edgesAS {pcds.ugraph}

The edges of the underlying or reflexivity graph of the Arc Slice Proximity Catch Digraph (AS-PCD) for 2D data - multiple triangle case

Description

An object of class "UndPCDs". Returns edges of the underlying or reflexivity graph of AS-PCD as left and right end points and related parameters and the quantities of these graphs. The vertices of these graphs 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, edges may exist for points only inside the convex hull of Yp points. It also provides various descriptions and quantities about the edges of the AS-PCD such as number of edges, edge 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. The different consideration of circumcenter vs any other interior center of the triangle is because the projections from circumcenter are orthogonal to the edges, while projections of M on the edges are on the extensions of the lines connecting M and the vertices. Each Delaunay triangle is first converted to an (nonscaled) basic triangle so that M will be the same type of center for each Delaunay triangle (this conversion is not necessary when M is CM).

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 edges, loops are not allowed so edges are only possible for points inside the convex hull of Yp points.

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

Usage

edgesAS(Xp, Yp, M = "CC", ugraph = c("underlying", "reflexivity"))

Arguments

Xp

A set of 2D points which constitute the vertices of the underlying or reflexivity graph 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" 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.

ugraph

The type of the graph based on AS-PCDs, "underlying" is for the underlying graph, and "reflexivity" is for the reflexivity graph (default is "underlying").

Value

A list with the elements

type

A description of the underlying or reflexivity graph of the digraph

parameters

Parameters of the underlying or reflexivity graph of the digraph, here, it is only the center M used to construct the vertex regions.

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 points

vert.name

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

LE, RE

Left and right end points of the edges of the underlying or reflexivity graph of AS-PCD for 2D data set Xp as vertices of the underlying or reflexivity graph of the digraph

mtitle

Text for "main" title in the plot of the underlying or reflexivity graph of the digraph

quant

Various quantities for the underlying or reflexivity graph of the digraph: number of vertices, number of partition points, number of intervals, number of edges, and edge 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 (2016). “Edge Density of New Graph Types Based on a Random Digraph Family.” Statistical Methodology, 33, 31-54.

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

edgesAStri, edgesPE, edgesCS, and arcsAS

Examples

#\donttest{
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-5;

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))

M<-c(1,1,1)

Edges<-edgesAS(Xp,Yp,M)
Edges
summary(Edges)
plot(Edges)
#}