Spatial Poisson Point Process Plotting with Voronoi Diagrams

Spatial Poisson point process is a common random point generation method typically used to simulate random distribution phenomena in space. In MATLAB, we can implement this process using built-in functions or simple code, and further plot Voronoi diagrams to analyze the spatial distribution characteristics of these random points.

### Spatial Poisson Point Process The core of spatial Poisson point process involves randomly generating point locations that satisfy uniform distribution or specific density distributions. In MATLAB, uniform random points can be generated using the `rand` function, while more complex point distributions can be created using advanced toolboxes like the Statistics and Machine Learning Toolbox. For Poisson-distributed points, the `poissrnd` function generates random numbers from a Poisson distribution, which can be combined with spatial coordinates to create clustered point patterns.

### Voronoi Diagrams Voronoi diagrams partition space into multiple regions where each region contains one generating point, and any point within that region is closer to its generating point than to any other. In MATLAB, the `voronoi` function quickly visualizes Voronoi diagrams, while `voronoin` computes geometric properties of Voronoi cells. The algorithm works by first creating a Delaunay triangulation using `delaunayTriangulation`, then computing perpendicular bisectors between adjacent points to form Voronoi boundaries.

### Implementation Approach Random point generation: Use `rand(n,2)` to generate n uniform random points in [0,1]×[0,1] space, or `poissrnd(lambda)` combined with spatial scaling for Poisson-distributed densities. Voronoi diagram plotting: Visualize using `voronoi(x,y)` where x and y are coordinate vectors, or perform detailed geometric analysis with `voronoin([x,y])` which returns vertices and cells information. Optimization and adjustment: Modify point density parameters or distribution patterns to observe Voronoi cell changes, enabling exploration of spatial partitioning characteristics through parameter sweeps and statistical analysis of cell areas.

This methodology finds extensive applications in geographic information systems, wireless network base station planning, ecology, and other fields where analyzing spatial coverage and neighborhood relationships of random point sets is crucial for optimization and pattern recognition.