rel.edge.std.triCM {pcds}

The index of the edge region in the standard equilateral triangle that contains a point

Description

Returns the index of the edge whose region contains point, p, in the standard equilateral triangle T_e=T(A=(0,0),B=(1,0),C=(1/2,\sqrt{3}/2)) 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,M), edge region 2 is T(A,C,M), and edge region 3 is T(A,B,M).

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

Usage

rel.edge.std.triCM(p)

Arguments

p

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

Value

A list with three elements

re

Index of the CM-edge region that contains point, p in the standard equilateral triangle T_e

tri

The vertices of the standard equilateral triangle T_e, where row labels are A, B, and C 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.triCM, rel.edge.basic.tri, and edge.reg.triCM

Examples

P<-c(.4,.2)
rel.edge.std.triCM(P)

A<-c(0,0); B<-c(1,0); C<-c(0.5,sqrt(3)/2);
Te<-rbind(A,B,C)
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
CM<-(A+B+C)/3

n<-20  #try also n<-40
Xp<-runif.std.tri(n)$gen.points

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

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

plot(Te,asp=1,xlab="",ylab="",axes=TRUE,pch=".",xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01))
points(Xp,pch=".")
polygon(Te)
L<-Te; 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(Te,CM)
xc<-txt[,1]+c(-.03,.03,.03,-.06)
yc<-txt[,2]+c(.02,.02,.02,.03)
txt.str<-c("A","B","C","CM")
text(xc,yc,txt.str)

p1<-(A+B+CM)/3
p2<-(B+C+CM)/3
p3<-(A+C+CM)/3

plot(Te,xlab="",ylab="",axes=TRUE,pch=".",xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01))
polygon(Te)
L<-Te; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

txt<-rbind(Te,CM,p1,p2,p3)
xc<-txt[,1]+c(-.03,.03,.03,-.06,0,0,0)
yc<-txt[,2]+c(.02,.02,.02,.03,0,0,0)
txt.str<-c("A","B","C","CM","re=3","re=1","re=2")
text(xc,yc,txt.str)