# The index of the $CM$-vertex region in a triangle that contains a given point Returns the index of the vertex whose region contains point `p` in the triangle `tri`$=(y_1,y_2,y_3)$ with vertex regions are constructed with center of mass $CM=(y_1+y_2+y_3)/3$ (see the plots in the example for illustrations). The vertices of triangle, `tri`, are labeled as $1,2,3$ according to the row number the vertex is recorded in `tri`. If the point, `p`, is not inside `tri`, 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 (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.triCM(p, tri) ``` ## Arguments - `p`: A 2D point for which $CM$-vertex region it resides in is to be determined in the triangle `tri`. - `tri`: A $3 \times 2$ matrix with each row representing a vertex of the triangle. ## Returns A `list` with two elements - **rv**: Index of the $CM$-vertex region that contains point, `p` in the triangle `tri`. - **tri**: The vertices of the triangle, where row number corresponds to the vertex index in `rv`. ## Examples ```r A<-c(1,1); B<-c(2,0); C<-c(1.6,2); Tr<-rbind(A,B,C); P<-c(1.4,1.2) rel.vert.triCM(P,Tr) n<-20 #try also n<-40 Xp<-runif.tri(n,Tr)$g Rv<-vector() for (i in 1:n) Rv<-c(Rv,rel.vert.triCM(Xp[i,],Tr)$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(Tr[,1],Xp[,1]) Ylim<-range(Tr[,2],Xp[,2]) xd<-Xlim[2]-Xlim[1] yd<-Ylim[2]-Ylim[1] plot(Tr,xlab="",ylab="",axes=TRUE,pch=".",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05)) polygon(Tr) points(Xp,pch=".") 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(Tr,CM,D1,D2,D3) xc<-txt[,1]+c(-.02,.02,.02,-.02,.02,-.01,-.01) yc<-txt[,2]+c(-.02,-.04,.06,-.02,.02,.06,-.06) txt.str<-c("rv=1","rv=2","rv=3","CM","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 `rel.vert.tri`, `rel.vert.triCC`, `rel.vert.basic.triCM`, `rel.vert.basic.triCC`, `rel.vert.basic.tri`, and `rel.vert.std.triCM` ## Author(s) Elvan Ceyhan

rel.vert.triCM function