# A Study on the Normalized Delineation of Airspace Sectors Based on Flight Conflict Dynamics
## 2 Flight Conflict Network
### Determine Network Edges  
In the flight conflict network, the connecting edges between nodes represent potential conflicts. Therefore, two conditions need to be satisfied for nodes in a flight conflict network to constitute edges: firstly, the corresponding aircraft of the nodes are relatively close to each other in terms of their positions; secondly, the speed barrier relationship is satisfied by the determination of the speed barrier model.  
Aircraft are required to maintain a certain vertical or horizontal separation from each other during operation, and an ellipsoidal flight protection zone is established for airborne targets, as shown in Figure 1, with the equatorial radius of the zone \(\mathtt{a}=\mathtt{b}=d_{l}\), and the polar radius \(\mathtt{c}=d_{v}\).  
Let the coordinates of aircraft B be a. The expression for the ellipsoid surface of the protected area is:  
\[\frac{(x-X_{\mathrm{B}})^{2}}{d_{l}{}^{2}}+\frac{(y-Y_{\mathrm{B}})^{2}}{d_{l} {}^{2}}+\frac{(z-Z_{\mathrm{B}})^{2}}{d_{\mathtt{v}}{}^{2}}=1 \tag{1}\]  
No conflict exists when the vertical separation between aircraft is greater than 300 m, so \(d_{l}=10\) km, \(d_{v}=300\) m is taken.  
The velocity barrier method limits the relative velocity range of potential conflict of the target in the two-dimensional plane according to the geometric rules, and judges that it will be in conflict when its velocity relative to the obstacle is in the range. Since the airspace in which the aircraft operates is a three-dimensional space, the velocity barrier method is extended to three dimensions and the protected area is ellipsoidal in general. As shown in Figure 2, the relative velocity \(v_{r}=v_{\mathrm{A}}-v_{\mathrm{B}}\) can be regarded as the velocity of aircraft A relative to B when aircraft B is at rest. When the \(v_{r}\) direction is within the angle range of the three-dimensional velocity barrier cone, it means that if the current state of motion is not deployed, aircraft A will enter the flight protection zone of B and a flight conflict will occur.  
Let the three-dimensional coordinates of aircraft A be \((X_{\mathrm{A}},Y_{\mathrm{A}},Z_{\mathrm{A}})\), and the direction vector of \(v_{r}\) be \((v_{x},v_{y},v_{z})\). Then, according to the pointwise equation of the three-dimensional straight line, we have:  
\[\frac{x-X_{\mathrm{A}}}{v_{x}}=\frac{y-Y_{\mathrm{A}}}{v_{y}}=\frac{z-Z_{ \mathrm{A}}}{v_{z}} \tag{2}\]  
where \((X_{\mathrm{A}},Y_{\mathrm{A}},Z_{\mathrm{A}})\) denotes the coordinates of aircraft A; \((v_{x},v_{y},v_{z})\) is the direction vector of \(v_{r}\). To associate the surface aspect (1) with the linear Equation (2), if there are two solutions, it means that there are two intersections, C1 and C2, between the flight protection  
Figure 1: Ellipsoidal flight protection zone.  
zone of aircraft B and the line where \(\mathbf{v}_{r}\) is located; at the same time, if there is \(\cos\gamma>0\) that holds, i.e., \(\gamma<90^{\circ}\), then the direction of \(v_{r}\) is inside the cone of the velocity barrier, and the nodes of the aircraft, A and B, form a connecting edge between each other.  
\[\cos\gamma=\cos\left<\overrightarrow{v_{r}},\mathbf{AB}\right>=\frac{\overrightarrow {v_{r}}\bullet AB}{\left|\overrightarrow{v_{r}}\right|\left|AB\right|} \tag{3}\]  
\(\gamma\) is the angle between the relative velocity \(v_{r}\) and the position vectors AB of aircraft A and B; \(\left|\text{AB}\right|\) denotes the distance between A and B.