pcds (version 0.1.2)

rv.bastriCM: The index of the \(CM\)-vertex region in a basic triangle that contains a point

Description

Returns the index of the vertex whose region contains point p in the basic triangle \(T_b=T((0,0),(1,0),(c_1,c_2))\) 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 center of mass CM=((1+c1)/3,c2/3) of \(T_b\). (see the plots in the example for illustrations).

The vertices of the basic triangle \(T_b\) 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 with the vertex, \(CM\), and midpoints of the edges adjacent to the vertex.

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.

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

Usage

rv.bastriCM(p, c1, c2)

Arguments

p

A 2D point for which \(CM\)-vertex region it resides in is to be determined in the basic triangle \(T_b\).

c1, c2

Positive real numbers which constitute the upper vertex of the basic triangle (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 \(CM\)-vertex region that contains point, p in the basic triangle \(T_b\)

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 \(3=(c_1,c_2)\).

References

#' @author Elvan Ceyhan

See Also

rv.triCM, rv.tri.cent, rv.triCC, rv.bastriCC, rv.bastri.cent, and rvTeCM

Examples

# NOT RUN {
c1<-.4; c2<-.6
P<-c(.4,.2)
rv.bastriCM(P,c1,c2)

c1<-.5; c2<-.8
P<-c(.4,.2)
rv.bastriCM(P,c1,c2)

P<-c(.8,.2)
rv.bastriCM(P,c1,c2)

P<-c(1,.2)
rv.bastriCM(P,c1,c2)

c1<-.4; c2<-.6
A<-c(0,0);B<-c(1,0);C<-c(c1,c2);
Tb<-rbind(A,B,C)
CM<-(A+B+C)/3
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

rv.bastriCM(A,c1,c2)
rv.bastriCM(CM,c1,c2)
rv.bastriCM(D1,c1,c2)
rv.bastriCM(D2,c1,c2)
rv.bastriCM(D3,c1,c2)

n<-10  #try also n<-20
dat<-runif.bastri(n,c1,c2)$g

Rv<-vector()
for (i in 1:n)
  Rv<-c(Rv,rv.bastriCM(dat[i,],c1,c2)$rv)
Rv

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

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

txt<-rbind(Tb,CM,Ds)
xc<-txt[,1]+c(-.03,.03,.02,-.01,.06,-.05,.0)
yc<-txt[,2]+c(.02,.02,.02,.04,.02,.02,-.03)
txt.str<-c("A","B","C","CM","D1","D2","D3")
text(xc,yc,txt.str)

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

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

txt<-rbind(Tb,CM,Ds)
xc<-txt[,1]+c(-.03,.03,.02,-.01,.04,-.03,.0)
yc<-txt[,2]+c(.02,.02,.02,.04,.02,.02,-.03)
txt.str<-c("A","B","C","CM","D1","D2","D3")
text(xc,yc,txt.str)

# }