Covering Circular Domains with Hexagonal Cells

Covering circular regions with hexagonal cell structures is a classic problem in geometric modeling and wireless communication network design. This hexagonal tiling approach achieves optimal spatial utilization and is commonly used in base station layout planning. Here we explore the core implementation concepts using MATLAB.

The key to hexagonal coverage lies in determining the arrangement pattern. Each regular hexagon can be viewed as composed of 6 equilateral triangles, whose geometric properties ensure the efficiency of cellular networks. Implementation requires focusing on two core parameters: hexagon edge length and circle center coordinates.

The construction process involves three main steps: First, calculate the required number of hexagonal layers based on the circle's radius. Using a radial layering method, expand hexagons outward from the center point layer by layer. Each additional layer increases the hexagon ring's radius by a fixed step size (related to the hexagon's height). In MATLAB, this can be implemented using nested loops with layer indices and angular displacement calculations.

Secondly, boundary conditions must be handled. Since hexagon grid coordinates are calculated based on regular hexagonal geometry, hexagons outside the circular boundary require clipping judgment. By computing the distance from each hexagon's center to the circle's center, we can efficiently filter cellular units that fall completely or partially within the circular domain. This can be implemented using MATLAB's vectorized distance calculations and logical indexing.

Finally, the visualization phase. MATLAB's patch function is ideal for drawing hexagons - by calculating the six vertex coordinates for each hexagon, all cellular units can be generated in batches using loops. To distinguish different layers, gradient colors or varying transparency can enhance visualization. The implementation typically involves creating vertex matrices and using patch's face-vertex data structure for efficient rendering.

In practical applications, this method can be extended to 3D space to form honeycomb column structures, or adapted for multi-scale coverage by adjusting hexagon dimensions. In 5G network planning, similar modeling approaches are frequently used for optimizing base station deployment and signal coverage analysis.