pcds (version 0.1.6)

NumArcsAS: Number of arcs of Arc Slice Proximity Catch Digraphs (AS-PCDs) - multiple triangle case

Description

Returns the number of arcs and various other quantities, vectors, and lists for Arc Slice Proximity Catch Digraph (AS-PCD) whose vertices are the data points in Xp in the multiple triangle case.

AS proximity regions are defined with respect to the Delaunay triangles based on Yp points and vertex regions in each triangle are based on the center M="CC" for circumcenter of each Delaunay triangle or \(M=(\alpha,\beta,\gamma)\) in barycentric coordinates in the interior of each Delaunay triangle; default is M="CC" i.e., circumcenter of each triangle.

Convex hull of Yp is partitioned by the Delaunay triangles based on Yp points (i.e., multiple triangles are the set of these Delaunay triangles whose union constitutes the convex hull of Yp points).

See (ceyhan:Phd-thesis,ceyhan:comp-geo-2010,ceyhan:mcap2012;textualpcds) for more on AS-PCDs. Also see (okabe:2000,ceyhan:comp-geo-2010,sinclair:2016;textualpcds) for more on Delaunay triangulation and the corresponding algorithm.

Usage

NumArcsAS(Xp, Yp, M = "CC")

Value

A list with the elements

num.arcs

Total number of arcs in all triangles, i.e., the number of arcs for the entire AS-PCD

num.in.conhull

Number of Xp points in the convex hull of Yp points

num.in.tris

The vector of number of Xp points in the Delaunay triangles based on Yp points

weight.vec

The vector of the areas of Delaunay triangles based on Yp points

tri.num.arcs

The vector of the number of arcs of the component of the AS-PCD in the Delaunay triangles based on Yp points

del.tri.ind

A matrix of indices of Delaunay triangles based on Yp points, each column corresponds to the vector of indices of the vertices of one of the Delaunay triangle.

data.tri.ind

A vector of indices of vertices of the Delaunay triangles in which data points reside, i.e., column number of del.tri.ind for each Xp point.

Arguments

Xp

A set of 2D points which constitute the vertices of the AS-PCD.

Yp

A set of 2D points which constitute the vertices of the Delaunay triangles.

M

The center of the triangle. "CC" stands for circumcenter of each Delaunay triangle or 3D point in barycentric coordinates which serves as a center in the interior of each Delaunay triangle; default is M="CC" i.e., the circumcenter of each triangle.

Author

Elvan Ceyhan

References

See Also

NumArcsAStri, NumArcsPE, and NumArcsCS

Examples

if (FALSE) {
nx<-15; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx),runif(nx))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

plotDeltri(Xp,Yp,xlab="",ylab="")

M<-"CC"  #try also M<-c(1,1,1)

NumArcsAS(Xp,Yp,M)
NumArcsAS(Xp,Yp[1:3,],M)
NumArcsAS(c(.4,.2),Yp,M)

Xp2=Xp+10
NumArcsAS(Xp2,Yp,M)

NumArcsAS(Xp,rbind(Yp,Yp),M)
}