# The closest points among a data set in the vertex regions to the respective faces in a tetrahedron An object of class `"Extrema"`. Returns the closest data points among the data set, `Xp`, to face $i$ in `M`-vertex region $i$ for $i=1,2,3,4$ in the tetrahedron $th=T(A,B,C,D)$. Vertex labels are $A=1$, $B=2$, $C=3$, and $D=4$ and corresponding face labels are $BCD=1$, $ACD=2$, $ABD=3$, and $ABC=4$. Vertex regions are based on center `M` which can be the center of mass (`"CM"`) or circumcenter (`"CC"`) of `th`. ```r cl2faces.vert.reg.tetra(Xp, th, M = "CM") ``` ## Arguments - `Xp`: A set of 3D points representing the set of data points. - `th`: A $4 \times 3$ matrix with each row representing a vertex of the tetrahedron. - `M`: The center to be used in the construction of the vertex regions in the tetrahedron, `th`. Currently it only takes `"CC"` for circumcenter and `"CM"` for center of mass; default=`"CM"`. ## Returns A `list` with the elements - **txt1**: Vertex labels are $A=1$, $B=2$, $C=3$, and $D=4$ (correspond to row number in Extremum Points). - **txt2**: A short description of the distances as `"Distances from Closest Points to Faces ..."`. - **type**: Type of the extrema points - **desc**: A short description of the extrema points - **mtitle**: The `"main"` title for the plot of the extrema - **ext**: The extrema points, here, closest points to faces in the respective vertex region. - **ind.ext**: The data indices of extrema points, `ext`. - **X**: The input data, `Xp`, can be a `matrix` or `data frame` - **num.points**: The number of data points, i.e., size of `Xp` - **supp**: Support of the data points, here, it is `th` - **cent**: The center point used for construction of vertex regions, it is circumcenter of center of mass for this function - **ncent**: Name of the center, it is circumcenter `"CC"` or center of mass `"CM"` for this function. - **regions**: Vertex regions inside the tetrahedron `th` provided as a list. - **region.names**: Names of the vertex regions as `"vr=1","vr=2","vr=3","vr=4"` - **region.centers**: Centers of mass of the vertex regions inside `th`. - **dist2ref**: Distances from closest points in each vertex region to the corresponding face. ## Examples ```r A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3) set.seed(1) tetra<-rbind(A,B,C,D)+matrix(runif(12,-.25,.25),ncol=3) n<-10 #try also n<-20 Cent<-"CC" #try also "CM" n<-20 #try also n<-100 Xp<-runif.tetra(n,tetra)$g #try also Xp<-cbind(runif(n),runif(n),runif(n)) Ext<-cl2faces.vert.reg.tetra(Xp,tetra,Cent) Ext summary(Ext) plot(Ext) clf<-Ext$ext if (Cent=="CC") {M<-circumcenter.tetra(tetra)} if (Cent=="CM") {M<-apply(tetra,2,mean)} Xlim<-range(tetra[,1],Xp[,1],M[1]) Ylim<-range(tetra[,2],Xp[,2],M[2]) Zlim<-range(tetra[,3],Xp[,3],M[3]) xd<-Xlim[2]-Xlim[1] yd<-Ylim[2]-Ylim[1] zd<-Zlim[2]-Zlim[1] plot3D::scatter3D(Xp[,1],Xp[,2],Xp[,3], phi =0,theta=40, bty = "g", main="Closest Pointsin CC-Vertex Regions \n to the Opposite Faces", xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05), zlim=Zlim+zd*c(-.05,.05), pch = 20, cex = 1, ticktype = "detailed") #add the vertices of the tetrahedron plot3D::points3D(tetra[,1],tetra[,2],tetra[,3], add=TRUE) 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) plot3D::points3D(clf[,1],clf[,2],clf[,3], pch=4,col="red", add=TRUE) plot3D::text3D(tetra[,1],tetra[,2],tetra[,3], labels=c("A","B","C","D"), add=TRUE) #for center of mass use #Cent<-apply(tetra,2,mean) 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<-rbind(M,M,M,M,M,M) plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3], add=TRUE,lty=2) ``` ## See Also `fr2vertsCCvert.reg`, `fr2edgesCMedge.reg.std.tri`, `fr2vertsCCvert.reg.basic.tri` and `kfr2vertsCCvert.reg` ## Author(s) Elvan Ceyhan

cl2faces.vert.reg.tetra function