# The index of the $CM$-vertex region in the standard equilateral triangle that contains a given point Returns the index of the vertex whose region contains point `p` in standard equilateral triangle $T_e=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ with vertex regions are constructed with center of mass CM (see the plots in the example for illustrations). The vertices of triangle, $T_e$, are labeled as $1,2,3$ according to the row number the vertex is recorded in $T_e$. If the point, `p`, is not inside $T_e$, then the function yields `NA` as output. The corresponding vertex region is the polygon with the vertex, $CM$, and midpoints of the edges adjacent to the vertex. See also (if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="ceyhan:Phd-thesis,ceyhan:comp-geo-2010,ceyhan:mcap2012;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r rel.vert.std.triCM(p) ``` ## Arguments - `p`: A 2D point for which $CM$-vertex region it resides in is to be determined in the standard equilateral triangle $T_e$. ## Returns A `list` with two elements - **rv**: Index of the vertex whose region contains point, `p`. - **tri**: The vertices of the triangle, $T_e$, where row number corresponds to the vertex index in `rv` with row $1=(0,0)$, row $2=(1,0)$, and row $3=(1/2,\sqrt{3}/2)$. ## Examples ```r A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2); Te<-rbind(A,B,C) n<-20 #try also n<-40 set.seed(1) Xp<-runif.std.tri(n)$gen.points rel.vert.std.triCM(Xp[1,]) Rv<-vector() for (i in 1:n) Rv<-c(Rv,rel.vert.std.triCM(Xp[i,])$rv) Rv CM<-(A+B+C)/3 D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2; Ds<-rbind(D1,D2,D3) Xlim<-range(Te[,1],Xp[,1]) Ylim<-range(Te[,2],Xp[,2]) xd<-Xlim[2]-Xlim[1] yd<-Ylim[2]-Ylim[1] plot(Te,asp=1,pch=".",xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05)) polygon(Te) points(Xp,pch=".",col=1) L<-matrix(rep(CM,3),ncol=2,byrow=TRUE); R<-Ds segments(L[,1], L[,2], R[,1], R[,2], lty = 2) txt<-rbind(Te,CM) xc<-txt[,1]+c(-.02,.03,.02,0) yc<-txt[,2]+c(.02,.02,.03,.05) txt.str<-c("A","B","C","CM") text(xc,yc,txt.str) text(Xp,labels=factor(Rv)) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `rel.vert.basic.triCM`, `rel.vert.tri`, `rel.vert.triCC`, `rel.vert.basic.triCC`, `rel.vert.triCM`, and `rel.vert.basic.tri` ## Author(s) Elvan Ceyhan

rel.vert.std.triCM function