spatstat.geom (version 3.3-3)

delaunayDistance: Distance on Delaunay Triangulation

Description

Computes the graph distance in the Delaunay triangulation of a point pattern.

Usage

delaunayDistance(X)

Value

A symmetric square matrix with non-negative integer entries.

Arguments

X

Spatial point pattern (object of class "ppp").

Definition of neighbours

Note that dirichlet(X) restricts the Dirichlet tessellation to the window containing X, whereas dirichletDistance uses the Dirichlet tessellation over the entire two-dimensional plane. Some points may be Delaunay neighbours according to delaunayDistance(X) although the corresponding tiles of dirichlet(X) do not share a boundary inside Window(X).

Author

Adrian Baddeley , Rolf Turner and Ege Rubak .

Details

The Delaunay triangulation of a spatial point pattern X is defined as follows. First the Dirichlet/Voronoi tessellation based on X is computed; see dirichlet. This tessellation is extended to cover the entire two-dimensional plane. Then two points of X are defined to be Delaunay neighbours if their Dirichlet/Voronoi tiles share a common boundary. Every pair of Delaunay neighbours is joined by a straight line to make the Delaunay triangulation.

The graph distance in the Delaunay triangulation between two points X[i] and X[j] is the minimum number of edges of the Delaunay triangulation that must be traversed to go from X[i] to X[j]. Two points have graph distance 1 if they are immediate neighbours.

This command returns a matrix D such that D[i,j] is the graph distance between X[i] and X[j].

See Also

delaunay, delaunayNetwork.

Examples

  X <- runifrect(20)
  M <- delaunayDistance(X)
  plot(delaunay(X), lty=3)
  text(X, labels=M[1, ], cex=2)