pcds (version 0.1.8)

rel.edge.basic.tri: The index of the edge region in a standard basic triangle form that contains a point

Description

Returns the index of the edge whose region contains point, p, in the standard basic triangle form \(T_b=T(A=(0,0),B=(1,0),C=(c_1,c_2))\) 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 standard basic triangle form \(T_b\).

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 \(T(A,B,M)\). In the standard basic triangle form \(T_b\) \(c_1\) 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 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-CS;textualpcds).

Usage

rel.edge.basic.tri(p, c1, c2, M)

Value

A list with three elements

re

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

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

p

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

M

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

Author

Elvan Ceyhan

References

See Also

rel.edge.triCM, rel.edge.tri, rel.edge.basic.tri, rel.edge.std.triCM, and edge.reg.triCM

Examples

# \donttest{
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)
rel.edge.basic.tri(P,c1,c2,M)

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

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

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

re<-vector()
for (i in 1:n)
  re<-c(re,rel.edge.basic.tri(Xp[i,],c1,c2,M)$re)
re

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