# The indicator for three 3D points constituting a dominating set for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard regular tetrahedron case Returns $I(\{$`p1,p2,pt3`} is a dominating set of the PE-PCD$)$ where the vertices of the PE-PCD are the 3D data set `Xp` in the standard regular tetrahedron $T_h=T((0,0,0),(1,0,0),(1/2,\sqrt{3}/2,0),(1/2,\sqrt{3}/6,\sqrt{6}/3))$, that is, returns 1 if {`p1,p2,pt3`} is a dominating set of PE-PCD, returns 0 otherwise. Point, `p1`, is in the region of vertex `rv1` (default is `NULL`), point, `p2`, is in the region of vertex `rv2` (default is `NULL`); point, `pt3`), is in the region of vertex `rv3`) (default is `NULL`); vertices (and hence `rv1, rv2` and `rv3`) are labeled as `1,2,3,4` in the order they are stacked row-wise in $T_h$. PE proximity region is constructed with respect to the tetrahedron $T_h$ with expansion parameter $r \ge 1$ and vertex regions are based on center of mass $CM$ (equivalent to circumcenter in this case). `ch.data.pnts` is for checking whether points `p1`, `p2` and `pt3` are all data points in `Xp` or not (default is `FALSE`), so by default this function checks whether the points `p1`, `p2` and `pt3` would constitute a dominating set if they actually were all 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.num3PEstd.tetra( p1, p2, pt3, Xp, r, rv1 = NULL, rv2 = NULL, rv3 = NULL, ch.data.pnts = FALSE ) ``` ## Arguments - `p1, p2, pt3`: Three 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. - `r`: A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$. - `rv1, rv2, rv3`: The indices of the vertices whose regions contains `p1`, `p2` and `pt3`, respectively. They take the vertex labels as `1,2,3,4` as in the row order of the vertices in $T_h$ (default is `NULL` for all). - `ch.data.pnts`: A logical argument for checking whether points `p1` and `p2` are data points in `Xp` or not (default is `FALSE`). ## Returns $I(\{$`p1,p2,pt3`} 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,pt3`} is a dominating set of PE-PCD, returns 0 otherwise ## Examples ```r set.seed(123) 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 20, 40, 100 (larger n may take a long time) Xp<-runif.std.tetra(n)$g #try also Xp<-cbind(runif(n),runif(n),runif(n)) r<-1.25 Idom.num3PEstd.tetra(Xp[1,],Xp[2,],Xp[3,],Xp,r) ind.gam3<-vector() for (i in 1:(n-2)) for (j in (i+1):(n-1)) for (k in (j+1):n) {if (Idom.num3PEstd.tetra(Xp[i,],Xp[j,],Xp[k,],Xp,r)==1) ind.gam3<-rbind(ind.gam3,c(i,j,k))} ind.gam3 #or try rv1<-rel.vert.tetraCC(Xp[1,],tetra)$rv; rv2<-rel.vert.tetraCC(Xp[2,],tetra)$rv; rv3<-rel.vert.tetraCC(Xp[3,],tetra)$rv Idom.num3PEstd.tetra(Xp[1,],Xp[2,],Xp[3,],Xp,r,rv1,rv2,rv3) #or try rv1<-rel.vert.tetraCC(Xp[1,],tetra)$rv; Idom.num3PEstd.tetra(Xp[1,],Xp[2,],Xp[3,],Xp,r,rv1) #or try rv2<-rel.vert.tetraCC(Xp[2,],tetra)$rv Idom.num3PEstd.tetra(Xp[1,],Xp[2,],Xp[3,],Xp,r,rv2=rv2) P1<-c(.1,.1,.1) P2<-c(.3,.3,.3) P3<-c(.4,.1,.2) Idom.num3PEstd.tetra(P1,P2,P3,Xp,r) Idom.num3PEstd.tetra(Xp[1,],c(1,1,1),Xp[3,],Xp,r,ch.data.pnts = FALSE) #gives an error message if ch.data.pnts = TRUE since not all points are data points in Xp ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `Idom.num3PEtetra` ## Author(s) Elvan Ceyhan

Idom.num3PEstd.tetra function