Algorithm for Computing Medial Axis of Arbitrary Simple Polygons

The medial axis (also known as the skeleton or skeleton line) of an arbitrary simple polygon is a classical problem in computational geometry. Its core objective is to find the set of all points inside the polygon that are equidistant to the boundary. The medial axis has wide applications in polygon simplification, path planning, shape analysis, and other fields. Below is an analysis of the algorithm concepts and key MATLAB implementation points.

### Algorithm Approaches Voronoi Diagram Method: The medial axis of a polygon can be approximated as a subset of its internal Voronoi diagram. Specific steps include: Sampling polygon vertices and edges into discrete point sets. Computing the Voronoi diagram for these points. Filtering Voronoi edges located inside the polygon to form the medial axis.

Distance Transform Method: Binarize the polygon (interior as 1, exterior as 0). Apply distance transformation to calculate the shortest distance from each interior point to the boundary. Extract local distance maxima points and connect them to form the medial axis.

### Key MATLAB Implementation Points Input Processing: Polygon vertices must be arranged in clockwise or counterclockwise order to ensure a simple polygon (no self-intersections). Use the `polyshape` object to validate polygon legality and handle vertex ordering automatically.

Voronoi Diagram Generation: Utilize the `voronoi` function to generate Voronoi diagrams for discrete points, combined with `inpolygon` for filtering internal edges. For better precision, consider using `voronoin` which returns Voronoi vertices and regions for more controlled processing.

Optimization and Post-processing: Remove short branches or noise based on length thresholds using morphological operations or custom filtering. Smooth the medial axis curve using interpolation methods like `spline` or `pchip` for better visual results.

### Extended Considerations Complex Polygons: For polygons containing holes, additional processing is required to account for hole boundary influences. This may involve separate Voronoi computations for outer and inner boundaries. Performance Optimization: For large-scale polygons, incorporate spatial partitioning techniques (such as quadtrees) to accelerate computation. The `boundary` function can help extract simplified boundaries before processing.

This algorithm relies on MATLAB's Computational Geometry toolbox, with the core concept being the extraction of topological skeletons through geometric transformations. Practical applications require balancing precision and efficiency - for instance, sampling density directly affects the smoothness of the medial axis. Consider using `pdist2` for efficient distance calculations when implementing the distance transform method.