Description Usage Arguments Details Value Author(s) See Also Examples
Produce a summary of a Direchlet (Voronoi) tessellation in terms of parameters relevant to Lewis's law and AboavWeaire's law. Note that “law” in the function name corresponds to “LewisAboavWeaire”.
1  lawSummary(object)

object 
An object of class 
Tiles are stripped away from the tessellation in “layers”.
Layer 1 consists of “boundary” tiles, i.e. tiles having
at least one vertex on the enclosing rectangle (determined by the
rw
argument of deldir()
). Layer 2 consists
of tiles which are neighbours of tiles in layer 1 (i.e. tiles
determined by points that are Delaunay neighbours of points
determining the tiles in layer 1). Layer 3 consists of tiles
which are neighbours of tiles in layer 2.
The parameters of interest in respect of the LewisAboavWeaire summary are then calculated in terms of the tiles that remain after the three layers have been stripped away, which will be referred to as “interior” tiles. These parameters are:
the areas of each of the interior tiles
the number of edges of each of the interior tiles
the number of edges of all neighbouring tiles of each of the interior tiles.
Note that the neighbouring tiles of the interior tiles may include tiles which are not themselves interior tiles (i.e. tiles which are in layer 3).
This function was created at the request of Kai Xu (Fisheries College, Jimei University, Xiamen, Fujian, China 361021).
If no tiles remain after the three layers have been stripped
away, then the returned value is NULL
. Otherwise the
returned value is a list with components calculated in terms of
the remaining (“interior”) tiles. These components are:
tile.vertices
A list whose entries are data frames
giving the coorinates of the vertices of the interior tiles.
tile.areas
A vector of the areas of the interior tiles
in the tessellation in question.
num.edges
A vector of the number of edges of each
such tile.
num.nbr.edges
A list with a component for each
point, in the set being tessellated, whose corresponding tile
is an interior tile. Each component of this
list is the vector of the number of edges of the interior tiles
determined by points which are Delaunay neighbours of
the point corresponding to the list component in question.
totnum.nbr.edges
A vector whose entries consist
of the sums of the vectors in the foregoing list.
The returned list also has attributes as follows:
i1
An integer vector whose entries are in the indices
of the tiles in layer 1.
i2
An integer vector whose entries are in the indices
of the tiles in layer 2.
i3
An integer vector whose entries are in the indices
of the tiles in layer 3.
i.kept
An integer vector whose entries are in the indices
of the tiles that are kept, i.e. those that remain after the three layers
have been stripped away.
Rolf Turner r.turner@auckland.ac.nz
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35  # A random pattern:
set.seed(42)
xy1 < data.frame(x=runif(400,0,20),y=runif(400,0,20))
dxy1 < deldir(xy1)
ldxy1 < lawSummary(dxy1)
tl1 < tile.list(dxy1)
plot(0,0,type="n",xlim=c(2,35),ylim=c(0,20),asp=1,xlab="x",ylab="y",bty="l")
plot(tl1,showpoints=FALSE,add=TRUE)
points(xy1[attr(ldxy1,"i1"),],pch=20,col="yellow")
points(xy1[attr(ldxy1,"i2"),],pch=20,col="blue")
points(xy1[attr(ldxy1,"i3"),],pch=20,col="green")
points(xy1[attr(ldxy1,"i.kept"),],pch=20,col="red")
legend("right",pch=20,col=c("yellow","blue","green","red"),
legend=c("layer 1","layer 2","layer 3","interior"))
# A highly structured pattern (example due to Kai Xu):
set.seed(115)
x < c(rep(1:20,10),rep((1:20)+0.5,10))
y < c(rep(1:10,each=20),rep((1:10)+0.5,each=20))*sqrt(3)
a < runif(400,0,2*pi)
b < runif(400,1,1)
x < x+0.1*cos(a)*b
y < y+0.1*sin(a)*b
xy2 < data.frame(x,y)
dxy2 < deldir(xy2)
ldxy2 < lawSummary(dxy2)
tl2 < tile.list(dxy2)
plot(0,0,type="n",xlim=c(2,35),ylim=c(0,20),asp=1,xlab="x",ylab="y",bty="l")
plot(tl2,showpoints=FALSE,add=TRUE)
points(xy2[attr(ldxy2,"i1"),],pch=20,col="yellow")
points(xy2[attr(ldxy2,"i2"),],pch=20,col="blue")
points(xy2[attr(ldxy2,"i3"),],pch=20,col="green")
points(xy2[attr(ldxy2,"i.kept"),],pch=20,col="red")
legend("right",pch=20,col=c("yellow","blue","green","red"),
legend=c("layer 1","layer 2","layer 3","interior"))

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