pcds (version 0.1.8)

IarcCSedge.reg.std.tri: The indicator for the presence of an arc from a point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case

Description

Returns \(I(\)p2 is in \(N_{CS}(p1,t))\) for points p1 and p2, that is, returns 1 if p2 is in \(N_{CS}(p1,t)\), returns 0 otherwise, where \(N_{CS}(x,t)\) is the CS proximity region for point \(x\) with expansion parameter \(t>0\). This function is equivalent to IarcCSstd.tri, except that it computes the indicator using the functions IarcCSstd.triRAB, IarcCSstd.triRBC and IarcCSstd.triRAC which are edge-region specific indicator functions. For example, IarcCSstd.triRAB computes \(I(\)p2 is in \(N_{CS}(p1,t))\) for points p1 and p2 when p1 resides in the edge region of edge \(AB\).

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 \(T_e\). re is the index of the edge region p1 resides, with default=NULL.

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:arc-density-CS,ceyhan:test2014;textualpcds).

Usage

IarcCSedge.reg.std.tri(p1, p2, t, M = c(1, 1, 1), re = NULL)

Value

\(I(\)p2 is in \(N_{CS}(p1,t))\) for p1, that is, returns 1 if p2 is in \(N_{CS}(p1,t)\), returns 0 otherwise

Arguments

p1

A 2D point whose CS proximity region is constructed.

p2

A 2D point. The function determines whether p2 is inside the CS proximity region of p1 or not.

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 \(T_e\).

re

The index of the edge region in \(T_e\) containing the point, either 1,2,3 or NULL (default is NULL).

Author

Elvan Ceyhan

References

See Also

IarcCStri 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<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

t<-1
IarcCSedge.reg.std.tri(Xp[1,],Xp[2,],t,M)
IarcCSstd.tri(Xp[1,],Xp[2,],t,M)

#or try
re<-rel.edge.std.triCM(Xp[1,])$re
IarcCSedge.reg.std.tri(Xp[1,],Xp[2,],t,M,re=re)
# }