Delaunay Triangulation
Delaunay triangulation connects a set of points into a network of non-overlapping triangles such that no point lies inside the circumscribed circle of any triangle. It is the mathematical dual of the Voronoi diagram and is fundamental to surface modeling, mesh generation, and spatial interpolation.
Delaunay triangulation, named after Boris Delaunay who formalized it in 1934, is a method of connecting a set of discrete points in the plane into a triangulated mesh that maximizes the minimum angle of all triangles. This property avoids excessively thin, elongated triangles, producing a well-conditioned mesh that is ideal for interpolation and surface modeling.
Construction and Properties
The Delaunay condition requires that the circumscribed circle (circumcircle) of every triangle contains no other points from the dataset. This criterion yields a unique triangulation (assuming no four points are cocircular) that has several desirable geometric properties: it maximizes the minimum angle, it is the dual graph of the Voronoi diagramVoronoi DiagramA Voronoi diagram is a geometric partition of space into regions based on distance to a set of generating points, whe..., and it contains the nearest neighbor graph and the minimum spanning tree as subgraphs. Common algorithms include incremental insertion, divide-and-conquer, and Fortune's sweep line, all achieving O(n log n) time complexity.
Applications
In GISGISGeographic Information Systems (GIS) enable users to analyze and visualize spatial data to uncover patterns, relation..., Delaunay triangulation forms the basis of Triangulated Irregular Networks (TINs) used for terrain surface modeling. Finite element analysis uses Delaunay meshes to discretize spatial domains for engineering simulations. Computer graphics employs it for mesh generation and texture mapping. Spatial interpolationSpatial InterpolationSpatial interpolation estimates unknown values at unsampled locations based on known values at measured points, creat... methods such as natural neighbor interpolation rely on the Delaunay structure to compute weighted averages. Navigation and robotics use Delaunay edges to construct path networks.
Relationship to Voronoi Diagrams
The Delaunay triangulation and Voronoi diagramVoronoi DiagramA Voronoi diagram is a geometric partition of space into regions based on distance to a set of generating points, whe... are dual structures: connecting the generating points of adjacent Voronoi cells produces the Delaunay triangulation, and the circumcenters of Delaunay triangles are the Voronoi vertices. This duality means that computing one structure efficiently yields the other, providing both proximity partitioning and optimal triangulation from a single computation.
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