# The index of the $CC$-vertex region in a tetrahedron that contains a point Returns the index of the vertex whose region contains point `p` in a tetrahedron $th=T(A,B,C,D)$ and vertex regions are based on the circumcenter $CC$ of `th`. (see the plots in the example for illustrations). The vertices of the tetrahedron `th` are labeled as $1=A$, $2=B$, $3=C$, and $4=C$ also according to the row number the vertex is recorded in `th`. If the point, `p`, is not inside `th`, then the function yields `NA` as output. The corresponding vertex region is the polygon whose interior points are closest to that vertex. If `th` is regular tetrahedron, 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;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r rel.vert.tetraCC(p, th) ``` ## Arguments - `p`: A 3D point for which $CC$-vertex region it resides in is to be determined in the tetrahedron `th`. - `th`: A $4 \times 3$ matrix with each row representing a vertex of the tetrahedron. ## Returns A `list` with two elements - **rv**: Index of the $CC$-vertex region that contains point, `p` in the tetrahedron `th` - **tri**: The vertices of the tetrahedron, where row number corresponds to the vertex index in `rv`. ## Examples ```r set.seed(123) A<-c(0,0,0)+runif(3,-.2,.2); B<-c(1,0,0)+runif(3,-.2,.2); C<-c(1/2,sqrt(3)/2,0)+runif(3,-.2,.2); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)+runif(3,-.2,.2); tetra<-rbind(A,B,C,D) n<-20 #try also n<-40 Xp<-runif.tetra(n,tetra)$g rel.vert.tetraCC(Xp[1,],tetra) Rv<-vector() for (i in 1:n) Rv<-c(Rv,rel.vert.tetraCC(Xp[i,],tetra)$rv) Rv CC<-circumcenter.tetra(tetra) CC Xlim<-range(tetra[,1],Xp[,1],CC[1]) Ylim<-range(tetra[,2],Xp[,2],CC[2]) Zlim<-range(tetra[,3],Xp[,3],CC[3]) xd<-Xlim[2]-Xlim[1] yd<-Ylim[2]-Ylim[1] zd<-Zlim[2]-Zlim[1] plot3D::scatter3D(tetra[,1],tetra[,2],tetra[,3], phi =0,theta=40, bty = "g", main="Scatterplot of data points \n and CC-vertex regions", xlim=Xlim+xd*c(-.05,.05), ylim=Ylim+yd*c(-.05,.05), zlim=Zlim+zd*c(-.05,.05), pch = 20, cex = 1, ticktype = "detailed") L<-rbind(A,A,A,B,B,C); R<-rbind(B,C,D,C,D,D) plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3], add=TRUE,lwd=2) #add the data points plot3D::points3D(Xp[,1],Xp[,2],Xp[,3],pch=".",cex=3, add=TRUE) plot3D::text3D(tetra[,1],tetra[,2],tetra[,3], labels=c("A","B","C","D"), add=TRUE) plot3D::text3D(CC[1],CC[2],CC[3], labels=c("CC"), add=TRUE) D1<-(A+B)/2; D2<-(A+C)/2; D3<-(A+D)/2; D4<-(B+C)/2; D5<-(B+D)/2; D6<-(C+D)/2; L<-rbind(D1,D2,D3,D4,D5,D6); R<-matrix(rep(CC,6),ncol=3,byrow=TRUE) plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3], add=TRUE,lty = 2) F1<-intersect.line.plane(A,CC,B,C,D) L<-matrix(rep(F1,4),ncol=3,byrow=TRUE); R<-rbind(D4,D5,D6,CC) plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3],col=2, add=TRUE,lty = 2) F2<-intersect.line.plane(B,CC,A,C,D) L<-matrix(rep(F2,4),ncol=3,byrow=TRUE); R<-rbind(D2,D3,D6,CC) plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3],col=3, add=TRUE,lty = 2) F3<-intersect.line.plane(C,CC,A,B,D) L<-matrix(rep(F3,4),ncol=3,byrow=TRUE); R<-rbind(D3,D5,D6,CC) plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3],col=4, add=TRUE,lty = 2) F4<-intersect.line.plane(D,CC,A,B,C) L<-matrix(rep(F4,4),ncol=3,byrow=TRUE); R<-rbind(D1,D2,D4,CC) plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3],col=5, add=TRUE,lty = 2) plot3D::text3D(Xp[,1],Xp[,2],Xp[,3], labels=factor(Rv), add=TRUE) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `rel.vert.tetraCM` and `rel.vert.triCC` ## Author(s) Elvan Ceyhan

rel.vert.tetraCC function