# 2D Triangulation of Signals Source by Pole-Polar Geometric Models
## 1 Introduction
### Standard Methods and Considerations
#### 1.2.3 Generalization of 2D Location Function Range-Based  
In 2D space \((x,y)\) are two unknowns for the location of a point \(P\). In Cartesian coordinates, the point \((x,y)\) can be represented in terms of its range by \((x-x_{1})^{2}+(y-y_{1})^{2}=(r_{1})^{2}\), where \((x_{1},y_{1})\) is a known reference point and \(r_{1}\) is the range (or radial distance) between the unknown point \((x,y)\) and the reference point \((x,y)\). To solve \((x,y)\) is necessary a second observation given by another reference point \((x_{2},y_{2})\) that generates the new circle equation \((x-x_{2})^{2}+(y-y_{2})^{2}=(r_{2})^{2}\). There are two solutions for these two quadratic equations. If these two solutions are the same, then these two circles are tangent and the solution for \(P\) is found. If these two circles are secant, \(P\) has two ambiguous solutions. In this case, to obtain the correct solution is necessary known a third reference point \((x_{3},y_{3})\) and its respective \(r_{3}\) range to \(P\). This is known as trilateration, the estimation of the position of a point unambiguously based on the measurements of distances from three or more known reference locations [7]. Otherwise, these three circles do not intercept at a single point because there are errors in ranges measurements. In this case, the solution for \(P\) is inaccurate and need to be estimated with error minimization. To provide this error minimization it is necessary known others reference points \((x_{k},y_{k})\) and its respective ranges \(r_{k}\)--see Equation (8). As the Cartesian plane is concerned, the Euclidean distance between two points \(P_{0}(x_{0},y_{0})\) and \(P_{1}(x_{1},y_{1})\) is given by \(d=\sqrt{(x_{1}-x_{0})^{2}+(y_{1}-y_{0})^{2}}\), so, by means of (6) or (7), the relationship \((x_{1}-x_{0})^{2}+(y_{1}-y_{0})^{2}=(v(t_{R}-t_{E}))^{2}=10^{2\frac{\rho_{0}- \rho(d)}{10\mathcal{L}}}\) is obtained, which is the circle equation with radius \(r=v(t_{R}-t_{E})=10^{\frac{\rho_{0}-\rho(d)}{10\mathcal{L}}}\) and center \((x_{0},y_{0})\). If \(P_{0}\) represent the receiver position and \(P_{1}\) the emitter position, the emitter belongs (best case) to this circle line and the receiver is at the circle center. Note that signal-based location methods need to know the value of the radial range of the signal \(r\) regardless of whether this value is estimated by strength data or the wave propagation speed of that signal. Despite of how the value of \(r\) is estimated, such methods do not change. Therefore, performing experiments using signal strength and signal propagation velocity data is irrelevant to compare the accuracy achieved by different methods.  
The radial distance (or range data) \(d=r\) between emitter and receiver is identified through (6) or (7). Using this concept to \(m>2\) receivers by one emitter, the following is obtained (8)  
\[\mathbf{F}(x,y)=\left\{\begin{array}{c}\left(x-x_{1}\right)^{2}+(y-y_{1})^{2}-(r _{1})^{2}=0\\ \left(x-x_{2}\right)^{2}+(y-y_{2})^{2}-(r_{2})^{2}=0\\ \vdots\\ \left(x-x_{m}\right)^{2}+(y-y_{m})^{2}-(r_{m})^{2}=0\end{array}\right. \tag{8}\]  
The Equation (8) is a nonlinear equations system to solve the emitter location \((x,y)\). If \(m=2\), this equations system is normal and can be solved either by a numerical or empirical method or applying the analytical solution showed in Figure 1a. If \(m>2\), the system is redundant and allows minimizing the solution error by means of Nonlinear Least-Squares (NLLS).