## 2 Preliminary Work
### Mathematical Analysis  
The general approach for calculating the VTW for regional targets involves discretizing the region into several points, computing the VTW for each discrete point at each time step, and then merging these to obtain the VTW for the global regional target. The combined coarse and precise search method is currently a widely used rapid algorithm [25; 35], which involves traversing each time step first with a large step angle calculation and then using a binary precise search to improve computational speed. However, further analysis reveals optimization potential in this method. As illustrated in Figure 3, the dashed lines represent the sub-point track, red star represent ground regions, black rectangular frames represent rectangular regions covering ground regions, and solid lines inside the rectangles represent valid sub-point tracks. When traversing the global time interval with large steps, angle calculations for each discrete point at every time step are required, involving nested traversal and a high number of computations. If this step can be shortened to a smaller time interval, as in the rectangle, reducing the nested traversal to only that interval, it will significantly reduce the number of computations. This involves first calculating the sub-point position using the orbital model, then conducting coarse search nested traversal within a smaller interval, and finally performing the binary search. In the following analysis, this approach is referred to as the \"new approach\", while the coarse and precise search is termed the \"original approach\". The computational complexity analysis of both approaches will be discussed below.  
According to the big O notation for calculating time complexity analysis [36], the computational complexities of the original approach and the new approach are denoted as \(T_{o}(n)\) and \(T_{x}(n)\), respectively, with the computation formulas as shown in Formula (4).  
\[\begin{array}{c}T_{o}(n)=\max(O(n_{a}\cdot m),O(\log n_{c}))\\ T_{x}(n)=\max(O(n_{a}),O(n_{b}\cdot m),O(\log n_{c}))\end{array} \tag{4}\]  
In the former formula, \(O(n_{a}\cdot m)\) represents the computational complexity of the coarse search phase in the original approach, where \(n_{a}\) denotes the number of sampling traversals within the global interval, \(m\) represents the number of discrete points in the polygon, and  
Figure 3: Schematic diagram of orbit filtering principle.  
\(O(log\ n_{c})\) indicates the complexity of the precise search phase within a short time interval, where \(n_{c}\) represents the number of binary searches within the short interval. In the latter formula, \(O(n_{a})\) denotes the complexity of the traversing and computing positions within the calculation interval in the new approach, while \(O(n_{b}\cdot m)\) represents the complexity of the coarse search phase, where \(n_{b}\) denotes the number of traversals within the smaller interval. The complexity of the binary search phase remains the same as before. From the above analysis, it can be inferred that the nested traversal is the part with the higher complexity. In comparison to the original approach, \(n_{b}\) is significantly less than \(n_{a}\), indicating that the coarse search phase in the new approach involves fewer nested traversals and a significant reduction in computation, thereby improving computational speed. The algorithm proposed in this paper is thus based on this mathematical and theoretical foundation, achieving an increase in computational speed by reducing the number of computations.