pcds (version 0.1.4)

re.bastri.cent: The index of the edge region in a basic triangle that contains a point

Description

Returns the index of the edge whose region contains point, pt, in the basic triangle \(T_b=T(A=(0,0),B=(1,0),C=(c_1,c_2))\) and edge regions based on center \(M=(m_1,m_2)\) in Cartesian coordinates or \(M=(\alpha,\beta,\gamma)\) in barycentric coordinates in the interior of the basic triangle \(T_b\).

Edges are labeled as 3 for edge \(AB\), 1 for edge \(BC\), and 2 for edge \(AC\). If the point, pt, 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 \(T(A,B,M)\). In the basic triangle \(T_b\) \(c_1\) is in \([0,1/2]\), \(c_2>0\) and \((1-c_1)^2+c_2^2 \le 1\).

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-CS;textualpcds).

Usage

re.bastri.cent(pt, c1, c2, M)

Value

A list with three elements

re

Index of the M-edge region that contains point, pt in the basic triangle \(T_b\).

tri

The vertices of the triangle, 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

Arguments

pt

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

M

A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the basic triangle \(T_b\).

Author

Elvan Ceyhan

References

See Also

re.triCM, re.tri.cent, re.bastri.cent, reTeCM, and redge.triCM

Examples

c1<-.4; c2<-.6
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C);
M<-c(.6,.2)

P<-c(.4,.2)
re.bastri.cent(P,c1,c2,M)

P<-c(1.4,.2)
re.bastri.cent(P,c1,c2,M)

c1<-.5; c2<-.8
P<-c(.4,.2)
re.bastri.cent(P,c1,c2,M)

P<-c(.8,.2)
re.bastri.cent(P,c1,c2,M)

c1<-.4; c2<-.6
A<-c(0,0);B<-c(1,0);C<-c(c1,c2);
Tb<-rbind(A,B,C)

re.bastri.cent(A,c1,c2,M)
re.bastri.cent(B,c1,c2,M)
re.bastri.cent(C,c1,c2,M)
re.bastri.cent(M,c1,c2,M)

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

M<-as.numeric(runif.bastri(1,c1,c2)$g)  #try also M<-c(.6,.2)

re<-vector()
for (i in 1:n)
  re<-c(re,re.bastri.cent(dat[i,],c1,c2,M)$re)
re

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=TRUE,pch=".",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
points(dat,pch=".")
polygon(Tb)
L<-Tb; R<-rbind(M,M,M)
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
text(dat,labels=factor(re))

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