## 4 SD-EAR in Detail
### Network Model
The situation is expressed in Figure 3.  
Consider one particular edge E from (x\({}_{\rm v}\), y\({}_{\rm v}\)) to (x\({}_{\rm v+1}\), y\({}_{\rm v+1}\)). The slope m\({}_{\rm 1}\) of E is mathematically expressed in Equation (1):  
\[\rm m_{\rm 1}=(y_{\rm v+1}-y_{\rm v})/(x_{\rm v+1}-x_{\rm v}) \tag{1}\]  
Let C be the center of the radio-circle of n\({}_{\rm i}\). The shortest distance from C to edge E is given by the length of perpendicular from C to E. Assume that the perpendicular drawn on E from C intersects E at point W. Then, the slope m\({}_{\rm 2}\) of CW is related to m1 as in Equation (2):  
\[\rm m_{\rm 2}=(-1)/\rm m_{\rm 1} \tag{2}\]  
CW passes through the point (x\({}_{\rm i}\), y\({}_{\rm i}\)). Based on this, the equation of CW appears in Equation (3):  
\[\rm y=m_{\rm 2}\,x+b_{\rm 2} \tag{3}\]  
Figure 3: Polygonal zone and radio-circle embedding. where:  
\[m_{2}=-(x_{v+1}-x_{v})/(y_{v+1}-y_{v}) \tag{4}\]  
and:  
\[b_{2}=(y_{i}+(x_{v+1}-x_{v})/(y_{v+1}-y_{v})\,x_{i}) \tag{5}\]  
The equation of the edge E from (x\({}_{v}\), y\({}_{v}\)) to (x\({}_{v+1}\), y\({}_{v+1}\)) is mathematically expressed in Equation (6):  
\[y=m_{1}\,x+b_{1} \tag{6}\]  
\[s.t.\,b_{1}=(x_{v+1}y_{v}-x_{v}\,y_{v+1})/(x_{v+1}-x_{v}) \tag{7}\]  
The coordinates (w\({}_{x}\), w\({}_{y}\)) of the point of intersection of edge E and CW (i.e., the point W) are given by:  
\[w_{x}=(b_{2}-b_{1})/(m_{1}-m_{2}) \tag{8}\]  
\[\rm m_{\rm 1}=(y_{\rm v+1}-y_{\rm v})/(x_{\rm v+1}-x_{\rm v}) \tag{1}\]0  
If the Cartesian distance between the two points (x\({}_{i}\), y\({}_{i}\)) and (w\({}_{x}\), w\({}_{y}\)) is greater than or equal to rad\({}_{i}\), then it can be concluded that the radio-circle of n\({}_{i}\) is embedded within the given zone. The mathematical version of the condition appears in Equation (10):  
\[\rm m_{\rm 1}=(y_{\rm v+1}-y_{\rm v})/(x_{\rm v+1}-x_{\rm v}) \tag{1}\]1  
If such a condition (as in Equation (10)) is satisfied by all of the edges of the polygon, then it can be concluded that the radio circle of n\({}_{i}\) is completely bounded by the polygonal zone. Situation 2: The zone Z\({}_{P}\) under inspection is circular, defined by cnter (x(Z\({}_{P}\)), y(Z\({}_{P}\))) and radius rad(Z\({}_{P}\)); the node to be embedded has center (x\({}_{i}\), y\({}_{i}\)) and a radio range rad\({}_{i}\) of n\({}_{i}\)  
The situation is expressed in \[\rm m_{\rm 1}=(y_{\rm v+1}-y_{\rm v})/(x_{\rm v+1}-x_{\rm v}) \tag{1}\]2  
In Figure 4, AQ is the radius of the zone; AM specifies the distance between the centers of the zone and node n\({}_{i}\). Please note that node n\({}_{i}\) is positioned at point M. MN is the radio range of the node. The mathematical expressions for AQ, AM, and MN appear in Equations (11)-(13):  
\[\rm m_{\rm 1}=(y_{\rm v+1}-y_{\rm v})/(x_{\rm v+1}-x_{\rm v}) \tag{1}\]3  
\[\rm m_{\rm 1}=(y_{\rm v+1}-y_{\rm v})/(x_{\rm v+1}-x_{\rm v}) \tag{1}\]4  
\[\rm m_{\rm 1}=(y_{\rm v+1}-y_{\rm v})/(x_{\rm v+1}-x_{\rm v}) \tag{1}\]5  
\[\rm m_{\rm 1}=(y_{\rm v+1}-y_{\rm v})/(x_{\rm v+1}-x_{\rm v}) \tag{1}\]6 Circular zone and radio-circle embedding. The required conditions for circle embedding appears in Equation (14):  
\[\rm m_{\rm 1}=(y_{\rm v+1}-y_{\rm v})/(x_{\rm v+1}-x_{\rm v}) \tag{1}\]7  
Situation 3: The zone under inspection is elliptic, as defined by Equation (15); the node to be embedded has center (\(\rm{x_{i}}\), \(\rm{y_{i}}\)) and a radio range of \(\rm{rad_{i}}\). \[\rm m_{\rm 1}=(y_{\rm v+1}-y_{\rm v})/(x_{\rm v+1}-x_{\rm v}) \tag{1}\]8  
s.t. a > b. The situation is represented in \[\rm m_{\rm 1}=(y_{\rm v+1}-y_{\rm v})/(x_{\rm v+1}-x_{\rm v}) \tag{1}\]9  
The node to be embedded has a center of O (\(\rm{x_{i}}\), \(\rm{y_{i}}\)) and a radio range of \(\rm{rad_{i}}\). (\(\rm{h}\),\(\rm{k}\)) is the center of the ellipse. We draw a line AB through O i.e., the center of the radio circle of \(\rm{n_{i}}\), parallel to the Y axis. The line intersects the ellipse at points A and B.