## 5 Proposed Methods
### Modeling Afct
#### 5.1.1 Conditions for Membership of Intersection Planes of Four Sensor Nodes to Make a Pencil  
_Lemma 4._ The intersection planes of four sensing cones make a pencil.  
_Proof._ In target tracking using the sensing information of three sensor nodes, the equation of two intersection planes is sufficient for calculating the axis of pencil for target tracking. Equation (36) shows the equation of two planes \(\pi_{12}\) and \(\pi_{13}\).  
\[\begin{cases}2(x_{2}-x_{1})x+2(y_{2}-y_{1})y-2(t_{2}-t_{1})v^{2}t-((t_{1}{}^{2 }-t_{2}{}^{2})v^{2}+(x_{2}{}^{2}-x_{1}{}^{2})+(y_{2}{}^{2}-y_{1}{}^{2}))=0\\ 2(x_{3}-x_{1})x+2(y_{3}-y_{1})y-2(t_{3}-t_{1})v^{2}t-((t_{1}{}^{2}-t_{3}{}^{2} )v^{2}+(x_{3}{}^{2}-x_{1}{}^{2})+(y_{3}{}^{2}-y_{1}{}^{2}))=0\end{cases} \tag{36}\]  
Equation (37) shows the intersection planes of the third and fourth cones:  
\[\begin{split}\{2(x_{3}-x_{4})x+2(y_{3}-y_{4})y-2(t_{3}-t_{4})v^{ 2}t-((t_{4}{}^{2}-t_{3}{}^{2})v^{2}+(x_{3}{}^{2}-x_{4}{}^{2})+(y_{3}{}^{2}-y_ {4}{}^{2}))=0\end{split} \tag{37}\]  
If we can find values for \(k_{1}\) and \(k_{2}\) that satisfy Equation (38), we prove that the new intersection plane of Equation (37) makes a pencil with three intersection planes for each pair of three cones of three previous sensing nodes [35].  
\[\left\{\begin{array}{c}x_{3}-x_{4}=\ k_{1}.\left(x_{2}-x_{1}\right)+k_{2}.\left(x _{3}-x_{1}\right)\\ y_{3}-y_{4}=\ k_{1}.\left(y_{2}-y_{1}\right)+k_{2}.\left(y_{3}-y_{1}\right)\\ t_{3}-t_{4}=\ k_{1}.\left(t_{2}-t_{1}\right)\ +\ k_{2}.\left(t_{3}-t_{1}\right)\\ ({t_{4}}^{2}-{t_{3}}^{2})v^{2}+({x_{3}}^{2}-{x_{4}}^{2})+({y_{3}}^{2}-{y_{4}}^ {2})=\\ k_{1}.\left(({t_{1}}^{2}-{t_{2}}^{2})v^{2}+({x_{2}}^{2}-{x_{1}}^{2})+({y_{2}}^{2 }-{y_{1}}^{2})\right)+\\ k_{2}.\left(({t_{4}}^{2}-{t_{3}}^{2})v^{2}+({x_{3}}^{2}-{x_{4}}^{2})+({y_{3}}^{2 }-{y_{4}}^{2})\right)\end{array}\right. \tag{38}\]  
The first three equations of simultaneous equations in Equation (38) hold if the normal vector of the new plane is a linear combination of normal vectors of previous planes. Since the sensing cone of the fourth sensing node in Figure 11 satisfies this condition, its intersection planes with other sensing cones belongs to the constructed pencil of previous three sensing cones. If the fourth vector \(\overline{P_{3}P_{4}}\) is a linear combination of the three previous vectors, it satisfies the condition of a pencil [40,41]. This occurs when the 3-dimensional sensing information of the fourth sensing node resides on the plane that three previous sensing nodes had made in 3-dimensional space.  
The 3-dimensional sensing information of three previous sensor nodes are shown in Figure 11 by \(P_{1}\), \(P_{2}\), and \(P_{3}\) points. Only one plane passes from three points that do not reside on a straight line in the 3-dimensional space. The probability that a random chosen point in the 3-dimensional space resides on a specific plane is nearly zero. Therefore, the probability that the sensing information of a fourth sensor in the form of cone makes intersection planes with three previous cones belonging to the pencil that three cones of three previous sensor nodes had made, is approximately zero.