# An Algorithm to Find the Shortest Path through Obstacles of Arbitrary Shapes and Positions in 2D
## 6 Results  
We now present examples of the shortest paths generation by our method for some real-world digital elevation maps (DEM), and for one artificial landscape with forbidden zones. These maps represent various regions of the world that were used as examples in [1]. The altitude of the UAV at which the trajectories lie has been set to some plausible values. The paths obtained are shown in Figure 12. The computations for each example were done with our implementation of the algorithm with the software Matlab 2020b and run on a portable Omen laptop from HP with an Intel Core i7-10750H CPU @ 2.60 GHz. Details on this CPU are given in Appendix A. The computation times reported are the averages of 10 runs of the same calculation of the solution.  
* Artificial canyon  
The map of Figure 18a represents a fictitious terrain. Its dimensions are 500 \(\times\) 500 pixels, and the size of the terrain is 5 km \(\times\) 5 km. To this terrain, we have added four circular no-fly zones. The landscape shown corresponds to a plane at an altitude of 100 m. The starting position is \([10,250]^{\mathrm{T}}\) and the target position is \([400,250]^{\mathrm{T}}\). The length of the shortest path is 4799.2 m. The calculation requires 0.1125 s.  
* Austrian Alps  
The map in Figure 18b represents the terrain in the Austrian Alps, with the town of Imst to the west and Telf to the east. It has 385 \(\times\) 500 pixels and represents a region of 30 km \(\times\) 39 km at an altitude of 1700 m. \(S\) = \([25,40]^{\mathrm{T}}\) and \(T\) = \([350,90]^{\mathrm{T}}\). Note that the path correctly goes through the passage at the start of the trajectory, in the upper left hand side corner of the map. The length of the shortest path is 36,825 m. The calculation requires 0.1732 s.  
* Canadian Rockies  
The map in Figure 18c represents the terrain through which flows the Fraser River near Lillooet, BC, Canada. It has 419 \(\times\) 500 pixels that correspond to a region of 8.7 km \(\times\) 10.4 km. The path is in a plane at 450 m of altitude with \(S\) = \([50,400]^{\mathrm{T}}\) and \(T\) = \([350,150]^{\mathrm{T}}\). The length of the shortest path is 12,354 m. The calculation requires 0.0793 s.  
* Norilsk, Russia  
The map in Figure 18d represents the terrain east of Norilsk in Russia. It has 416 \(\times\) 500 pixels that correspond to a region of 24.4 km \(\times\) 29.3 km. The path is in a plane at 600 m of altitude with \(S\) = \([175,75]^{\mathrm{T}}\) and \(T\) = \([375,325]^{\mathrm{T}}\). The length of the shortest path is 27,265 m. The calculation requires 0.0621 s.  
* Lima, Peru  
The map in Figure 18e represents a region around Lima, Peru. It has 380 \(\times\) 500 pixels that correspond to a terrain of 15.5 km \(\times\) 20.4 km. The path is in a plane at 500 m of altitude, with \(S\) = \([195,45]^{\mathrm{T}}\) and \(T\) = \([50,470]^{\mathrm{T}}\). The length of the shortest path is 25,061 m. The calculation requires 0.5501 s.  
* Grand Canyon, USA  
The map in Figure 18f represents a region around the Grand Canyon in Colorado, USA. It has 332 \(\times\) 500 pixels that correspond to a terrain of 4.9 km \(\times\) 7.3 km. The path is in a plane at 100 m of altitude with \(S\) = \([190,495]^{\mathrm{T}}\) and \(T\) = \([190,75]^{\mathrm{T}}\). The length of the shortest path is 9416 m. The calculation requires 0.1141 s.