# The indicator for a point being a dominating point for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one triangle 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 triangle `tri`, that is, returns 1 if `p` is a dominating point of PE-PCD, and returns 0 otherwise. Point, `p`, is in the vertex region of vertex `rv` (default is `NULL`); vertices are labeled as $1,2,3$ in the order they are stacked row-wise in `tri`. PE proximity region is constructed with respect to the triangle `tri` with expansion parameter $r \ge 1$ and vertex regions are based on center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle `tri` or based on the circumcenter of `tri`; default is $M=(1,1,1)$, i.e., the center of mass of `tri`. `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:masa-2007,ceyhan:dom-num-NPE-Spat2011,ceyhan:mcap2012;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r Idom.num1PEtri(p, Xp, tri, r, M = c(1, 1, 1), rv = NULL, ch.data.pnt = FALSE) ``` ## Arguments - `p`: A 2D point that is to be tested for being a dominating point or not of the PE-PCD. - `Xp`: A set of 2D points which constitutes the vertices of the PE-PCD. - `tri`: A $3 \times 2$ matrix with each row representing a vertex of the triangle. - `r`: A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$. - `M`: A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri` or the circumcenter of `tri` which may be entered as "CC" as well; default is $M=(1,1,1)$, i.e., the center of mass of `tri`. - `rv`: Index of the vertex whose region contains point `p`, `rv` takes the vertex labels as $1,2,3$ as in the row order of the vertices in `tri`. - `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, and 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 #try also n<-20 set.seed(1) Xp<-runif.tri(n,Tr)$g M<-as.numeric(runif.tri(1,Tr)$g) #try also M<-c(1.6,1.0) r<-1.5 #try also r<-2 Idom.num1PEtri(Xp[1,],Xp,Tr,r,M) Idom.num1PEtri(c(1,2),c(1,2),Tr,r,M) Idom.num1PEtri(c(1,2),c(1,2),Tr,r,M,ch.data.pnt = TRUE) gam.vec<-vector() for (i in 1:n) {gam.vec<-c(gam.vec,Idom.num1PEtri(Xp[i,],Xp,Tr,r,M))} ind.gam1<-which(gam.vec==1) ind.gam1 #or try Rv<-rel.vert.tri(Xp[1,],Tr,M)$rv Idom.num1PEtri(Xp[1,],Xp,Tr,r,M,Rv) Ds<-prj.cent2edges(Tr,M) if (dimension(M)==3) {M<-bary2cart(M,Tr)} #need to run this when M is given in barycentric coordinates Xlim<-range(Tr[,1],Xp[,1],M[1]) Ylim<-range(Tr[,2],Xp[,2],M[2]) xd<-Xlim[2]-Xlim[1] yd<-Ylim[2]-Ylim[1] plot(Tr,pch=".",xlab="",ylab="",axes=TRUE, xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05)) polygon(Tr) points(Xp,pch=1,col=1) L<-rbind(M,M,M); R<-Ds segments(L[,1], L[,2], R[,1], R[,2], lty=2) points(rbind(Xp[ind.gam1,]),pch=4,col=2) #rbind is to insert the points correctly if there is only one dominating point txt<-rbind(Tr,M,Ds) xc<-txt[,1]+c(-.02,.03,.02,-.02,.04,-.03,.0) yc<-txt[,2]+c(.02,.02,.05,-.03,.04,.06,-.07) txt.str<-c("A","B","C","M","D1","D2","D3") text(xc,yc,txt.str) P<-c(1.4,1) Idom.num1PEtri(P,P,Tr,r,M) Idom.num1PEtri(Xp[1,],Xp,Tr,r,M) Idom.num1PEtri(c(1,2),Xp,Tr,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.num1PEbasic.tri` and `Idom.num1AStri` ## Author(s) Elvan Ceyhan

Idom.num1PEtri function