## 5 Proposed Methods
### Modeling Afct
#### 5.1.2 Properties of the Fourth Sensing Node  
We proved that the probability that the sensing information of the fourth sensor node belongs to the same pencil of three previous sensor nodes is zero. The information of four different sensing motes that do not reside on the same plane, make four different pencils. Because the coordinates of all four points originated from the same base, triple combination of four sensing motes' information satisfies the condition of a pencil. Four unbounded cones can have six different pairs of combinations with each other that make six intersection planes. Triple combination of sensing information of four sensing nodes makes four different pencils as it is shown in Figure 12.  
Figure 11: Four points residing on a plane.  
**Definition 5**.: All planes passing through a common point are known as _bundle of planes_[42].  
An interesting property of these four pencils is that they all pass through a unique common point most of the times. This point is the correct spatio-temporal information of the target object. As it is shown in Figure 12, uncommon intersection planes of each pencil make a bundle of planes. Figure 12 is an extension of Figure 6 and shows the target tracking results with sensing information of four sensor nodes. Figure 6 shows that only one pencil is generated when we use the information of three individual sensing modes for target tacking. If the axis line of this pencil intersects with the down nappes, there will be two different spatio-temporal information of the target object and the correct one cannot be distinguished. This is also shown in Figure 12 where two answers are located on the \(\xi_{123}\) axis line, which is the axis line of sheaf of planes \(\pi_{12},\pi_{13}\) and \(\pi_{23}\) made by the information of three sensing modes 1, 2 and 3. Using the sensing information of a fourth sensing mode helps to distinguish the correct answer. This figure shows that by using the information of a fourth sensing mode, the axis line of four pencils of planes passes through a common point, which is the correct spatio-temporal information of the target object.  
Figure 13 is the extension of Figure 7. Four sensing cones can have four different triple combinations. All four triple combinations of four cones have three intersection curves that meet each other at two different points. Figure 13 shows that all intersection curves meet each other at one unique common point that represents the real spatio-temporal information of the target object. In 3-coverage target tracking, if the axis line of a pencil intersects with the up and down nappes, we can find the correct answer with 100% confidence. Otherwise, when the axis line of a pencil only intersects with  
Figure 12: Triple combination of four different sensing motes’ information yielding four different pencils that pass through a unique common point.  
the down nappes, we need the information of a fourth sensing mote, which means we need to have four-sensing coverage. The information of the fourth sensing mote must not reside on the same plane as the planes of the 3-dimensional reported information of three previous sensing motes.