cl2edges.vert.reg.basic.tri: The closest points among a data set in the vertex regions to...

In pcds: Proximity Catch Digraphs and Their Applications

The closest points among a data set in the vertex regions to the corresponding edges in a standard basic triangle

Description

An object of class "Extrema". Returns the closest data points among the data set, Xp, to edge i in M-vertex region i for i=1,2,3 in the standard basic triangle T_b=T(A=(0,0),B=(1,0),C=(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 labels are A=1, B=2, and C=3, and corresponding edge labels are BC=1, AC=2, and AB=3.

Vertex regions are 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 T_b or based on the circumcenter of T_b.

Any given triangle can be mapped to the standard 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, standard basic triangle is useful for simulation studies under the uniformity hypothesis.

See also (\insertCiteceyhan:Phd-thesis,ceyhan:comp-geo-2010,ceyhan:dom-num-NPE-Spat2011;textualpcds).

Usage

cl2edges.vert.reg.basic.tri(Xp, c1, c2, M)

Arguments

Xp	A set of 2D points representing the set of data points.
c1, c2	Positive real numbers which constitute the vertex of the standard basic triangle 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 T_b or the circumcenter of T_b.

Value

A list with the elements

txt1	Vertex labels are A=1, B=2, and C=3 (correspond to row number in Extremum Points).
txt2	A short description of the distances as "Distances to Edges in the Respective \eqn{M}-Vertex Regions".
type	Type of the extrema points
desc	A short description of the extrema points
mtitle	The "main" title for the plot of the extrema
ext	The extrema points, here, closest points to edges in the corresponding vertex region.
X	The input data, Xp, can be a matrix or data frame
num.points	The number of data points, i.e., size of Xp
supp	Support of the data points, here, it is T_b.
cent	The center point used for construction of vertex regions
ncent	Name of the center, cent, it is "M" or "CC" for this function
regions	Vertex regions inside the triangle, T_b.
region.names	Names of the vertex regions as "vr=1", "vr=2", and "vr=3"
region.centers	Centers of mass of the vertex regions inside T_b.
dist2ref	Distances of closest points in the vertex regions to corresponding edges.

Author(s)

Elvan Ceyhan

References

\insertAllCited

See Also

cl2edgesCMvert.reg, cl2edgesMvert.reg, and cl2edges.std.tri

Examples

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

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

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.3)

Ext<-cl2edges.vert.reg.basic.tri(Xp,c1,c2,M)
Ext
summary(Ext)
plot(Ext)

cl2e<-Ext

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]

plot(Tb,pch=".",xlab="",ylab="",
main="Closest Points in M-Vertex Regions \n to the Opposite Edges",
axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
points(Xp,pch=1,col=1)
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(cl2e$ext,pch=3,col=2)

xc<-Tb[,1]+c(-.02,.02,0.02)
yc<-Tb[,2]+c(.02,.02,.02)
txt.str<-c("A","B","C")
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)

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