IarcCSset2pnt.std.tri {pcds}

The indicator for the presence of an arc from a point in set S to the point p for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case

Description

Returns I(p in N_{CS}(x,t) for some x in S), that is, returns 1 if p is in \cup_{x in S} N_{CS}(x,t), returns 0 otherwise, CS proximity region is constructed with respect to the standard equilateral triangle T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2)) with the expansion parameter t>0 and edge regions are based on 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 (which is equivalent to circumcenter of T_e).

Edges of T_e, AB, BC, AC, are also labeled as edges 3, 1, and 2, respectively. If p is not in S and either p or all points in S are outside T_e, it returns 0, but if p is in S, then it always returns 1 regardless of its location (i.e., loops are allowed).

See also (Ceyhan (2012)).

Usage

IarcCSset2pnt.std.tri(S, p, t, M = c(1, 1, 1))

Arguments

S

A set of 2D points. Presence of an arc from a point in S to point p is checked by the function.

p

A 2D point. Presence of an arc from a point in S to point p is checked by the function.

t

A positive real number which serves as the expansion parameter in CS proximity region in the standard equilateral triangle T_e=T((0,0),(1,0),(1/2,\sqrt{3}/2)).

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.

Value

I(p is in \cup_{x in S} N_{CS}(x,t)), that is, returns 1 if p is in S or inside N_{CS}(x,t) for at least one x in S, returns 0 otherwise. CS proximity region is constructed with respect to the standard equilateral triangle T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2)) with M-edge regions.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

See Also

IarcCSset2pnt.tri, IarcCSstd.tri, IarcCStri, and IarcPEset2pnt.std.tri

Examples

A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-10

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<-.5

S<-rbind(Xp[1,],Xp[2,])  #try also S<-c(.5,.5)
IarcCSset2pnt.std.tri(S,Xp[3,],t,M)
IarcCSset2pnt.std.tri(S,Xp[3,],t=1,M)
IarcCSset2pnt.std.tri(S,Xp[3,],t=1.5,M)

S<-rbind(c(.1,.1),c(.3,.4),c(.5,.3))
IarcCSset2pnt.std.tri(S,Xp[3,],t,M)