six.extremaTe: The closest points among a data set in the standard equilateral triangle to the median lines in the six half edge regions

Description

An object of class "Extrema". Returns the six closest points among the data set, Xp, in the standard equilateral triangle \(T_e=T(A=(0,0),B=(1,0),C=(1/2,\sqrt{3}/2))\) in half edge regions. In particular, in regions \(r_1\) and \(r_6\), it finds the closest point in each region to the line segment \([A,CM]\) in regions \(r_2\) and \(r_3\), it finds the closest point in each region to the line segment \([B,CM]\) and in regions \(r_4\) and \(r_5\), it finds the closest point in each region to the line segment \([C,CM]\) where \(CM=(A+B+C)/3\) is the center of mass.

See the example for this function or example for index.six.Te function. If there is no data point in region \(r_i\), then it returns "NA NA" for \(i\)-th row in the extrema. ch.all.intri is for checking whether all data points are in \(T_e\) (default is FALSE).

Usage

six.extremaTe(Xp, ch.all.intri = FALSE)

Value

A list with the elements

txt1: Region labels as r1-r6 (correspond to row number in Extremum Points).
txt2: A short description of the distances as "Distances to Line Segments (A,CM), (B,CM), and (C,CM) in the six regions r1-r6".
type: Type of the extrema points
mtitle: The "main" title for the plot of the extrema
ext: The extrema points, here, closest points in each of regions r1-r6 to the line segments joining vertices to the center of mass, \(CM\).
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 \(T_e\).
cent: The center point used for construction of edge regions.
ncent: Name of the center, cent, it is center of mass "CM" for this function.
regions: The six regions, r1-r6 and edge regions inside the triangle, \(T_e\), provided as a list.
region.names: Names of the regions as "r1"-"r6" and names of the edge regions as "er=1", "er=2", and "er=3".
region.centers: Centers of mass of the regions r1-r6 and of edge regions inside \(T_e\).
dist2ref: Distances from closest points in each of regions r1-r6 to the line segments joining vertices to the center of mass, \(CM\).

Arguments

Xp: A set of 2D points among which the closest points in the standard equilateral triangle to the median lines in 6 half edge regions.
ch.all.intri: A logical argument for checking whether all data points are in \(T_e\) (default is FALSE).

Author

Elvan Ceyhan

Examples

Run this code

# \donttest{
n<-20  #try also n<-100
Xp<-runif.std.tri(n)$gen.points

Ext<-six.extremaTe(Xp)
Ext
summary(Ext)
plot(Ext)

sixt<-Ext

A<-c(0,0); B<-c(1,0); C<-c(0.5,sqrt(3)/2);
Te<-rbind(A,B,C)
CM<-(A+B+C)/3
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

h1<-c(1/2,sqrt(3)/18); h2<-c(2/3, sqrt(3)/9); h3<-c(2/3, 2*sqrt(3)/9);
h4<-c(1/2, 5*sqrt(3)/18); h5<-c(1/3, 2*sqrt(3)/9); h6<-c(1/3, sqrt(3)/9);

r1<-(h1+h6+CM)/3;r2<-(h1+h2+CM)/3;r3<-(h2+h3+CM)/3;
r4<-(h3+h4+CM)/3;r5<-(h4+h5+CM)/3;r6<-(h5+h6+CM)/3;

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

plot(A,pch=".",xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),
ylim=Ylim+yd*c(-.05,.05))
polygon(Te)
L<-Te; R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
polygon(rbind(h1,h2,h3,h4,h5,h6))
points(Xp)
points(sixt$ext,pty=2,pch=4,col="red")

txt<-rbind(Te,r1,r2,r3,r4,r5,r6)
xc<-txt[,1]+c(-.02,.02,.02,0,0,0,0,0,0)
yc<-txt[,2]+c(.02,.02,.03,0,0,0,0,0,0)
txt.str<-c("A","B","C","1","2","3","4","5","6")
text(xc,yc,txt.str)
# }