rel.edge.basic.triCM {pcds}

The index of the CM-edge region in a standard basic triangle form that contains a point

Description

Returns the index of the edge whose region contains point, p, in the standard basic triangle form T_b=T(A=(0,0),B=(1,0),C=(c_1,c_2) where c_1 is in [0,1/2], c_2>0 and (1-c_1)^2+c_2^2 \le 1 with edge regions based on center of mass CM=(A+B+C)/3.

Edges are labeled as 3 for edge AB, 1 for edge BC, and 2 for edge AC. If the point, p, is not inside tri, then the function yields NA as output. Edge region 1 is the triangle T(B,C,CM), edge region 2 is T(A,C,CM), and edge region 3 is T(A,B,CM).

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

See also (Ceyhan (2005, 2010); Ceyhan et al. (2007)).

Usage

rel.edge.basic.triCM(p, c1, c2)

Arguments

p

A 2D point for which CM-edge region it resides in is to be determined in the standard basic triangle form T_b.

c1, c2

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

Value

A list with three elements

re

Index of the CM-edge region that contains point, p in the standard basic triangle form T_b

tri

The vertices of the triangle, where row labels are A=(0,0), B=(1,0), and C=(c_1,c_2) with edges are labeled as 3 for edge AB, 1 for edge BC, and 2 for edge AC.

desc

Description of the edge labels

Author(s)

Elvan Ceyhan

References

Ceyhan E (2005). An Investigation of Proximity Catch Digraphs in Delaunay Tessellations, also available as technical monograph titled Proximity Catch Digraphs: Auxiliary Tools, Properties, and Applications. Ph.D. thesis, The Johns Hopkins University, Baltimore, MD, 21218.

Ceyhan E (2010). “Extension of One-Dimensional Proximity Regions to Higher Dimensions.” Computational Geometry: Theory and Applications, 43(9), 721-748.

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.

Ceyhan E, Priebe CE, Marchette DJ (2007). “A new family of random graphs for testing spatial segregation.” Canadian Journal of Statistics, 35(1), 27-50.

See Also

rel.edge.triCM, rel.edge.tri, rel.edge.basic.tri, rel.edge.std.triCM, and edge.reg.triCM

Examples

c1<-.4; c2<-.6
P<-c(.4,.2)
rel.edge.basic.triCM(P,c1,c2)

A<-c(0,0);B<-c(1,0);C<-c(c1,c2);
Tb<-rbind(A,B,C)
CM<-(A+B+C)/3

rel.edge.basic.triCM(A,c1,c2)
rel.edge.basic.triCM(B,c1,c2)
rel.edge.basic.triCM(C,c1,c2)
rel.edge.basic.triCM(CM,c1,c2)

n<-20  #try also n<-40
Xp<-runif.basic.tri(n,c1,c2)$g

re<-vector()
for (i in 1:n)
  re<-c(re,rel.edge.basic.triCM(Xp[i,],c1,c2)$re)
re

Xlim<-range(Tb[,1],Xp[,1])
Ylim<-range(Tb[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tb,xlab="",ylab="",axes=TRUE,pch=".",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
points(Xp,pch=".")
polygon(Tb)
L<-Tb; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)
text(Xp,labels=factor(re))

txt<-rbind(Tb,CM)
xc<-txt[,1]+c(-.03,.03,.02,0)
yc<-txt[,2]+c(.02,.02,.02,-.04)
txt.str<-c("A","B","C","CM")
text(xc,yc,txt.str)