IedgeASbasic.tri {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) - standard basic triangle case

Description

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

AS proximity region is constructed in the standard basic triangle T_b=T((0,0),(1,0),(c_1,c_2)) where c_1 is in [0,1/2], c_2>0 and (1-c_1)^2+c_2^2 \leq 1.

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 standard basic triangle T_b or based on circumcenter of T_b; default is M="CC", i.e., circumcenter of T_b.

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

Any given triangle can be mapped to the standard basic triangle by a combination of rigid body motions (i.e., translation, rotation and reflection) and scaling, preserving uniformity of the points in the original triangle. Hence, standard basic triangle is useful for simulation studies under the uniformity hypothesis.

See also (Ceyhan (2005, 2010)).

Usage

IedgeASbasic.tri(
  p1,
  p2,
  c1,
  c2,
  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.

c1, c2

Positive real numbers which constitute the vertex of the standard basic triangle adjacent to the shorter edges; c_1 must be in [0,1/2], c_2>0 and (1-c_1)^2+c_2^2 \le 1.

M

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

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 the standard basic 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

IedgeAStri, IedgeCSbasic.tri, IedgePEbasic.tri and IarcASbasic.tri

Examples

#\donttest{
c1<-.4; c2<-.6
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C);

M<-as.numeric(pcds::runif.basic.tri(1,c1,c2)$g)
set.seed(4)
P1<-as.numeric(pcds::runif.basic.tri(1,c1,c2)$g)
P2<-as.numeric(pcds::runif.basic.tri(1,c1,c2)$g)
IedgeASbasic.tri(P1,P2,c1,c2,M)
IedgeASbasic.tri(P1,P2,c1,c2,M,ugraph = "reflexivity")

P1<-c(.4,.2)
P2<-c(.5,.26)
IedgeASbasic.tri(P1,P2,c1,c2,M)
IedgeASbasic.tri(P1,P2,c1,c2,M,ugraph="r")
#}