Thiessen Polygons

Thiessen polygons, named after American meteorologist Alfred H. Thiessen, are a method of spatial tessellation that partitions a geographic area into non-overlapping polygons based on proximity to a set of discrete input points. Each polygon encloses all locations that are closer to its generating point than to any other point in the dataset. Also known as Voronoi diagrams (after mathematician Georgy Voronoi), this geometric construction is one of the most fundamental spatial structures in computational geometry and finds wide application in GISGISGeographic Information Systems (GIS) enable users to analyze and visualize spatial data to uncover patterns, relation... analysis.

Thiessen polygon construction begins with a set of input points distributed across a study area. For each pair of adjacent points, the perpendicular bisector of the line segment connecting them is calculated. The intersection of these bisectors defines the polygon boundaries, with each boundary segment equidistant between two points. The resulting tessellation covers the entire study area with non-overlapping, contiguous polygons. In computational geometry, the Voronoi diagramVoronoi DiagramA Voronoi diagram is a geometric partition of space into regions based on distance to a set of generating points, whe... is the dual of the Delaunay triangulationDelaunay TriangulationDelaunay triangulation connects a set of points into a network of non-overlapping triangles such that no point lies i...: connecting the generating points of adjacent Voronoi cells produces the Delaunay triangle network. GISGISGeographic Information Systems (GIS) enable users to analyze and visualize spatial data to uncover patterns, relation... software typically computes Thiessen polygons using optimized algorithms based on Fortune's sweep line method or incremental insertion.

Thiessen polygons serve diverse analytical purposes in geospatial science. Meteorology and hydrology assign rainfall measurements from weather stations to surrounding areas using Thiessen polygons, calculating area-weighted average precipitation for watersheds. This Thiessen polygon method remains a standard technique in hydrological analysis despite the availability of more sophisticated interpolation methods. Service area delineation uses Thiessen polygons to define the territory closest to each facility, such as assigning households to the nearest fire station, school, or post office. Market analysis partitions geographic markets based on proximity to retail locations, creating trade areas that approximate natural customer catchments. Ecology uses Thiessen polygons to define territories around nesting sites or resource points. Telecommunications assigns service coverage areas to cell towers based on proximity before detailed propagation modeling.

Thiessen polygons provide a mathematically rigorous and unique partition of space based on proximity, ensuring that every location is assigned to exactly one polygon. The method requires only point locations as input, with no additional parameters or assumptions about data distribution. The resulting polygons have useful geometric properties: every point within a polygon is closer to its generating point than to any other. Thiessen polygons are computationally efficient to generate and provide an intuitive visual representation of proximity relationships. They serve as the geometric foundation for natural neighbor interpolation and other spatial analysis methods.

Thiessen polygons assume isotropic travel, meaning that proximity is measured as straight-line distance, which may not reflect real-world accessibility affected by roads, barriers, or terrain. The tessellation is extremely sensitive to the input point configuration; adding or removing a single point can dramatically change polygon boundaries. Edge polygons extend to the boundary of the study area and may be unrealistically large for peripheral points. Thiessen polygons assign all area to the nearest single point, which may oversimplify situations where multiple facilities serve overlapping areas.

Weighted Voronoi diagrams incorporate facility capacity or attractiveness to create more realistic service area models. Network-based Voronoi diagrams calculate proximity along road networks rather than Euclidean distanceEuclidean DistanceEuclidean distance is the straight-line distance between two points in a plane, computed using the Pythagorean theore.... Adaptive Voronoi tessellation dynamically updates as point locations change, supporting real-time logistics applications. Integration with agent-based models enables simulation of customer behavior within Voronoi-defined service territories.