{Pkj}\) and \(t_{Qkj}\) that passes by each \(C_{k}\). 5:endfor 6: For all \(t_{Pkj}\) and \(t_{Qkj}\) tangent-lines, by (d-3), computes the intersections points, \((x_{+},y_{+})\), among all tangent-lines. Store 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 tangent-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 4**TLI--Tangent Lines Intersections Model Algorithm