# The indicator for a point being a dominating point for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case with $t=1$ Returns $I($`p` is a dominating point of the CS-PCD$)$ where the vertices of the CS-PCD are the 2D data set `Xp` in the standard equilateral triangle $T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2))$, that is, returns 1 if `p` is a dominating point of CS-PCD, returns 0 otherwise. Point, `p`, is in the edge region of edge re (default is `NULL`); vertices are labeled as $1,2,3$ in the order they are stacked row-wise in $T_e$, and the opposite edges are labeled with label of the vertices (that is, edge numbering is `1,2`, and `3` for edges $AB$, $BC$, and $AC$). CS proximity region is constructed with respect to $T_e$ with expansion parameter $t=1$ and edge regions are based on center of mass $CM=(1/2,\sqrt{3}/6)$. `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,ceyhan:mcap2012;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r Idom.num1CSt1std.tri(p, Xp, re = NULL, ch.data.pnt = FALSE) ``` ## Arguments - `p`: A 2D point that is to be tested for being a dominating point or not of the CS-PCD. - `Xp`: A set of 2D points which constitutes the vertices of the CS-PCD. - `re`: The index of the edge region in $T_e$ containing the point, either `1,2,3` or `NULL` (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 CS-PCD$)$ where the vertices of the CS-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); B<-c(1,0); C<-c(1/2,sqrt(3)/2); CM<-(A+B+C)/3 Te<-rbind(A,B,C); n<-10 set.seed(1) Xp<-runif.std.tri(n)$gen.points Idom.num1CSt1std.tri(Xp[3,],Xp) Idom.num1CSt1std.tri(c(1,2),c(1,2)) Idom.num1CSt1std.tri(c(1,2),c(1,2),ch.data.pnt = TRUE) gam.vec<-vector() for (i in 1:n) {gam.vec<-c(gam.vec,Idom.num1CSt1std.tri(Xp[i,],Xp))} ind.gam1<-which(gam.vec==1) ind.gam1 Xlim<-range(Te[,1],Xp[,1]) Ylim<-range(Te[,2],Xp[,2]) xd<-Xlim[2]-Xlim[1] yd<-Ylim[2]-Ylim[1] plot(Te,pch=".",xlab="",ylab="",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05)) polygon(Te) points(Xp) L<-Te; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE); 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(Te,CM) xc<-txt[,1]+c(-.02,.02,.01,.05) yc<-txt[,2]+c(.02,.02,.03,.02) txt.str<-c("A","B","C","CM") text(xc,yc,txt.str) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `Idom.num1CSstd.tri` ## Author(s) Elvan Ceyhan

Idom.num1CSt1std.tri function