# The index of the $CC$-vertex region in a triangle that contains a point Returns the index of the vertex whose region contains point `p` in a triangle `tri`$=(A,B,C)$ and vertex regions are based on the circumcenter $CC$ of `tri`. (see the plots in the example for illustrations). The vertices of the triangle `tri` are labeled as $1=A$, $2=B$, and $3=C$ also 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 whose interior points are closest to that vertex. If `tri` is equilateral triangle, then $CC$ and $CM$ (center of mass) coincide. 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.triCC(p, tri) ``` ## Arguments - `p`: A 2D point for which $CC$-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 $CC$-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(1.5,2); Tr<-rbind(A,B,C); P<-c(1.3,1.2) rel.vert.triCC(P,Tr) CC<-circumcenter.tri(Tr) #the circumcenter D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2; Ds<-rbind(D1,D2,D3) Xlim<-range(Tr[,1],CC[1]) Ylim<-range(Tr[,2],CC[2]) xd<-Xlim[2]-Xlim[1] yd<-Ylim[2]-Ylim[1] plot(Tr,asp=1,xlab="",ylab="",pch=".",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05)) polygon(Tr) L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds segments(L[,1], L[,2], R[,1], R[,2], lty = 2) txt<-rbind(Tr,CC,Ds) xc<-txt[,1]+c(-.07,.08,.06,.12,-.1,-.1,-.09) yc<-txt[,2]+c(.02,-.02,.03,.0,.02,.06,-.04) txt.str<-c("A","B","C","CC","D1","D2","D3") text(xc,yc,txt.str) RV1<-(A+.5*(D3-A)+A+.5*(D2-A))/2 RV2<-(B+.5*(D3-B)+B+.5*(D1-B))/2 RV3<-(C+.5*(D2-C)+C+.5*(D1-C))/2 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) n<-20 #try also n<-40 Xp<-runif.tri(n,Tr)$g Rv<-vector() for (i in 1:n) Rv<-c(Rv,rel.vert.triCC(Xp[i,],Tr)$rv) Rv Xlim<-range(Tr[,1],Xp[,1]) Ylim<-range(Tr[,2],Xp[,2]) xd<-Xlim[2]-Xlim[1] yd<-Ylim[2]-Ylim[1] plot(Tr,asp=1,xlab="",ylab="", main="Illustration of CC-Vertex Regions\n in a Triangle", pch=".",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05)) polygon(Tr) points(Xp,pch=".") L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds segments(L[,1], L[,2], R[,1], R[,2], lty = 2) text(Xp,labels=factor(Rv)) txt<-rbind(Tr,CC,Ds) xc<-txt[,1]+c(-.07,.08,.06,.12,-.1,-.1,-.09) yc<-txt[,2]+c(.02,-.02,.03,.0,.02,.06,-.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 `rel.vert.tri`, `rel.vert.triCM`, `rel.vert.basic.triCM`, `rel.vert.basic.triCC`, `rel.vert.basic.tri`, and `rel.vert.std.triCM` ## Author(s) Elvan Ceyhan

rel.vert.triCC function