arcsAS {pcds}

The arcs of Arc Slice Proximity Catch Digraph (AS-PCD) for a 2D data set - multiple triangle case

Description

An object of class "PCDs". Returns arcs of AS-PCD as tails (or sources) and heads (or arrow ends) and related parameters and the quantities of the digraph. The vertices of the AS-PCD 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, i.e., AS proximity regions are defined only for Xp points inside the convex hull of Yp points. That is, arcs may exist for points only inside the convex hull of Yp points. It also provides various descriptions and quantities about the arcs of the AS-PCD such as number of arcs, arc density, etc.

Vertex regions 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. M must be entered in barycentric coordinates unless it is the circumcenter.

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). For the number of arcs, loops are not allowed so arcs are only possible for points inside the convex hull of Yp points.

See (Ceyhan (2005, 2010)) for more on AS PCDs. Also see (Okabe et al. (2000); Ceyhan (2010); Sinclair (2016)) for more on Delaunay triangulation and the corresponding algorithm.

Usage

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

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 triangulation. The Delaunay triangles partition the convex hull of Yp points.

M

The center of the triangle. "CC" represents the 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. M must be entered in barycentric coordinates unless it is the circumcenter.

Value

A list with the elements

type

A description of the type of the digraph

parameters

Parameters of the digraph, here, it is the center used to construct the vertex regions, default is circumcenter, denoted as "CC", otherwise given in barycentric coordinates.

tess.points

Tessellation points, i.e., points on which the tessellation of the study region is performed, here, tessellation is the Delaunay triangulation based on Yp points.

tess.name

Name of the tessellation points tess.points

vertices

Vertices of the digraph, Xp.

vert.name

Name of the data set which constitute the vertices of the digraph

S

Tails (or sources) of the arcs of AS-PCD for 2D data set Xp in the multiple triangle case as the vertices of the digraph

E

Heads (or arrow ends) of the arcs of AS-PCD for 2D data set Xp in the multiple triangle case as the vertices of the digraph

mtitle

Text for "main" title in the plot of the digraph

quant

Various quantities for the digraph: number of vertices, number of partition points, number of intervals, number of arcs, and arc density.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2005). An Investigation of Proximity Catch Digraphs in Delaunay Tessellations, also available as technical monograph titled Proximity Catch Digraphs: Auxiliary Tools, Properties, and Applications. Ph.D. thesis, The Johns Hopkins University, Baltimore, MD, 21218.

Ceyhan E (2010). “Extension of One-Dimensional Proximity Regions to Higher Dimensions.” Computational Geometry: Theory and Applications, 43(9), 721-748.

Ceyhan E (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

Okabe A, Boots B, Sugihara K, Chiu SN (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, New York.

Sinclair D (2016). “S-hull: a fast radial sweep-hull routine for Delaunay triangulation.” 1604.01428.

See Also

arcsAStri, arcsPEtri, arcsCStri, arcsPE, and arcsCS

Examples

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-5;  #try also nx=20; nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
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))

M<-c(1,1,1)  #try also M<-c(1,2,3)

Arcs<-arcsAS(Xp,Yp,M) #try also the default M with Arcs<-arcsAS(Xp,Yp)
Arcs
summary(Arcs)
plot(Arcs)

arcsAS(Xp,Yp[1:3,],M)