# The indicator for two 3D points constituting a dominating set for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one tetrahedron case Returns $I(${`p1,p2`} is a dominating set of the PE-PCD$)$ where the vertices of the PE-PCD are the 3D data set `Xp` in the tetrahedron `th`, that is, returns 1 if {`p1,p2`} is a dominating set of PE-PCD, returns 0 otherwise. Point, `p1`, is in the region of vertex `rv1` (default is `NULL`) and point, `p2`, is in the region of vertex `rv2` (default is `NULL`); vertices (and hence `rv1` and `rv2`) 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. `ch.data.pnts` is for checking whether points `p1` and `p2` are both data points in `Xp` or not (default is `FALSE`), so by default this function checks whether the points `p1` and `p2` would constitute a dominating set if they actually were both 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.num2PEtetra( p1, p2, Xp, th, r, M = "CM", rv1 = NULL, rv2 = NULL, ch.data.pnts = FALSE ) ``` ## Arguments - `p1, p2`: Two 3D points to be tested for constituting a dominating set 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"`. - `rv1, rv2`: The indices of the vertices whose regions contains `p1` and `p2`, respectively. They take the vertex labels as `1,2,3,4` as in the row order of the vertices in `th` (default is `NULL` for both). - `ch.data.pnts`: A logical argument for checking whether both points `p1` and `p2` are data points in `Xp` or not (default is `FALSE`). ## Returns $I(${`p1,p2`} is a dominating set of the PE-PCD$)$ where the vertices of the PE-PCD are the 3D data set `Xp`), that is, returns 1 if {`p1,p2`} is a dominating set of PE-PCD, 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 set.seed(1) Xp<-runif.tetra(n,tetra)$g #try also Xp<-cbind(runif(n),runif(n),runif(n)) M<-"CM"; #try also M<-"CC"; r<-1.5 Idom.num2PEtetra(Xp[1,],Xp[2,],Xp,tetra,r,M) Idom.num2PEtetra(c(-1,-1,-1),Xp[2,],Xp,tetra,r,M) ind.gam2<-ind.gamn2<-vector() for (i in 1:(n-1)) for (j in (i+1):n) {if (Idom.num2PEtetra(Xp[i,],Xp[j,],Xp,tetra,r,M)==1) {ind.gam2<-rbind(ind.gam2,c(i,j)) } } ind.gam2 #or try rv1<-rel.vert.tetraCC(Xp[1,],tetra)$rv;rv2<-rel.vert.tetraCC(Xp[2,],tetra)$rv Idom.num2PEtetra(Xp[1,],Xp[2,],Xp,tetra,r,M,rv1,rv2) #or try rv1<-rel.vert.tetraCC(Xp[1,],tetra)$rv; Idom.num2PEtetra(Xp[1,],Xp[2,],Xp,tetra,r,M,rv1) #or try rv2<-rel.vert.tetraCC(Xp[2,],tetra)$rv Idom.num2PEtetra(Xp[1,],Xp[2,],Xp,tetra,r,M,rv2=rv2) P1<-c(.1,.1,.1) P2<-c(.4,.1,.2) Idom.num2PEtetra(P1,P2,Xp,tetra,r,M) Idom.num2PEtetra(c(-1,-1,-1),Xp[2,],Xp,tetra,r,M,ch.data.pnts = FALSE) #gives an error message if ch.data.pnts = TRUE #since not both points, p1 and p2, are data points in Xp ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `Idom.num2PEstd.tetra`, `Idom.num2PEtri` and `Idom.num2PEbasic.tri` ## Author(s) Elvan Ceyhan

Idom.num2PEtetra function