# 2D Triangulation of Signals Source by Pole-Polar Geometric Models
## 1 Introduction  
Signals can be of various natures, including sound waves, visible and non-visible light, or other electromagnetic spectrum energies (radio waves or radar waves, for instance). The position of a signal emitter or receiver device can be estimated (located) by triangulating the signal data that it sends/acquires. The signal data comprise: the nature of the own signal (as light (electromagnetic energy), sound and vibration); its intrinsic attributes (such as power, frequency, and amplitude); its extrinsic attributes (as the time when the signal arrives at each particular receiver; the strength and the propagation angle of a signal acquired at each particular receiver). Current approaches used to locate an emitter or a receiver can be deterministic, statistical or machine-learning methods. Thus, we propose to perform this triangulation by geometric models that explore elements of pole-polar geometry. Here, we are presenting five geometric models to solve the point location problem: (1) based on centroid of a set of polar-points (PPC); (2) based on a convex hull region defined by a set of interestpoints (CHC); (3) based on centroid of a set of interest points obtained by polar-lines intersections (PLI); (4) based on centroid of a set of interest points obtained by tangent lines intersections, (TLI); (5) based on centroid of a set of interest points obtained by tangent lines intersections with minimal angles (MAI). The first one has computational cost \(O(n)\), whereas the others cost \(O(nlogn)\). The \(n\) value is the number of points of interest. For a system composed of \(m\) receivers, the maximum of polar-points of interest generated is \(p=\left( \begin{array}{c}m\\ 2\end{array} \right)=m!/(2(m-2)!)\). For PPC, CHC and PLI models \(n=p\). For TLI and MAI models \(n=2p\). In terms of \(m\), the cost of proposed methods is \(O\big{(}m^{2}\big{)}\) in accordance with the cost to perform \(\left( \begin{array}{c}m\\ 2\end{array} \right)\) combinations. We use IEEE 802.11 Network infrastructures to evaluate the precision of these geometric models, with the aim to acquire strength signal data from emitter-receiver system. The acquired data was not preprocessed. The acquired raw data are used to provide better analysis of the errors committed by the five geometric methods proposed in this work (PPC, CHC, PLI, TLI, and MAI) in comparisons with the errors committed by Newton-Rapson (NRm), Least-Square (LSm), and Weighted Least Squares (WLSm) which have computational cost \(\sim O\big{(}m^{3}\big{)}\). This paper does not address operations or methods related with signal data preprocessing. If you have particular interest in these operations, see Section 3.2 in this work. This means that there are errors in the acquired data used in this work. Such errors are due to the signal multipath; the presence of obstacles and the co-presence of other electromagnetic sources (see Section 1.1). The innovation proposed in this work is to use geometric methods to solve a geometric problem (estimation of the location of signal source or a receiver device). Obviously, in order to guarantee the accuracy of a location, it is necessary to ensure data with high quality. Signal location estimation by range-based methods needs to solve a nonlinear equations system (quadratic equations). To facilitate this solution, the equations of this system are linearized by subtracting equations, that is, any equation of this system is chosen and is used to subtract all other ones from this system, and thus all terms of all equations that have degree two are eliminated from these equations and the system becomes linear. However, the realization of this subtraction also eliminates from the system the equation chosen to perform this linearization. For example, if there were five equations before the linearization, after the linearization there will be four equations in the system. It is important to consider that this equation that was eliminated contemplates data of radial range of the signal acquired by some device of the location network.