# The indicator for a 3D point being a dominating point for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one tetrahedron case Returns $I($`p` is a dominating point of the PE-PCD$)$ where the vertices of the PE-PCD are the 2D data set `Xp` in the tetrahedron `th`, that is, returns 1 if `p` is a dominating point of PE-PCD, returns 0 otherwise. Point, `p`, is in the vertex region of vertex `rv` (default is `NULL`); vertices are labeled as `1,2,3,4` in the order they are stacked row-wise in `th`. PE proximity region is constructed with respect to the tetrahedron `th` with expansion parameter $r \ge 1$ and vertex regions are based on center of mass (`M="CM"`) or circumcenter (`M="CC"`) only. and vertex regions are based on center of mass $CM$ (equivalent to circumcenter in this case). `ch.data.pnt` is for checking whether point `p` is a data point in `Xp` or not (default is `FALSE`), so by default this function checks whether the point `p` would be a dominating point if it actually were in the data set. 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 Idom.num1PEtetra(p, Xp, th, r, M = "CM", rv = NULL, ch.data.pnt = FALSE) ``` ## Arguments - `p`: A 3D point that is to be tested for being a dominating point or not of the PE-PCD. - `Xp`: A set of 3D points which constitutes the vertices of the PE-PCD. - `th`: A $4 \times 3$ matrix with each row representing a vertex of the tetrahedron. - `r`: A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$. - `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"`. - `rv`: Index of the vertex whose region contains point `p`, `rv` takes the vertex labels as `1,2,3,4` as in the row order of the vertices in standard tetrahedron, default is `NULL`. - `ch.data.pnt`: A logical argument for checking whether point `p` is a data point in `Xp` or not (default is `FALSE`). ## Returns $I($`p` is a dominating point of the PE-PCD$)$ where the vertices of the PE-PCD are the 2D data set `Xp`, that is, returns 1 if `p` is a dominating point, returns 0 otherwise ## 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) tetra<-rbind(A,B,C,D) n<-5 #try also n<-20 Xp<-runif.tetra(n,tetra)$g #try also Xp<-cbind(runif(n),runif(n),runif(n)) M<-"CM"; cent<-apply(tetra,2,mean) #center of mass #try also M<-"CC"; cent<-circumcenter.tetra(tetra) #circumcenter r<-2 P<-c(.4,.1,.2) Idom.num1PEtetra(Xp[1,],Xp,tetra,r,M) Idom.num1PEtetra(P,Xp,tetra,r,M) #or try RV<-rel.vert.tetraCC(Xp[1,],tetra)$rv Idom.num1PEtetra(Xp[1,],Xp,tetra,r,M,rv=RV) Idom.num1PEtetra(c(-1,-1,-1),Xp,tetra,r,M) Idom.num1PEtetra(c(-1,-1,-1),c(-1,-1,-1),tetra,r,M) gam.vec<-vector() for (i in 1:n) {gam.vec<-c(gam.vec,Idom.num1PEtetra(Xp[i,],Xp,tetra,r,M))} ind.gam1<-which(gam.vec==1) ind.gam1 g1.pts<-Xp[ind.gam1,] Xlim<-range(tetra[,1],Xp[,1],cent[1]) Ylim<-range(tetra[,2],Xp[,2],cent[2]) Zlim<-range(tetra[,3],Xp[,3],cent[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", 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) if (length(g1.pts)!=0) {plot3D::points3D(g1.pts[,1],g1.pts[,2],g1.pts[,3], pch=4,col="red", add=TRUE)} plot3D::text3D(tetra[,1],tetra[,2],tetra[,3], labels=c("A","B","C","D"), 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<-rbind(cent,cent,cent,cent,cent,cent) plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3], add=TRUE,lty=2) P<-c(.4,.1,.2) Idom.num1PEtetra(P,Xp,tetra,r,M) Idom.num1PEtetra(c(-1,-1,-1),Xp,tetra,r,M,ch.data.pnt = FALSE) #gives an error message if ch.data.pnt = TRUE since p is not a data point ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `Idom.num1PEstd.tetra`, `Idom.num1PEtri` and `Idom.num1PEbasic.tri` ## Author(s) Elvan Ceyhan

Idom.num1PEtetra function