Besides points, such diagrams use lines and polygons as seeds. By augmenting the diagram with line segments that connect to nearest points on the seeds, a planar subdivision of the environment is obtained. The navigation mesh has been generalized to support 3D multi-layered environments, such as an airport or a multi-storey building. In biology , Voronoi diagrams are used to model a number of different biological structures, including cells [18] and bone microarchitecture. In this usage, they are generally referred to as Thiessen polygons. In ecology , Voronoi diagrams are used to study the growth patterns of forests and forest canopies, and may also be helpful in developing predictive models for forest fires.

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The region of influence is called a Voronoi region and the collection of all the Voronoi regions is the Voronoi diagram. The Voronoi diagram is an N-D geometric construct, but most practical applications are in 2-D and 3-D space. The properties of the Voronoi diagram are best understood using an example. Use the 2-D voronoi function to plot the voronoi diagram for a set of points.

The Voronoi diagram of a set of points X is closely related to the Delaunay triangulation of X. To see this relationship, construct a Delaunay triangulation of the point set X and superimpose the triangulation plot on the Voronoi diagram. Also, the vertices of the Voronoi edges are located at the circumcenters of the Delaunay triangles.

To find the index of this triangle, query the triangulation. The triangle contains the location -1, 0. You can compute the Voronoi diagram from the Delaunay triangulation and vice versa. Observe that the Voronoi regions associated with points on the convex hull are unbounded for example, the Voronoi region associated with X The edges in this region "end" at infinity.

While the Voronoi diagram provides a nearest-neighbor decomposition of the space around each point in the set, it does not directly support nearest-neighbor queries.

However, the geometric constructions used to compute the Voronoi diagram are also used to perform nearest-neighbor searches. In practice, Voronoi computation is not practical in dimensions beyond 6-D for moderate to large data sets, due to the exponential growth in required memory. The voronoi plot function plots the Voronoi diagram for a set of points in 2-D space. The voronoiDiagram method supports computation of the Voronoi topology for discrete points 2-D or 3-D.

The voronoiDiagram method is recommended for 2-D or 3-D topology computations as it is more robust and gives better performance for large data sets. This method supports incremental insertion and removal of points and complementary queries, such as nearest-neighbor point search. The voronoin function and the voronoiDiagram method represent the topology of the Voronoi diagram using a matrix format. See Triangulation Matrix Format for further details on this data structure.

Given a set of points, X, obtain the topology of the Voronoi diagram as follows: Using the voronoin function.

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