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GraphTheory[GeometricGraphs]

  

DelaunayGraph

  

construct Delaunay graph

 

Calling Sequence

Parameters

Options

Description

Definitions

Examples

References

Compatibility

Calling Sequence

DelaunayGraph(P, opts)

Parameters

P

-

Matrix or list of lists representing set of points

opts

-

(optional) one or more options as specified below

Options

• 

triangulation : list of three-element lists.

  

Supply a previously computed Delaunay triangulation of P. The input must be a valid Delaunay triangulation in the format returned by ComputationalGeometry[DelaunayTriangulation]: a list of three-element lists of integers, representing triangles in a triangulation of P.

• 

vertices : list of integers, strings or symbols

  

Specifies the vertices to be used in the generated graph.

• 

weighted : true or false

  

If weighted=true, the result is a weighted graph whose edge weights correspond to the Euclidean distance between points. The default is false.

Description

• 

The DelaunayGraph(P, opts) command returns a Delaunay graph for the point set P.

• 

The parameter P must be a Matrix or list of lists representing a set of points.

Definitions

• 

Let P be a set of points in n dimensions.

• 

A Delaunay graph is a undirected graph corresponding to a Delaunay triangulation on P. Its vertices correspond to the points in P and an edge exists between points p and q from P whenever p and q form the corners of a triangle in a Delaunay triangulation.

• 

The Delaunay graph is an example of geometric spanner, a graph for which the weighted graph distance connecting any two vertices p and q is a bounded multiple of the Euclidean distance between p and q.

• 

The Gabriel graph on P is a subgraph of the Delaunay graph on P.

Examples

Generate a set of random two-dimensional points and draw the Delaunay graph.

withGraphTheory:

withGeometricGraphs:

pointsLinearAlgebra:-RandomMatrix60,2,generator=0..100.,datatype=float8

(1)

DGDelaunayGraphpoints

DGGraph 1: an undirected unweighted graph with 60 vertices and 167 edge(s)

(2)

DrawGraphDG

References

  

Chew, L. Paul (1986), "There is a planar graph almost as good as the complete graph", Proc. 2nd Annual Symposium on Computational Geometry, pp. 169–177. doi:10.1145/10515.10534

Compatibility

• 

The GraphTheory[GeometricGraphs][DelaunayGraph] command was introduced in Maple 2020.

• 

For more information on Maple 2020 changes, see Updates in Maple 2020.

See Also

DelaunayTriangulation

GeometricGraphs