# Aerial Surveillance Leveraging Delaunay Triangulation and Multiple-UAV Imaging Systems
## 2 Materials and Methods  
### Delaunay Triangulation  
One of the well-known problems in computational geometry involves computing the Delaunay triangulation of a given set of points. This mathematical challenge was first explored by the French mathematician [PERSON] (or [PERSON]) in 1934 [15]. Since its inception, numerous algorithms have been developed by scientists to address this problem. Notably, in 1985, [PERSON] and [PERSON] [16] introduced a Divide and Conquer algorithm that demonstrated improved performance, achieving the optimal bound of O (n log n). This means that the running time of the algorithm grows logarithmically with the size of the input, represented by \"n\". Another widely employed method is the Incremental Construction algorithm, which gradually builds the triangulation by adding points one at a time, adjusting the existing triangulation accordingly. Bowyer-Watson and incremental insertion algorithms are examples of this category [17]. Moreover, advancements in randomized algorithms, such as the randomized incremental construction, have brought about probabilistic techniques to efficiently compute Delaunay triangulation. Each algorithm presents unique trade-offs in terms of computational complexity, memory usage, and applicability to specific scenarios, catering to the diverse needs of computational geometry practitioners and researchers.  
DT App  
The Android application \"Drones DT\", which was developed as a part of this research initiative using Kotlin and JAVA in Android Studio, is a notable accomplishment. This app features the implementation of the Divide and Conquer algorithm for Delaunay triangulation, demonstrating an optimal time complexity of O(n log n). The strategic use of the algorithm enhances the app's efficiency, particularly when dealing with large datasets. Through the fusion of Kotlin and JAVA, the app offers a user-friendly experience, making it a valuable tool for those involved in tasks related to point sets and triangulation, especially in the context of aerial surveillance. The application interface and screens are shown in Figure 1 below. The successful development of \"Drones DT\" not only showcases technological prowess but also contributes to the advancement of mobile applications in the field of computational geometry.  
The DT Drones application as shown above streamlines the process of leveraging Delaunay triangulation for effective aerial surveillance. The user-friendly interface guides users through three simple steps. First, by tapping to add drones, users can seamlessly input any number of drones they have at their disposal. Following this, tapping to add  
Figure 1: The screens and interface of DT Drones application. The points in red color indicates the drones and the point in green shows the target.  
a target location provides a straightforward way to specify the area of interest. The final step involves pressing the DT icon to initiate Delaunay triangulation calculations. The application then displays the triangulated result of the target locations, along with suggested locations for the drones. This intuitive and step-by-step approach ensures ease of use, allowing users to efficiently harness the power of Delaunay triangulation for optimal drone deployment in the context of surveillance and reconnaissance. The flowchart of the (Drones DT) application is shown in Appendix A.