cl2edges.vert.reg.basic.tri {pcds}

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 (Ceyhan (2005, 2010)).

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

Ceyhan E (2005). An Investigation of Proximity Catch Digraphs in Delaunay Tessellations, also available as technical monograph titled Proximity Catch Digraphs: Auxiliary Tools, Properties, and Applications. Ph.D. thesis, The Johns Hopkins University, Baltimore, MD, 21218.

Ceyhan E (2010). “Extension of One-Dimensional Proximity Regions to Higher Dimensions.” Computational Geometry: Theory and Applications, 43(9), 721-748.

Ceyhan E (2011). “Spatial Clustering Tests Based on Domination Number of a New Random Digraph Family.” Communications in Statistics - Theory and Methods, 40(8), 1363-1395.

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)