# lawSummary: Produce a Lewis-Aboav-Weaire summary of a tessellation. In deldir: Delaunay Triangulation and Dirichlet (Voronoi) Tessellation

## Description

Produce a summary of a Direchlet (Voronoi) tessellation in terms of parameters relevant to Lewis's law and Aboav-Weaire's law. Note that “law” in the function name corresponds to “Lewis-Aboav-Weaire”.

## Usage

 `1` ```lawSummary(object) ```

## Arguments

 `object` An object of class `"deldir"` as returned by the function `deldir()`.

## Details

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 Lewis-Aboav-Weaire 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).

## Value

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.

## Author(s)

Rolf Turner r.turner@auckland.ac.nz

`tile.list()` `tile.centroids()`
 ``` 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")) ```