# The index of the $CM$-vertex region in a standard basic triangle form that contains a point Returns the index of the vertex whose region contains point `p` in the standard basic triangle form $T_b=T((0,0),(1,0),(c_1,c_2))$ where $c_1$ is in $[0,1/2]$, $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ and vertex regions are based on the center of mass CM=((1+c1)/3,c2/3) of $T_b$. (see the plots in the example for illustrations). The vertices of the standard basic triangle form $T_b$ are labeled as $1=(0,0)$, $2=(1,0)$,and $3=(c_1,c_2)$ also according to the row number the vertex is recorded in $T_b$. If the point, `p`, is not inside $T_b$, 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. Any given triangle can be mapped to the standard basic triangle form 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 form is useful for simulation studies under the uniformity hypothesis. See also (if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="ceyhan:Phd-thesis,ceyhan:comp-geo-2010,ceyhan:mcap2012,ceyhan:arc-density-PE;textual",package="pcds",cached_env=.Rdpack.currefs) ) ```r rel.vert.basic.triCM(p, c1, c2) ``` ## Arguments - `p`: A 2D point for which $CM$-vertex region it resides in is to be determined in the standard basic triangle form $T_b$. - `c1, c2`: Positive real numbers which constitute the upper vertex of the standard basic triangle form (i.e., the vertex adjacent to the shorter edges of $T_b$); $c_1$ must be in $[0,1/2]$, $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$. ## Returns A `list` with two elements - **rv**: Index of the $CM$-vertex region that contains point, `p` in the standard basic triangle form $T_b$ - **tri**: The vertices of the triangle, where row number corresponds to the vertex index in `rv` with row $1=(0,0)$, row $2=(1,0)$, and row $3=(c_1,c_2)$. ## Examples ```r c1<-.4; c2<-.6 P<-c(.4,.2) rel.vert.basic.triCM(P,c1,c2) A<-c(0,0);B<-c(1,0);C<-c(c1,c2); Tb<-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) n<-20 #try also n<-40 Xp<-runif.basic.tri(n,c1,c2)$g Rv<-vector() for (i in 1:n) Rv<-c(Rv,rel.vert.basic.triCM(Xp[i,],c1,c2)$rv) Rv Xlim<-range(Tb[,1],Xp[,1]) Ylim<-range(Tb[,2],Xp[,2]) xd<-Xlim[2]-Xlim[1] yd<-Ylim[2]-Ylim[1] plot(Tb,xlab="",ylab="",axes="T",pch=".",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05)) points(Xp,pch=".") polygon(Tb) L<-Ds; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE) segments(L[,1], L[,2], R[,1], R[,2], lty = 2) text(Xp,labels=factor(Rv)) txt<-rbind(Tb,CM,Ds) xc<-txt[,1]+c(-.03,.03,.02,-.01,.06,-.05,.0) yc<-txt[,2]+c(.02,.02,.02,.04,.02,.02,-.03) txt.str<-c("A","B","C","CM","D1","D2","D3") text(xc,yc,txt.str) plot(Tb,xlab="",ylab="",axes="T",pch=".",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05)) polygon(Tb) L<-Ds; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE) segments(L[,1], L[,2], R[,1], R[,2], lty = 2) RV1<-(A+D3+CM+D2)/4 RV2<-(B+D3+CM+D1)/4 RV3<-(C+D2+CM+D1)/4 txt<-rbind(RV1,RV2,RV3) xc<-txt[,1] yc<-txt[,2] txt.str<-c("rv=1","rv=2","rv=3") text(xc,yc,txt.str) txt<-rbind(Tb,CM,Ds) xc<-txt[,1]+c(-.03,.03,.02,-.01,.04,-.03,.0) yc<-txt[,2]+c(.02,.02,.02,.04,.02,.02,-.03) txt.str<-c("A","B","C","CM","D1","D2","D3") text(xc,yc,txt.str) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) #' @author Elvan Ceyhan ## See Also `rel.vert.triCM`, `rel.vert.tri`, `rel.vert.triCC`, `rel.vert.basic.triCC`, `rel.vert.basic.tri`, and `rel.vert.std.triCM`

rel.vert.basic.triCM function