Gam2ASbastri: The indicator for two points being a dominating set for Arc Slice Proximity Catch Digraphs (AS-PCDs) - basic triangle case

Description

Returns \(I(\){pt1,pt2} is a dominating set of AS-PCD\()\) where vertices of AS-PCD are the 2D data set Dt), that is, returns 1 if {pt1,pt2} is a dominating set of AS-PCD, returns 0 otherwise.

AS proximity regions are defined with respect to the basic triangle \(T_b=T(c(0,0),c(1,0),c(c1,c2))\), In the basic triangle, \(T_b\), \(c_1\) is in \([0,1/2]\), \(c_2>0\) and \((1-c_1)^2+c_2^2 \le 1\).

Any given triangle can be mapped to the 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 basic triangle is useful for simulation studies under the uniformity hypothesis.

Point, pt1, is in the vertex region of vertex rv1 (default is NULL) and point, pt2, is in the vertex region of vertex rv2 (default is NULL); vertices are labeled as \(1,2,3\) in the order they are stacked row-wise.

Vertex regions are based on the center M="CC" for circumcenter of \(T_b\); or \(M=(m_1,m_2)\) in Cartesian coordinates or \(M=(\alpha,\beta,\gamma)\) in barycentric coordinates in the interior of \(T_b\); default is M="CC".

ch.data.pnts is for checking whether points pt1 and pt2 are data points in Dt or not (default is FALSE), so by default this function checks whether the points pt1 and pt2 would be a dominating set if they actually were in the data set.

See also (ceyhan:Phd-thesis,ceyhan:comp-geo-2010,ceyhan:mcap2012;textualpcds).

Usage

Gam2ASbastri(
  pt1,
  pt2,
  Dt,
  c1,
  c2,
  M = "CC",
  rv1 = NULL,
  rv2 = NULL,
  ch.data.pnts = FALSE
)

Arguments

pt1, pt2

Two 2D points to be tested for constituting a dominating set of the AS-PCD.

A set of 2D points which constitutes the vertices of the AS-PCD.

c1, c2

Positive real numbers which constitute the vertex of the basic triangle adjacent to the shorter edges; \(c_1\) must be in \([0,1/2]\), \(c_2>0\) and \((1-c_1)^2+c_2^2 \le 1\).

The center of the triangle. "CC" stands for circumcenter of the triangle \(T_b\) or a 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle \(T_b\); default is M="CC" i.e., the circumcenter of \(T_b\).

rv1, rv2

The indices of the vertices whose regions contains pt1 and pt2, respectively. They take the vertex labels as \(1,2,3\) as in the row order of the vertices in \(T_b\) (default is NULL for both).

ch.data.pnts

A logical argument for checking whether points pt1 and pt2 are data points in Dt or not (default is FALSE).

Value

\(I(\){pt1,pt2} is a dominating set of the AS-PCD\()\) where the vertices of AS-PCD are the 2D data set Dt), that is, returns 1 if {pt1,pt2} is a dominating set of AS-PCD, returns 0 otherwise

References

Examples

Run this code

# NOT RUN {
c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C)
n<-20

set.seed(1)
dat<-runif.bastri(n,c1,c2)$g

M<-as.numeric(runif.bastri(1,c1,c2)$g)  #try also M<-c(.6,.2)

Gam2ASbastri(dat[1,],dat[2,],dat,c1,c2,M)
Gam2ASbastri(dat[1,],dat[1,],dat,c1,c2,M)  #one point can not a dominating set of size two

Gam2ASbastri(c(.2,.4),c(.2,.5),rbind(c(.2,.4),c(.2,.5)),c1,c2,M)

ind.gam2<-vector()
for (i in 1:(n-1))
  for (j in (i+1):n)
  {if (Gam2ASbastri(dat[i,],dat[j,],dat,c1,c2,M)==1)
   ind.gam2<-rbind(ind.gam2,c(i,j))}
ind.gam2

#or try
rv1<-rv.bastriCC(dat[1,],c1,c2)$rv
rv2<-rv.bastriCC(dat[2,],c1,c2)$rv
Gam2ASbastri(dat[1,],dat[2,],dat,c1,c2,M,rv1,rv2)
Gam2ASbastri(c(.2,.4),dat[2,],dat,c1,c2,M,rv1,rv2)

#or try
rv1<-rv.bastriCC(dat[1,],c1,c2)$rv
Gam2ASbastri(dat[1,],dat[2,],dat,c1,c2,M,rv1)

#or try
Rv2<-rv.bastriCC(dat[2,],c1,c2)$rv
Gam2ASbastri(dat[1,],dat[2,],dat,c1,c2,M,rv2=Rv2)

Gam2ASbastri(c(.3,.2),c(.35,.25),dat,c1,c2,M)

dat.fr<-data.frame(a=dat)
Gam2ASbastri(c(.3,.2),c(.35,.25),dat.fr,c1,c2,M)

# }