pcds.ugraph (version 0.1.1)

IedgeCSstd.tri: The indicator for the presence of an edge from a point to another for the underlying or reflexivity graphs of Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case

Description

Returns \(I(\)p1p2 is an edge in the underlying or reflexivity graph of CS-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 CS-PCD underlying graph, that is, returns 1 if p2 is in \(N_{CS}(p1,t)\) or p1 is in \(N_{CS}(p2,t)\), 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 CS-PCD reflexivity graph, that is, returns 1 if p2 is in \(N_{CS}(p1,t)\) and p1 is in \(N_{CS}(p2,t)\), returns 0 otherwise.

In both cases \(N_{CS}(x,t)\) is the CS proximity region for point \(x\) with expansion parameter \(t > 0\). CS 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 edge 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:Phd-thesis,ceyhan:comp-geo-2010,ceyhan:stamet2016;textualpcds.ugraph).

Usage

IedgeCSstd.tri(
  p1,
  p2,
  t,
  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 CS-PCDs in the standard equilateral triangle, and 0 otherwise.

Arguments

p1

A 2D point whose CS 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 graphs of CS-PCDs.

t

A positive real number which serves as the expansion parameter in CS proximity region.

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 CS-PCDs, "underlying" is for the underlying graph, and "reflexivity" is for the reflexivity graph (default is "underlying").

Author

Elvan Ceyhan

References

See Also

IedgeCSbasic.tri, IedgeCStri, and IarcCSstd.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)

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

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