Voronoi Diagrams and Triangular Graphs in MATLAB: Implementation and Applications

In MATLAB, Voronoi diagrams and triangulation are fundamental tools in computational geometry, widely used for spatial partitioning, path planning, and data visualization. A Voronoi diagram divides a plane into multiple regions where each region contains all points closest to a specific seed point, while triangulation connects a set of points into non-overlapping triangular meshes.

MATLAB provides built-in functions like `voronoi` and `delaunay` for rapid generation of these structures. The `voronoi(x,y)` function creates Voronoi diagrams from point coordinates, ideal for base station coverage simulation or nearest-neighbor queries. The `delaunay(x,y)` function performs Delaunay triangulation, commonly used in finite element analysis or surface reconstruction. For advanced applications, the `voronoiDiagram` and `delaunayTriangulation` classes offer finer control, enabling vertex information extraction or constrained boundary insertion through methods like `vertices` and `edges` properties.

The core distinction lies in their geometric properties: Voronoi diagrams represent spatial partitioning "competition" between seed points, while triangulation emphasizes point "connectivity." MATLAB's visualization functions like `triplot` for displaying triangular meshes and interactive tools like `boundary` for convex hull creation significantly enhance result interpretation, helping users intuitively understand complex geometric structures. The underlying Delaunay algorithm ensures optimal triangle shapes by maximizing minimum angles, preventing skinny triangles in the mesh generation process.