# 2D Triangulation of Signals Source by Pole-Polar Geometric Models
## 1 Introduction
### Useful 2D Geometric Definitions  
This subsection describes some geometric important definitions for understanding the proposed geometric models. Such definitions are labeled as, for example, (d-9) which means the 9 th definition and its uses are thus referenced in the text. References [22; 23] provide these definitions and further information in this regard. Figure 1: **(a)** Analytical solution to \((x,y)\) when there is an intersection between circles. (**b**) Possible tangential straight lines \(t_{1}\) and \(t_{2}\) for the circle line obtained by an external point \(E(x_{e},y_{e})\) (pole-point). Polar-line \(p\) and its geometric relationship with the tangent lines and a given conic section (circle in this case). Point and Line--A coordinate \(P(x,y)\) define the point \(P\) in the Cartesian plane \((\mathbb{R}^{2})\). The line that passes through the points \(P(x_{1},y_{1})\) and \(Q(x_{2},y_{2})\), \(P\
eq Q\), is denoted \(\overset{\leftrightarrow}{PQ}\). The line segment \(\overline{PQ}\) (with endpoints \(P\) and \(Q\)) is the portion of the line \(\overset{\leftrightarrow}{PQ}\) between points \(P\) and \(Q\). **(d-1)** The Euclidean distance \(D_{\overline{PQ}}\) between two points \(P(x_{1},y_{1})\) and \(Q(x_{2},y_{2})\) is given by  
\[D_{\overline{PQ}}=\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}.\]
**(d-2)** For constants \(A\), \(B\), \(C\) (\(A\) and \(B\) not both zero) all points \((x,y)\) satisfying the equation \(Ax+By+C=0\) define the implicit line equation in the Cartesian plane. For two points \(P(x_{1},y_{1})\) and \(Q(x_{2},y_{2})\), a particular \(\overset{\leftrightarrow}{PQ}\) line equation is obtained by  
\[A=y_{1}-y_{2};\ B=x_{2}-x_{1};\ C=-Ax_{1}-By_{1}.\]
**(d-3)** Two particular lines \(A_{1}x+B_{1}y+C_{1}=0\) and \(A_{2}x+B_{2}y+C_{2}=0\) have an interception at the point \((x_{+},y_{+})\), if \(d=A_{1}B_{2}-A_{2}B_{1}\
eq 0\), given by  
\[(x_{+},y_{+})=\left(\frac{B_{1}C_{2}-B_{2}C_{1}}{d},\frac{A_{2}C_{1}-A_{1}C_{2 }}{d}\right),\]  
if \(A_{1}A_{2}+B_{1}B_{2}=0\), lines are perpendicular. If \(A_{1}B_{2}=A_{2}B_{1}\), lines are parallel or coincident. **(d-4)** The angle \(\theta\) formed between two particular lines is given by  
\[\tan\theta=\frac{A_{1}B_{2}-A_{2}B_{1}}{A_{1}A_{2}+B_{1}B_{2}}.\]
**(d-5)** The line equation \(A_{p}x+B_{p}y+C_{p}=0\) that passes through point \(P\big{(}x_{p},y_{p}\big{)}\) and is perpendicular to the line \(Ax+By+C=0\) is defined as  
\[A_{p}=-B;\ B_{p}=A;\ C_{p}=Ax_{p}-By_{p}.\]  
Circle--In the Cartesian plane the equation \((x-x_{c})^{2}+(y-y_{c})^{2}=r^{2}\) defines the implicit circle equation centered at the point \(C(x_{c},y_{c})\) with radius \(r\). Let \(E(x_{e},y_{e})\) be an external point to a circle. By using \(E\) we can obtain two tangent lines, \(t_{1}\) and \(t_{2}\), to the circle (Figure 0(b)), which pass through points \(P(x_{1},y_{1})^{*}\) and \(Q(x_{2},y_{2})^{*}\), respectively. Points \(P\) and \(Q\) can be computed by applying the geometric concept of pole-polar definition. Pole-Polar Geometry--Pole-point and polar-line are, respectively, a point and a line that have a unique reciprocal relationship with respect to a given conic section. If the point lies on the conic section, its polar-line is the tangent line to the conic section at that point [24]. If the pole-point is external to the conic section, the polar-line intercepts the conic section exactly at the points that allow passing tangent lines from this pole-point (Figure 0(b)). Our interest is to pass two tangent lines, \(t_{1}\) and \(t_{2}\), through a circle centered at point \(C(x_{c},y_{c})\) with radius \(r\). Moreover, these lines must pass through a known external point \(E(x_{e},y_{e})\) (or pole-point) to this circle (Figure 0(b)). We need to locate the coordinates of the polar-points \(P(x_{1},y_{1})^{*}\) and \(Q(x_{2},y_{2})^{*}\), which define the polar-line \(p\) and lies to the tangents lines \(t_{1}\) and \(t_{2}\). Additionally we must find the equation of the polar-line \(p(Ax+By+C=0)\).