Document Type : Research Article

Authors

• Ali Shokri ¹
• Roman Rafig Maharramov ²
• Mutallim Mirzaahmed Mutallimov ^{3, 4, 5}
• Elshan Giyas Hashimov Hashimov ⁴
• lkin Aladdin Maharramov ⁴

¹ Department of Applied Mathematics, Sahand University of Technology, Sahand New-Town, Tabriz, Iran

² Military Scientific-Research Institute, National Defense University of the Ministry of Defense, Baku, Azerbaijan

³ Institute of Applied Mathematics, Baku State University, Baku, Azerbaijan

⁴ Azerbaijan Technical University, Baku, Azerbaijan

⁵ Institute of Information Technologies, Ministry of Science and Education of the Republic of Azerbaijan, Baku, Azerbijan

Abstract

In this paper, we address the problem of covering a given bounded domain in the plane using simple geometric figures. The proposed approach is based on a discretization of the domain, which leads to a corresponding discrete optimization problem. To solve this problem, we introduce a novel iterative algorithm that minimizes a given objective function by generating successive neighboring nodal points. As the covering elements, circular sectors with centers located outside the domain are considered. The objective is to determine the locations of the sector centers and their radii in such a way that the entire domain is completely covered, while the ratio of the total area of the covering sectors to the area of the domain is minimized. Finally, the algorithm is demonstrated on a representative example, and the resulting coverings are illustrated.

Highlights

• A novel discretization-based framework is developed for covering bounded planar domains using simple geometric figures.
• A new iterative optimization algorithm is proposed that minimizes a covering objective function through successive neighboring nodal point updates.
• Circular sectors with centers located outside the domain are employed as covering elements, enabling flexible and efficient coverage strategies.
• The algorithm optimizes both the locations and radii of sector centers to achieve complete domain coverage while minimizing the total covering area.
• Numerical demonstrations confirm the effectiveness of the method for fixed sector angles and prescribed radii, achieving maximal coverage with a minimal number of devices.
• The proposed approach is applicable to practical scenarios such as multilayer target coverage in military operations and is extensible to higher-dimensional covering problems.

Keywords

• Optimal coverage
• Discretization of the domain
• Discrete optimization

Main Subjects

• Linear and Non-Linear Programming