IedgePEstd.tri {pcds.ugraph}

The indicator for the presence of an edge from a point to another for the underlying or reflexivity graph of Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard equilateral triangle case

Description

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

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

In both cases N_{PE}(x,r) is the PE proximity region for point x with expansion parameter r \ge 1. PE proximity region is defined with respect to the standard equilateral triangle T_e=T(v=1,v=2,v=3)=T((0,0),(1,0),(1/2,\sqrt{3}/2)) 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 T_e; default is M=(1,1,1) i.e., the center of mass of T_e.

If p1 and p2 are distinct and either of them are outside T_e, 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, 2010)).

Usage

IedgePEstd.tri(
  p1,
  p2,
  r,
  M = c(1, 1, 1),
  ugraph = c("underlying", "reflexivity")
)

Arguments

p1

A 2D point whose PE proximity region is constructed.

p2

A 2D point. The function determines whether there is an edge from p1 to p2 or not in the underlying or reflexivity graph of PE-PCDs.

r

A positive real number which serves as the expansion parameter in PE proximity region; must be \ge 1.

M

A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the standard equilateral triangle T_e; default is M=(1,1,1) i.e. the center of mass of T_e.

ugraph

The type of the graph based on PE-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 PE-PCDs in the standard equilateral triangle, 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 (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.

See Also

IedgePEbasic.tri, IedgePEtri, and IarcPEstd.tri

Examples

#\donttest{
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)
n<-3

set.seed(1)
Xp<-pcds::runif.std.tri(n)$gen.points

M<-as.numeric(pcds::runif.std.tri(1)$g)

IedgePEstd.tri(Xp[1,],Xp[3,],r=1.5,M)
IedgePEstd.tri(Xp[1,],Xp[3,],r=1.5,M,ugraph="reflexivity")

P1<-c(.4,.2)
P2<-c(.5,.26)
r<-2
IedgePEstd.tri(P1,P2,r,M)
IedgePEstd.tri(P1,P2,r,M,ugraph = "reflexivity")
#}