# The furthest points from vertices in each $CC$-vertex region in a standard basic triangle An object of class `"Extrema"`. Returns the furthest data points among the data set, `Xp`, in each $CC$-vertex region from the corresponding vertex in the standard basic triangle $T_b=T(A=(0,0),B=(1,0),C=(c_1,c_2))$. 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. `ch.all.intri` is for checking whether all data points are inside $T_b$ (default is `FALSE`). See also (if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="ceyhan:Phd-thesis,ceyhan:mcap2012;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r fr2vertsCCvert.reg.basic.tri(Xp, c1, c2, ch.all.intri = FALSE) ``` ## Arguments - `Xp`: A set of 2D 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 $(1-c_1)^2+c_2^2 \le 1$ - `ch.all.intri`: A logical argument for checking whether all data points are inside $T_b$ (default is `FALSE`). ## Returns 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 from furthest points to ..."`. - **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, furthest points from vertices in each 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 edge regions. - **ncent**: Name of the center, `cent`, it is circumcenter `"CC"` for this function. - **regions**: Vertex regions inside the triangle, $T_b$, provided as a `list`. - **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 from furthest points in each vertex region to the corresponding vertex. ## Examples ```r c1<-.4; c2<-.6; A<-c(0,0); B<-c(1,0); C<-c(c1,c2); Tb<-rbind(A,B,C) n<-20 set.seed(1) Xp<-runif.basic.tri(n,c1,c2)$g Ext<-fr2vertsCCvert.reg.basic.tri(Xp,c1,c2) Ext summary(Ext) plot(Ext) f2v<-Ext CC<-circumcenter.basic.tri(c1,c2) #the circumcenter D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2; Ds<-rbind(D1,D2,D3) Xlim<-range(Tb[,1],Xp[,1]) Ylim<-range(Tb[,2],Xp[,2]) xd<-Xlim[2]-Xlim[1] yd<-Ylim[2]-Ylim[1] plot(A,pch=".",asp=1,xlab="",ylab="", main="Furthest Points in CC-Vertex Regions \n from the Vertices", xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05)) polygon(Tb) L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds segments(L[,1], L[,2], R[,1], R[,2], lty=2) points(Xp) points(rbind(f2v$ext),pch=4,col=2) txt<-rbind(Tb,CC,D1,D2,D3) xc<-txt[,1]+c(-.03,.03,0.02,.07,.06,-.05,.01) yc<-txt[,2]+c(.02,.02,.03,.01,.02,.02,-.04) txt.str<-c("A","B","C","CC","D1","D2","D3") text(xc,yc,txt.str) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `fr2vertsCCvert.reg`, `fr2edgesCMedge.reg.std.tri`, and `kfr2vertsCCvert.reg` ## Author(s) Elvan Ceyhan

fr2vertsCCvert.reg.basic.tri function