## 5 Proposed Methods
### Modeling Afct
#### 5.1.3 Proving the Properties of Four Sensing Nodes  
The use of the sensing information of four sensor nodes in 2-dimensional acoustic target tracking results in a unique correct result.  
We proved in Lemma 2 that the intersection planes of three sensing cones make a pencil. Now we extend that work and prove that the intersection planes of four sensing cones make a bundle of planes. We assume that we have four sensing nodes and their pair-wise intersection planes are \(\pi_{ij}\) where \(i,j\in\left\{1,2,3,4\right\},i\
eq j\). Four sensing cones will have six intersection planes. Four sensing cones can have four triple combinations that make four pencils and so will have four different axis lines. Using Lemma 2, we can conclude that each triple combination of sensing cones of four sensor nodes makes a pencil.  
Now we must prove that four different pencils intersect with each other on a common point. Let us assume a pencil \(\omega_{ijk}\) that is constructed from the sensing information of sensor nodes \(i,j,and\)\(k\), and another pencil \(\omega_{ijl}\) that is constructed from the sensing information of sensor nodes \(i,j,and\)\(l\). Pencil \(\omega_{ijl}\) consists of three intersection planes \(\pi_{ij}\), \(\pi_{il}\), and \(\pi_{jl}\), and pencil \(\omega_{ijk}\) consists of three intersection planes \(\pi_{ij}\), \(\pi_{lk}\) and \(\pi_{jk}\). Each pair of these pencils has a common intersection plane. The common intersection plane of a pencil \(\omega_{ijk}\) and \(\omega_{ijl}\) is the \(\pi_{ij}\) plane. Now we must prove that two uncommon  
Figure 13: Intersection curves of four sensing cones intersected on a common point.  
intersection planes of two different pencils make a bundle of planes. We prove that intersection planes \(\pi_{ll}\) and \(\pi_{jl}\) from the pencil \(\omega_{ljl}\) make a bundle of planes with intersection planes \(\pi_{ik}\) and \(\pi_{jk}\) of the pencil \(\omega_{ljk}\). Three planes can meet each other on a common point and make a bundle of planes if their equations fall in Equation (39), where \((\lambda,\mu,\upsilon)\!\in\!\Re\), and their equations are defined up to a common no vanishing factor [42].  
\[\lambda(A_{1}x+B_{1}y+\ C_{1}z+\ D_{1})+\mu(A_{2}x+B_{2}y+\ C_{2}z+\ D_{2})+ \ \upsilon(A_{3}x+B_{3}y+\ C_{3}z+\ D_{3})=0 \tag{39}\]  
The equations of intersection planes \(\pi_{il}\), \(\pi_{jl}\) and \(\pi_{ik}\) are as follows, respectively:  
\[\begin{cases}\pi_{il}\!:\!2(x_{l}-x_{i})x+2(y_{l}-y_{i})y-2(t_{l}-t_{i})v^{2}t +({t_{i}}^{2}-{t_{i}}^{2})v^{2}+({x_{l}}^{2}-{x_{i}}^{2})+({y_{l}}^{2}-{y_{i}} ^{2})=0\\ \pi_{jl}\!:\!2\big{(}x_{l}-x_{j}\big{)}x+2\big{(}y_{l}-y_{j}\big{)}y-2\big{(}t _{l}-t_{j}\big{)}v^{2}t+\big{(}{t_{j}}^{2}-{t_{l}}^{2}\big{)}v^{2}+\big{(}{x_{ l}}^{2}-{x_{j}}^{2}\big{)}+\big{(}{y_{l}}^{2}-{y_{j}}^{2}\big{)}=0\\ \pi_{ik}\!:\!2(x_{k}-x_{i})x+2(y_{k}-y_{i})y-2(t_{k}-t_{i})v^{2}t+({t_{i}}^{2 }-{t_{k}}^{2})v^{2}+({x_{k}}^{2}-{x_{i}}^{2})+({y_{k}}^{2}-{y_{i}}^{2})=0\end{cases} \tag{40}\]  
We can write the equations of planes in Equation (40) in a simplified form as in Equation (22) and then substitute them in Equation (39) as follows:  
\[\begin{split}\lambda\!:\!\big{(}A_{il}x+B_{il}y+C_{il}z+D_{il} \big{)}+\ \mu\left(A_{jl}x+B_{jl}y+C_{jl}z+D_{jl}\right)+\ \upsilon\left(A_{ik}x+B_{ik}y+C_{ik}z\right.\\ +D_{ik}\big{)}=0\end{split} \tag{41}\]  
If we assume \(\lambda=-1,\mu=+1,and\ \upsilon=1\) then the above relation is changed to the following:  
\[-(A_{ij}x+B_{ij}y+C_{ij}z+D_{ij})+\ (A_{ik}x+B_{ik}y+C_{ik}z+D_{ik})=0 \tag{42}\]  
Equation (42) is a linear combination of planes \(\pi_{ij}\) and \(\pi_{ik}\) equations. We proved in general form that planes \(\pi_{ij}\), \(\pi_{ik}\), and \(\pi_{jk}\) make a pencil. We know that Equation (42) is like Equation (26) and it is the equation of \(\pi_{jk}\) plane. The condition of Equation (41) thus becomes true and planes of Equation (40) satisfy the condition of Equation (39) and make a bundle of planes.