pcds (version 0.1.6)

funsCSEdgeRegs: Each function is for the presence of an arc from a point in one of the edge regions to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case

Description

Three indicator functions: IndNCSTeRAB, IndNCSTeRBC and IndNCSTeRAC.

The function IndNCSTeRAB returns I(p2 is in \(N_{CS}(p1,t)\) for p1 in \(RAB\) (edge region for edge \(AB\), i.e., edge 3) in the standard equilateral triangle \(T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2))\);

IndNCSTeRBC returns I(p2 is in \(N_{CS}(p1,t)\) for p1 in \(RBC\) (edge region for edge \(BC\), i.e., edge 1) in \(T_e\); and

IndNCSTeRAC returns I(p2 is in \(N_{CS}(p1,t)\) for p1 in \(RAC\) (edge region for edge \(AC\), i.e., edge 2) in \(T_e\). That is, each function returns 1 if p2 is in \(N_{CS}(p1,t)\), returns 0 otherwise.

CS proximity region is defined with respect to \(T_e\) whose vertices are also labeled as \(T_e=T(v=1,v=2,v=3)\) with expansion parameter \(t>0\) 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 \(T_e\)

If p1 and p2 are distinct and p1 is outside the corresponding edge region and p2 is outside \(T_e\), it returns 0, but if they are identical, then it returns 1 regardless of their location (i.e., it allows loops).

See also (ceyhan:Phd-thesis,ceyhan:comp-geo-2010;textualpcds).

Usage

IndNCSTeRAB(p1, p2, t, M)
IndNCSTeRBC(p1, p2, t, M)
IndNCSTeRAC(p1, p2, t, M)

Value

Each function returns \(I(\)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 \(T_e\).

Author

Elvan Ceyhan

See Also

IndNCSTeRABt1, IndNCSTeRBCt1 and IndNCSTeRACt1

Examples

if (FALSE) {
#Examples for IndNCSTeRAB
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
CM<-(A+B+C)/3
T3<-rbind(A,B,CM);

set.seed(1)
Xp<-runif.std.tri(3)$gen.points

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

t<-1

IndNCSTeRAB(Xp[1,],Xp[2,],t,M)
IndNCSTeRAB(c(.2,.5),Xp[2,],t,M)
}

if (FALSE) {
#Examples for IndNCSTeRBC
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
CM<-(A+B+C)/3
T1<-rbind(B,C,CM);

set.seed(1)
Xp<-runif.std.tri(3)$gen.points

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

t<-1

IndNCSTeRBC(Xp[1,],Xp[2,],t,M)
IndNCSTeRBC(c(.2,.5),Xp[2,],t,M)
}

if (FALSE) {
#Examples for IndNCSTeRAC
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
CM<-(A+B+C)/3
T2<-rbind(A,C,CM);

set.seed(1)
Xp<-runif.std.tri(3)$gen.points

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

t<-1

IndNCSTeRAC(Xp[1,],Xp[2,],t,M)
IndNCSTeRAC(c(.2,.5),Xp[2,],t,M)
}