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The closest neighbor b to any point p is on an edge bp in the Delaunay triangulation since the nearest neighbor graph is a subgraph of the Delaunay triangulation. The Delaunay triangulation is a geometric spanner : In the plane ( d = 2 ), the shortest path between two vertices, along Delaunay edges, is known to be no longer than 1.998 times the ...
The dual graph for a Voronoi diagram (in the case of a Euclidean space with point sites) corresponds to the Delaunay triangulation for the same set of points. The closest pair of points corresponds to two adjacent cells in the Voronoi diagram.
In computational geometry, the Bowyer–Watson algorithm is a method for computing the Delaunay triangulation of a finite set of points in any number of dimensions. The algorithm can be also used to obtain a Voronoi diagram of the points, which is the dual graph of the Delaunay triangulation.
The Delaunay triangulation, which has an edge between any pair of points whenever there exists an empty circle having the pair as a chord. The Urquhart graph, formed from the Delaunay triangulation by removing the longest edge of each triangle. For each remaining edge, the vertices of the Delaunay triangles that use that edge cannot lie within ...
The Delaunay triangulation of a set of points in the plane contains the Gabriel graph, the nearest neighbor graph and the minimal spanning tree of . Triangulations have a number of applications, and there is an interest to find the "good" triangulations of a given point set under some criteria as, for instance minimum-weight triangulations .
A triangulation is a planar straight line graph to which no more edges may be added, so called because every face is necessarily a triangle; a special case of this is the Delaunay triangulation, a graph defined from a set of points in the plane by connecting two points with an edge whenever there exists a circle containing only those two points.
The input to the constrained Delaunay triangulation problem is a planar straight-line graph, a set of points and non-crossing line segments in the plane.The constrained Delaunay triangulation of this input is a triangulation of its convex hull, including all of the input segments as edges, and using only the vertices of the input.
Each edge or triangle of the Delaunay triangulation may be associated with a characteristic radius: the radius of the smallest empty circle containing the edge or triangle. For each real number α , the α -complex of the given set of points is the simplicial complex formed by the set of edges and triangles whose radii are at most 1/ α .