rel.vert.basic.tri: The index of the vertex region in a standard basic triangle...
In pcds: Proximity Catch Digraphs and Their Applications

rel.vert.basic.tri

R Documentation

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

Description

Returns the index of the related vertex in the standard basic triangle form whose region contains point p. The standard basic triangle form is 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..

Vertex regions are based on the general 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. Vertices of the standard basic triangle form T_b are labeled according to the row number the vertex is recorded, i.e., as 1=(0,0), 2=(1,0),and 3=(c_1,c_2).

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, M, and projections from M to the edges on the lines joining vertices and M. That is, rv=1 has vertices (0,0),D_3,M,D_2; rv=2 has vertices (1,0),D_1,M,D_3; and rv=3 has vertices (c_1,c_2),D_2,M,D_1 (see the illustration in the examples).

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 (\insertCiteceyhan:Phd-thesis,ceyhan:comp-geo-2010,ceyhan:mcap2012;textualpcds).

Usage

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

Arguments

`p`	A 2D point for which `M`-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 vertex of the standard basic triangle form adjacent to the shorter edges; `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.

Value

A list with two elements

`rv`	Index of the vertex whose region contains point, `p`; index of the vertex is the same as the row number in the standard basic triangle form, `T_b`
`tri`	The vertices of the standard basic triangle form, `T_b`, where row number corresponds to the vertex index `rv` with `rv=1` is row `1=(0,0)`, `rv=2` is row `2=(1,0)`, and `rv=3` is row `3=(c_1,c_2)`.

Author(s)

Elvan Ceyhan

References

\insertAllCited

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)
rel.vert.basic.tri(P,c1,c2,M)

n<-20  #try also n<-40
set.seed(1)
Xp<-runif.basic.tri(n,c1,c2)$g

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

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

Ds<-prj.cent2edges.basic.tri(c1,c2,M)

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

if (dimension(M)==3) {M<-bary2cart(M,Tb)}
#need to run this when M is given in barycentric coordinates

plot(Tb,pch=".",xlab="",ylab="",axes=TRUE,
xlim=Xlim+xd*c(-.1,.1),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
points(Xp,pch=".",col=1)
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

xc<-Tb[,1]+c(-.04,.05,.04)
yc<-Tb[,2]+c(.02,.02,.03)
txt.str<-c("rv=1","rv=2","rv=3")
text(xc,yc,txt.str)

txt<-rbind(M,Ds)
xc<-txt[,1]+c(-.02,.04,-.03,0)
yc<-txt[,2]+c(-.02,.02,.02,-.03)
txt.str<-c("M","D1","D2","D3")
text(xc,yc,txt.str)
text(Xp,labels=factor(Rv))

pcds documentation built on May 29, 2024, 9:36 a.m.