# 2D Triangulation of Signals Source by Pole-Polar Geometric Models
## 1 Introduction
### Useful 2D Geometric Definitions
**(d-6)** The general equation of a conic in the Cartesian coordinate system is given by \(a_{xx}x^{2}+2a_{xy}xy+a_{yy}y^{2}+2b_{x}x+2b_{y}y+w=0\). We need the equation of the polar-line \(p(Ax+By+C=0)\) that can be obtained by a known pole-point \(E(x_{e},y_{e})\). The required coefficients of the respective polar-line \(p\) is given by: \(A=a_{xx}x_{e}+a_{xy}y_{e}+b_{x}\); \(B=a_{xy}x_{e}+a_{yy}y_{e}+b_{y}\); \(C=b_{x}x_{e}+b_{y}y_{e}+w\). In this work, the expected conic section is a circle. For the circle case, the following simplifications are helpful: \(a_{xx}=1\); \(a_{xy}=0\); \(a_{yy}=1\); \(b_{x}=-x_{c}\); \(b_{y}=-y_{c}\) and \(w=x_{c}^{2}+y_{c}^{2}-r^{2}\). The next step consists in placing points \(P(x_{2},y_{2})^{*}\) and \(Q(x_{2},y_{2})^{*}\), which are obtained by computing the intersection between the polar-line \(p\) and the circle line (Figure 1b). To compute the intersection of a line, \(Ax+By+C=0\), with a circle, \((x-x_{c})^{2}+(y-y_{c})^{2}=r^{2}\), the following conditions must be considered: \(d_{lc}=\frac{\left|Ax_{c}+By_{c}+C\right|}{\sqrt{A^{2}+B^{2}}}\) is the distance between the line and the circle center point \(C(x_{c},y_{c})\),  
\(\bigcirc\): if \(d_{lc}>r\), there is no intersection point;  
\(\bigcirc\): if \(d_{lc}=r\), the line is tangent to the circle and has one intersection point;  
\(\bigcirc\): if \(d_{lc}<r\), the line is secant to the circle and has two intersection points. The algebraic solution for this intersection is an equation of degree two. Another way to solve this intersection is applying some geometric relationships, as follows. To find the intersections points \(P(x_{1},y_{1})^{*}\) and \(Q(x_{2},y_{2})^{*}\), which are the polar-points, we have first to drop a perpendicular line (by d-5) from the center \(C(x_{c},y_{c})\) of the circle to the line \(p\). Let \(T(x_{t},y_{t})\) be the intersection point and \(\overset{\leftrightarrow}{CT}\) be the line that passes through \(C\) and \(T\) (Figure 1b). The equation of line \(p(Ax+By+C=0)\) is known (by d-6). Thus, the equation of line \(\overset{\leftrightarrow}{CT}\) is \(\overset{\leftrightarrow}{CT}(-Bx+Ay+Ax_{c}-By_{c})\). This way, the point \(T(x_{t},y_{t})\) can be computed by intersection between \(p\) and \(\overset{\leftrightarrow}{CT}\) lines (by d-3). \(D_{\overline{CT}}\) represents the Euclidean distance between points \(C\) and \(T\); \(D_{\overline{PT}}\) refers to the distance between points \(P\) and \(T\); \(D_{\overline{QT}}\) stands for the distance between points \(Q\) and \(T\); and \(D_{\overline{CP}}=D_{\overline{C\overline{C}}}=r\) (by d-1). The triangles \(\Delta CPT\) and \(\Delta CQT\) are right-angled, and hence we prove that  
\[D_{\overline{CT}}^{2}+D_{\overline{PT}}^{2}=r^{2},\]  
and  
\[D_{\overline{CT}}^{2}+D_{\overline{QT}}^{2}=r^{2}.\]
**(d-7)**: So, \(D_{\overline{QT}}=D_{\overline{PT}}=h=\sqrt{r^{2}-D_{\overline{CT}}^{2}}\); now, if we translate the point \(T\) by \(h\) units in both directions along line \(p\), the points \(P\) and \(Q\) are determined as follows  
\[(x_{1},y_{1})^{*}=\left(x_{t}-\frac{Bh}{\sqrt{A^{2}+B^{2}}},y_{t}+\frac{Ah}{ \sqrt{A^{2}+B^{2}}}\right),\]  
and, if \(h\
eq 0\)  
\[(x_{2},y_{2})^{*}=\left(x_{t}+\frac{Bh}{\sqrt{A^{2}+B^{2}}},y_{t}-\frac{Ah}{ \sqrt{A^{2}+B^{2}}}\right).\]  
The suggested geometric models that use the convex hull algorithm (CHC, PLI, TLI, and MAI) need to minimize the region of interest (ROI) and exclude bad points from final results. This minimization of ROI is obtained by obtaining a convex polygon defined on a set of previously computed points. A set \(S\) is convex if \(\forall x,y\in S\Rightarrow\overline{xy}\subseteq S\). Any region (polygon) with a \"dent\" is not convex [24]. The convex hull of a set of points is the smallest convex set containing these points [24; 25; 26]. **(d-8)**: The convex hull algorithm is used in this work to specify a Region of Interest (ROI).