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Let be a metric space with distance function .Let be a set of indices and let () be a tuple (indexed collection) of nonempty subsets (the sites) in the space .The Voronoi cell, or Voronoi region, , associated with the site is the set of all points in whose distance to is not greater than their distance to the other sites , where is any index different from .
A Voronoi diagram is a special kind of decomposition of a metric space determined by distances to a specified discrete set of objects in the space, e.g., by a discrete set of points. This diagram is named after Georgy Voronoi, also called a Voronoi tessellation, a Voronoi decomposition, or a Dirichlet tessellation after Peter Gustav Lejeune ...
In mathematics, a weighted Voronoi diagram in n dimensions is a generalization of a Voronoi diagram. The Voronoi cells in a weighted Voronoi diagram are defined in terms of a distance function. The distance function may specify the usual Euclidean distance, or may be some other, special distance function. In weighted Voronoi diagrams, each site ...
For a given set of points in space, a Voronoi diagram is a decomposition of space into cells, one for each given point, so that anywhere in space, the closest given point is inside the cell. This is equivalent to nearest neighbor interpolation, by assigning the function value at the given point to all the points inside the cell. [ 3 ]
As Fortune describes in ref., [1] a modified version of the sweep line algorithm can be used to construct an additively weighted Voronoi diagram, in which the distance to each site is offset by the weight of the site; this may equivalently be viewed as a Voronoi diagram of a set of disks, centered at the sites with radius equal to the weight of the site. the algorithm is found to have ...
Example Here x is the positive pole of V p and y its negative. As the cell corresponding to q is unbounded, only the negative pole z exists.. Let be the Voronoi diagram for a set of sites , and let be the Voronoi cell of corresponding to a site .
Worley noise, also called Voronoi noise and cellular noise, is a noise function introduced by Steven Worley in 1996. Worley noise is an extension of the Voronoi diagram that outputs a real value at a given coordinate that corresponds to the Distance of the nth nearest seed (usually n=1) and the seeds are distributed evenly through the region.
The Delaunay triangulation of a discrete point set P in general position corresponds to the dual graph of the Voronoi diagram for P. The circumcenters of Delaunay triangles are the vertices of the Voronoi diagram. In the 2D case, the Voronoi vertices are connected via edges, that can be derived from adjacency-relationships of the Delaunay ...