# Research on Airspace Conflict Detection Method Based on Spherical Discrete Grid Representation
## 3 Construction of Airspace Grid Model Based on Icosahedron Spherical Rhombus Subdivision  
### Construction of Subdivision Model  
In the quadtree subdivision method based on the spherical diamond grid, the degree of grid deformation is mainly judged by the ratio of the long and short axes of the grid, the ratio of the maximum and minimum area, and the mean square error of the grid perimeter. The ratio of the long and short axes of the subdivided rhombus grid at each level remains basically unchanged at about 1.4, and the ratio of the long and short axes of the rest of the subdivision methods is greater than 2 and will change with the change of the subdivision level. The maximum and minimum area ratio of the icosahedral rhombic grid will not change with the increase of the grid subdivision level, and is constant at 1.3, which is also the best among various subdivision methods [23]. The grid has the advantages of compactness and continuity, and the grid unit has one and only one identification point. It conforms to the ideal grid subdivision evaluation criteria proposed by [PERSON] [24] and is suitable for global multi-level subdivision and spatial coding. According to the geometric properties of the icosahedron, there are three methods of placing the spherical surface. In order to facilitate the simplified transformation from latitude and longitude coordinates to coded coordinates, as well as subsequent grid division and coding, this paper uses the 0-degree meridian as the symmetry axis to place the method. Correspondence between icosahedron and sphere [25]. As shown in Figure 4:For the great circle arc subdivision of the icosahedron, the spherical distance calculation methods include the great circle formula, the simplified great circle formula, and the Haversine formula. Compared with the accuracy of the above calculation methods, the great circle formula has a very serious rounding error under single precision, while the great circle formula The simplified formula of has a large error when the distance between the two poles or two points is large, and the precision of the Haversine formula is similar to that of the Cartesian coordinate system method, both of which are very high [26]. The distance \(D\) between two points on the sphere is obtained from [PERSON]'s formula, where \(\lambda\) denotes longitude, \(\varphi\) denotes latitude, and \(R\) is the radius of the Earth. \[D=2R\text{arcsin}\left(\sqrt{\sin^{2}\left(\frac{\varphi_{2}-\varphi_{1}}{2} \right)+\cos(\varphi_{1})\cos(\varphi_{2})\sin^{2}\left(\frac{\lambda_{2}- \lambda_{1}}{2}\right)}\right) \tag{1}\]  
The coordinates of the subdivision levels and boundary points and the edge lengths of the diamond grid are determined by calculation, and the correspondence between the scales of each level is shown in Table 1.  
The initial 10 rhombuses are at the 0 th level, and the length of a single rhombus edge is about 7061 km according to the spherical arc length calculation formula, and the length of a single rhombus edge is about 7 km when dissecting to the 10 th level, and the minimum distance of non-adjacent rhombus grids is one rhombus grid interval, i.e., the vertical distance from the rhombus vertex to the opposite edge is about 6 km. when CD, the  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Subdivision Level** & **Rhombus Side** & **Number of Global** & **Rhombic Spherical** & **Distance from the Vertex of the Rhombus to the Opposite Side/km** \\  & **Length/km** & **Grids** & **Area/km\({}^{2}\)** & **Distance from the Rhombus to the Opposite Side/km** \\ \hline
0 & 7061 & 10 & 51,006,786 & 6115 \\
1 & 3630 & 40 & 12,751,696 & 3057 \\
2 & 1865 & 160 & 3,187,924 & 1529 \\
3 & 960 & 640 & 796,981 & 765 \\
4 &  
airspace is expanded by one layer, i.e., the interval of two enclosing The minimum distance is 11.94 km, which is greater than 10 km.