# The indicator for a point being a dominating point for Arc Slice Proximity Catch Digraphs (AS-PCDs) - one triangle case Returns I(`p` is a dominating point of the AS-PCD whose vertices are the 2D data set `Xp`), that is, returns 1 if `p` is a dominating point of AS-PCD, returns 0 otherwise. Point, `p`, is in the region of vertex `rv` (default is `NULL`); vertices are labeled as $1,2,3$ in the order they are stacked row-wise in `tri`. AS proximity regions are defined with respect to the triangle `tri` and vertex regions are based on the 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 circumcenter of `tri`; default is `M="CC"`, i.e., circumcenter 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:comp-geo-2010,ceyhan:mcap2012;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r Idom.num1AStri(p, Xp, tri, M = "CC", 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 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`. - `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 AS-PCD whose vertices are the 2D data set `Xp`), that is, returns 1 if `p` is a dominating point of the 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.num1AStri(Xp[1,],Xp,Tr,M) Idom.num1AStri(Xp[1,],Xp[1,],Tr,M) Idom.num1AStri(c(1.5,1.5),c(1.6,1),Tr,M) Idom.num1AStri(c(1.6,1),c(1.5,1.5),Tr,M) gam.vec<-vector() for (i in 1:n) {gam.vec<-c(gam.vec,Idom.num1AStri(Xp[i,],Xp,Tr,M))} ind.gam1<-which(gam.vec==1) ind.gam1 #or try Rv<-rel.vert.triCC(Xp[1,],Tr)$rv Idom.num1AStri(Xp[1,],Xp,Tr,M,Rv) Idom.num1AStri(c(.2,.4),Xp,Tr,M) Idom.num1AStri(c(.2,.4),c(.2,.4),Tr,M) Xp2<-rbind(Xp,c(.2,.4)) Idom.num1AStri(Xp[1,],Xp2,Tr,M) if (dimension(M)==3) {M<-bary2cart(M,Tr)} #need to run this when M is given in barycentric coordinates CC<-circumcenter.tri(Tr) #the circumcenter if (isTRUE(all.equal(M,CC)) || identical(M,"CC")) {cent<-CC D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2; Ds<-rbind(D1,D2,D3) cent.name<-"CC" } else {cent<-M cent.name<-"M" Ds<-prj.cent2edges(Tr,M) } Xlim<-range(Tr[,1],Xp[,1]) Ylim<-range(Tr[,2],Xp[,2]) xd<-Xlim[2]-Xlim[1] yd<-Ylim[2]-Ylim[1] plot(A,pch=".",xlab="",ylab="",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05)) polygon(Tr) points(Xp) L<-rbind(cent,cent,cent); R<-Ds segments(L[,1], L[,2], R[,1], R[,2], lty=2) points(rbind(Xp[ind.gam1,]),pch=4,col=2) txt<-rbind(Tr,cent,Ds) xc<-txt[,1] yc<-txt[,2] txt.str<-c("A","B","C",cent.name,"D1","D2","D3") text(xc,yc,txt.str) Idom.num1AStri(c(1.5,1.1),Xp,Tr,M) Idom.num1AStri(c(1.5,1.1),Xp,Tr,M) Idom.num1AStri(c(1.5,1.1),Xp,Tr,M,ch.data.pnt=FALSE) #gives an error message if ch.data.pnt=TRUE since point p is not a data point in Xp ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `Idom.num1ASbasic.tri` ## Author(s) Elvan Ceyhan

Idom.num1AStri function