## 4 Numerical Results  
In Theorem 3.4 we have shown that for the same sample set, IRA will perform no worse than the resolution complete method from [15], but at the cost of solving a larger ATSP problem when there exist samples that are contained in multiple regions. In this section, we use Monte Carlo simulation to investigate the level of performance improvement that can be gained as well as the degree of increase in the size of the resulting ATSP by using IRA compared with the RCM.  
**Figure 6.** Simulation results for \(100\) Monte Carlo trials where both IRA and RCM optimized over the same \(50\) sample poses. (**a**) The color represents the average of the ratio of the tour length under IRA to the tour length under the RCM planning algorithm. Here the red regions indicate near parity in performance while the blue regions indicate that IRA produced tours that are approximately half the length of tours produced by the RCM algorithm; (**b**) The color represents the average of the ratio of the size of the ATSP solved under IRA to the size of the ATSP solved under the RCM planning algorithm. Here the blue regions indicate near parity in size while the red regions indicate that IRA increased the size of the ATSP by as much as four times.  
The centers of circular regions of variable but homogeneous diameters are randomly placed in a square of variable side length. By varying both the size of regions and the area in which the centers of the regions are confined we are able to vary the degree of overlap. The turning radius of the UAV \(\rho\) is set to unit radius. To solve for the tours we used the symmetric TSP solver _linkern_ available at [23], which uses the Chained Lin-Kernighan Heuristic from [24]. The radii of the circular regions were varied over \(\{0.5,0.75,1.00,\ldots,5.5\}\), and the length of the sides of the square were varied over \(\{5,5.5,\ldots,15\}\).  
For the first test, we ran \(100\) trials where \(10\) regions were randomly placed in the bounding box and \(50\) samples were drawn from the boundaries of the regions. The results can be seen in Figure 6, where the average ratio of the length of the tours found by the IRA to those found by RCM are displayed for each test configuration (Figure 6(a)) as well as the average ratio of the size of the resulting ATSP (Figure 6(b)). In a second test, we repeated the same test parameters where IRA optimized over \(50\) sample poses, but allowed the RCM to optimize over the same \(50\) samples plus an additional sample for each duplicated node in the IRA. These extra samples ensured that both algorithms solved the same size ATSP. The results can be seen in Figure 7, where the average ratio of the length of the tours found by the IRA to those found by RCM are displayed for each test configuration. In both instances, it is clear that for small regions and large bounding box (bottom right of plots), there is little to no overlap, and the two algorithms perform equivalently. The tests of interest are when the regions grow, and the bounding area shrinks (moving from bottom right to top left). For these cases we see that on average, IRA finds tours that are nearly half the length of RCM. It should be noted that as the density increases, there becomes a point where a single sample will be contained in all regions (the top left corner). In this case, RCM would still visit \(n\) samples (one from each region) while IRA would only visit the single sample contained in all regions. In practice there is no need for planning once it is recognized that a single loiter circle will visit all the regions, thus both algorithms were assigned the length of one loiter radius. It should also be noted that the size of the resulting ATSP is increased by only as much as \(4\times\), which is significantly less than the worst case analysis would predict \((10\times)\).