rel.vert.basic.triCC {pcds}

The index of the CC-vertex region in a standard basic triangle form that contains a point

Description

Returns the index of the vertex whose region contains point p in the standard basic triangle form T_b=T((0,0),(1,0),(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 and vertex regions are based on the circumcenter CC of T_b. (see the plots in the example for illustrations).

The vertices of the standard basic triangle form T_b are labeled as 1=(0,0), 2=(1,0),and 3=(c_1,c_2) also according to the row number the vertex is recorded in T_b. If the point, p, is not inside T_b, then the function yields NA as output. The corresponding vertex region is the polygon whose interior points are closest to that vertex.

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)).

Usage

rel.vert.basic.triCC(p, c1, c2)

Arguments

p

A 2D point for which CC-vertex 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 two elements

rv

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

tri

The vertices of the triangle, where row number corresponds to the vertex index in rv with row 1=(0,0), row 2=(1,0), and row 3=(c_1,c_2).

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.

See Also

rel.vert.triCM, rel.vert.tri, rel.vert.triCC, rel.vert.basic.triCM, rel.vert.basic.tri, and rel.vert.std.triCM

Examples

c1<-.4; c2<-.6;  #try also c1<-.5; c2<-.5;

P<-c(.3,.2)
rel.vert.basic.triCC(P,c1,c2)

A<-c(0,0);B<-c(1,0);C<-c(c1,c2);
Tb<-rbind(A,B,C)
CC<-circumcenter.basic.tri(c1,c2)  #the circumcenter
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

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

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

txt<-rbind(Tb,CC,Ds)
xc<-txt[,1]+c(-.03,.03,0.02,-.01,.05,-.05,.01)
yc<-txt[,2]+c(.02,.02,.03,.06,.03,.03,-.03)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)

RV1<-(A+D3+CC+D2)/4
RV2<-(B+D3+CC+D1)/4
RV3<-(C+D2+CC+D1)/4

txt<-rbind(RV1,RV2,RV3)
xc<-txt[,1]
yc<-txt[,2]
txt.str<-c("rv=1","rv=2","rv=3")
text(xc,yc,txt.str)

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

Rv<-vector()
for (i in 1:n)
  Rv<-c(Rv,rel.vert.basic.triCC(Xp[i,],c1,c2)$rv)
Rv

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

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

txt<-rbind(Tb,CC,Ds)
xc<-txt[,1]+c(-.03,.03,0.02,-.01,.05,-.05,.01)
yc<-txt[,2]+c(.02,.02,.03,.06,.03,.03,-.04)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)