IedgeAStri {pcds.ugraph}

The indicator for the presence of an edge from a point to another for the underlying or reflexivity graph of Arc Slice Proximity Catch Digraphs (AS-PCDs) - one triangle case

Description

Returns I(p1p2 is an edge in the underlying or reflexivity graph of AS-PCDs ) for points p1 and p2 in a given triangle.

More specifically, when the argument ugraph="underlying", it returns the edge indicator for the AS-PCD underlying graph, that is, returns 1 if p2 is in N_{AS}(p1) **or** p1 is in N_{AS}(p2), returns 0 otherwise. On the other hand, when ugraph="reflexivity", it returns the edge indicator for the AS-PCD reflexivity graph, that is, returns 1 if p2 is in N_{AS}(p1) **and** p1 is in N_{AS}(p2), returns 0 otherwise.

In both cases AS proximity region is constructed with respect to the triangle tri and vertex regions are based on the center, M=(m_1,m_2) in Cartesian coordinates or M=(\alpha,\beta,\gamma) in barycentric coordinates in the interior of the triangle tri or based on circumcenter of tri; default is M="CC", i.e., circumcenter of tri.

If p1 and p2 are distinct and either of them are outside tri, it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

See also (Ceyhan (2005, 2016)).

Usage

IedgeAStri(p1, p2, tri, M = "CC", ugraph = c("underlying", "reflexivity"))

Arguments

p1

A 2D point whose AS proximity region is constructed.

p2

A 2D point. The function determines whether there is an edge from p1 to p1 or not in the underlying or reflexivity graph of AS-PCDs.

tri

A 3 \times 2 matrix with each row representing a vertex of the triangle.

M

The center of the triangle. "CC" stands for circumcenter of the triangle tri or a 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of tri; default is M="CC", i.e., the circumcenter of tri.

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

Returns 1 if there is an edge between points p1 and p2 in the underlying or reflexivity graph of AS-PCDs in a given triangle tri, and 0 otherwise.

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 (2016). “Edge Density of New Graph Types Based on a Random Digraph Family.” Statistical Methodology, 33, 31-54.

See Also

IedgeASbasic.tri, IedgePEtri, IedgeCStri and IarcAStri

Examples

#\donttest{
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
M<-as.numeric(pcds::runif.tri(1,Tr)$g)

n<-3
set.seed(1)
Xp<-pcds::runif.tri(n,Tr)$g

IedgeAStri(Xp[1,],Xp[3,],Tr,M)
IedgeAStri(Xp[1,],Xp[3,],Tr,M,ugraph = "reflexivity")

set.seed(1)
P1<-as.numeric(pcds::runif.tri(1,Tr)$g)
P2<-as.numeric(pcds::runif.tri(1,Tr)$g)
IedgeAStri(P1,P2,Tr,M)
IedgeAStri(P1,P2,Tr,M,ugraph="r")
#}