# 2D Triangulation of Signals Source by Pole-Polar Geometric Models
## 4 Conclusions  
Our results show that there is a geometrical relationship between the emitter location and elements of pole-polar geometry. For this aim, we propose geometric models to solve a geometric problem: 2D location of a signal source and, if accurate data are available, an exact solution is found with cost \(O(n)\) (Section 2.1). If the data quality is satisfactory, the proposed models obtain the best estimative for the problem of locating the signal emitter source and are more robust when they operate on data with errors than the traditional WLSm, NRm, and LSm methods.  
This work contributes to cope with the problem of applying 2D point location in a geometric relationship by reducing the number of arithmetic operations needed by the current conventional methods in use and the inherent propagation errors in the acquired data. The proposed geometric models have low computational cost \(O(nlogn)\). The obtained results are consistent even when there is incoherence in the data or when there is incoherence geometric in the arrangement of the receivers system.  
When analyzing these results, we observe that the solutions given by the geometric models are located around the region of intersections of the radial signals ranges for all cases. Thus, a linear combination of all these solutions is also a solution. Consequently, we have computed the mean among the five solutions to derive this linear combination with no special motivation. This new solution minimizes the global error (Table 2).  
Five geometric models have been presented and experimented with displayed solutions that are located around the region of intersection of the radial signals range for all cases. A comparison of results with numerical methods Weighted Least-Square, Least-Square, and Newton-Rapson has been carried out and results achieved by the proposed geometric models have improved, especially in cases of moderate data incoherence.  
The proposed geometric models have the following limitations: (1) 2D solution (we are developing a 3D solution). (2) Proposed methods for estimating location require that the device to be located be inserted into the coverage area of the APs. We are developing a solution to this limitation in conjunction with the 3D solution. (3) APs may be arranged collinearly only for a few specific cases.  
To develop this work we have built a simulator in MatLab code and used the implemented functions to construct a prototype that processes real data [36]. All code and data used are at anyone's disposal. Go to www.coding2 change.org/articles/GeometricModels.zip. Consult User's Manual for more instructions.  
[PERSON] and [PERSON] conceived, designed and performed the experiments; proposed the geometrical models; analyzed the data and results; developed the MatLab code; wrote the paper. [PERSON] and [PERSON] conceived the experiments; analyzed the results; provided mathematical contributions; wrote the paper; acted as peer reviewers.  
This research has been supported by the Pololas project (TIN2016-76956-C3-2-R) of the Spanish Ministry of Economy and Competitiveness and by the particular research plan of the University of Seville (VPPI-US).  
Special thanks to the University of Seville--Spain and the State University of Maring--Brazil.  
The authors declare no conflict of interest. The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.