pcds (version 0.1.8)

rel.vert.basic.triCM: The index of the \(CM\)-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 center of mass CM=((1+c1)/3,c2/3) 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 with the vertex, \(CM\), and midpoints of the edges adjacent to the 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:Phd-thesis,ceyhan:comp-geo-2010,ceyhan:mcap2012,ceyhan:arc-density-PE;textualpcds)

Usage

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

Value

A list with two elements

rv

Index of the \(CM\)-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 \(3=(c_1,c_2)\).

Arguments

p

A 2D point for which \(CM\)-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 \((1-c_1)^2+c_2^2 \le 1\).

References

#' @author Elvan Ceyhan

See Also

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

Examples

# \donttest{
c1<-.4; c2<-.6
P<-c(.4,.2)
rel.vert.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
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

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.triCM(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,xlab="",ylab="",axes="T",pch=".",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
points(Xp,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(Xp,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)
# }