pcds.ugraph (version 0.1.1)

IedgePEbasic.tri: 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 basic 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 basic triangle.

More specifically, when the argument ugraph="underlying", it returns the edge indicator 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 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 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 \((1-c_1)^2+c_2^2 \le 1\).

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 \(M=(1,1,1)\) i.e., the center of mass 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:Phd-thesis,ceyhan:comp-geo-2010,ceyhan:stamet2016;textualpcds.ugraph).

Usage

IedgePEbasic.tri(
  p1,
  p2,
  r,
  c1,
  c2,
  M = c(1, 1, 1),
  ugraph = c("underlying", "reflexivity")
)

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 basic triangle, and 0 otherwise.

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 \(\ge 1\)

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

A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the standard basic triangle or circumcenter of \(T_b\) which may be entered as "CC" as well; default is \(M=(1,1,1)\) i.e., the center of mass of \(T_b\).

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").

Author

Elvan Ceyhan

References

See Also

IedgePEtri, IedgeASbasic.tri, IedgeCSbasic.tri and IarcPEbasic.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)

r<-1.5
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)
IedgePEbasic.tri(P1,P2,r,c1,c2,M)
IedgePEbasic.tri(P1,P2,r,c1,c2,M,ugraph = "reflexivity")

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