# The closest points among a data set in the standard equilateral triangle to the median lines in the six half edge regions 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`). ```r six.extremaTe(Xp, ch.all.intri = FALSE) ``` ## 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`). ## Returns 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 sixregions 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, $CM$. ## Examples ```r 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) ``` ## See Also `index.six.Te` and `cl2edges.std.tri` ## Author(s) Elvan Ceyhan

six.extremaTe function