# The indicator for two points constituting a dominating set for Arc Slice Proximity Catch Digraphs (AS-PCDs) - one triangle case Returns $I(${`p1,p2`} is a dominating set of the AS-PCD$)$ where vertices of the AS-PCD are the 2D data set `Xp`), that is, returns 1 if {`p1,p2`} is a dominating set of AS-PCD, returns 0 otherwise. AS proximity regions are defined with respect to the triangle `tri`. 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 $1,2,3$ in the order they are stacked row-wise in `tri`. Vertex regions are based on the center `M="CC"` for circumcenter of `tri`; or $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle `tri`; default is `M="CC"` the circumcenter of `tri`. `ch.data.pnts` is for checking whether points `p1` and `p2` are data points in `Xp` or not (default is `FALSE`), so by default this function checks whether the points `p1` and `p2` would constitute dominating set if they 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,ceyhan:mcap2012;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r Idom.num2AStri( p1, p2, Xp, tri, M = "CC", rv1 = NULL, rv2 = NULL, ch.data.pnts = FALSE ) ``` ## Arguments - `p1, p2`: Two 2D points to be tested for constituting a dominating set of the AS-PCD. - `Xp`: A set of 2D points which constitutes the vertices of the AS-PCD. - `tri`: Three 2D points, stacked row-wise, each row representing a vertex of the triangle. - `M`: The center of the triangle. `"CC"` stands for circumcenter of the triangle `tri` or a 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle $T_b$; default is `M="CC"` i.e., the circumcenter of `tri`. - `rv1, rv2`: The indices of the vertices whose regions contains `p1` and `p2`, respectively. They take the vertex labels as $1,2,3$ as in the row order of the vertices in `tri` (default is `NULL` for both). - `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`} is a dominating set of the AS-PCD$)$ where vertices of the AS-PCD are the 2D data set `Xp`), that is, returns 1 if {`p1,p2`} is a dominating set of AS-PCD, returns 0 otherwise ## Examples ```r A<-c(1,1); B<-c(2,0); C<-c(1.5,2); Tr<-rbind(A,B,C); n<-10 set.seed(1) Xp<-runif.tri(n,Tr)$g M<-as.numeric(runif.tri(1,Tr)$g) #try also M<-c(1.6,1.2) Idom.num2AStri(Xp[1,],Xp[2,],Xp,Tr,M) Idom.num2AStri(Xp[1,],Xp[1,],Xp,Tr,M) #same two points cannot be a dominating set of size 2 Idom.num2AStri(c(.2,.4),Xp[2,],Xp,Tr,M) Idom.num2AStri(c(.2,.4),c(.2,.5),Xp,Tr,M) Idom.num2AStri(c(.2,.4),c(.2,.5),rbind(c(.2,.4),c(.2,.5)),Tr,M) #or try rv1<-rel.vert.triCC(c(.2,.4),Tr)$rv rv2<-rel.vert.triCC(c(.2,.5),Tr)$rv Idom.num2AStri(c(.2,.4),c(.2,.5),rbind(c(.2,.4),c(.2,.5)),Tr,M,rv1,rv2) ind.gam2<-vector() for (i in 1:(n-1)) for (j in (i+1):n) {if (Idom.num2AStri(Xp[i,],Xp[j,],Xp,Tr,M)==1) ind.gam2<-rbind(ind.gam2,c(i,j))} ind.gam2 #or try rv1<-rel.vert.triCC(Xp[1,],Tr)$rv rv2<-rel.vert.triCC(Xp[2,],Tr)$rv Idom.num2AStri(Xp[1,],Xp[2,],Xp,Tr,M,rv1,rv2) #or try rv1<-rel.vert.triCC(Xp[1,],Tr)$rv Idom.num2AStri(Xp[1,],Xp[2,],Xp,Tr,M,rv1) #or try Rv2<-rel.vert.triCC(Xp[2,],Tr)$rv Idom.num2AStri(Xp[1,],Xp[2,],Xp,Tr,M,rv2=Rv2) Idom.num2AStri(c(1.3,1.2),c(1.35,1.25),Xp,Tr,M) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `Idom.num2ASbasic.tri` ## Author(s) Elvan Ceyhan

Idom.num2AStri function