Voronoi Diagram and Triangulation Implementation in MATLAB

Voronoi diagrams and triangulation graphs represent two fundamental computational geometry tools extensively used in mathematics and computer science applications. The Voronoi diagram partitions a plane into distinct regions (Voronoi cells) based on the distance to a specified set of seed points, where each region contains all points closer to its corresponding seed than to any other. This structure finds significant applications in computer graphics for terrain generation, spatial analysis, and nearest-neighbor computations. In MATLAB, the voronoi() function generates Voronoi diagrams from point sets, while voronoin() handles n-dimensional cases, with options to customize diagram properties through parameters like line styles and point markers. Triangulation graphs, particularly Delaunay triangulation, form by connecting points to create triangles where no point lies inside the circumcircle of any triangle. This dual relationship with Voronoi diagrams makes triangulation essential for mesh generation, surface reconstruction, and network analysis. MATLAB provides robust triangulation support through functions like delaunay() for 2D/3D triangulation, delaunayTriangulation() for object-oriented manipulation, and triplot() for visualization. The triangulation can model network connectivity by representing nodes as triangle vertices and connections as edges. MATLAB's computational geometry toolbox offers extensive libraries including boundary tracing (boundary()), convex hull calculations (convhull()), and interpolation techniques (scatteredInterpolant()) that complement these structures. Implementation typically involves: 1) Data preprocessing using unique() to remove duplicates, 2) Core computation with voronoi()/delaunay() functions, 3) Visualization customization through ColorMap adjustments and alpha channel settings, and 4) Geometric analysis using regionprops() for area calculations or inpolygon() for point containment checks. These tools enable researchers to solve complex spatial problems efficiently, enhancing work accuracy through algorithmically optimized distance calculations and topology-preserving transformations.