# The indices of the vertex regions in a triangle that contains the points in a give data set Returns the indices of the vertices whose regions contain the points in data set `Xp` in a triangle `tri`$=(A,B,C)$ and vertex regions are based on the center `cent` which yields nondegenerate asymptotic distribution of the domination number of PE-PCD for uniform data in `tri` for expansion parameter `r` in $(1,1.5]$. Vertices of triangle `tri` are labeled as $1,2,3$ according to the row number the vertex is recorded if a point in `Xp` is not inside `tri`, then the function yields `NA` as output for that entry. The corresponding vertex region is the polygon with the vertex, `cent`, and projection points on the edges. The center label `cent` values `1,2,3` correspond to the vertices $M_1$, $M_2$, and $M_3$; with default 1 (see the examples for an illustration). See also (if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="ceyhan:Phd-thesis,ceyhan:dom-num-NPE-Spat2011,ceyhan:comp-geo-2010,ceyhan:mcap2012;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r rel.verts.tri.nondegPE(Xp, tri, r, cent = 1) ``` ## Arguments - `Xp`: A set of 2D points representing the set of data points for which indices of the vertex regions containing them are to be determined. - `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 in $(1,1.5]$ for this function. - `cent`: Index of the center (as $1,2,3$ corresponding to $M_1,\,M_2,\,M_3$) which gives nondegenerate asymptotic distribution of the domination number of PE-PCD for uniform data in `tri` for expansion parameter `r` in $(1,1.5]$; default `cent=1`. ## Returns A `list` with two elements - **rv**: Indices (i.e., a `vector` of indices) of the vertices whose region contains points in `Xp` in the triangle `tri` - **tri**: The vertices of the triangle, where row number corresponds to the vertex index in `rv`. ## Examples ```r A<-c(1,1); B<-c(2,0); C<-c(1.5,2); Tr<-rbind(A,B,C); r<-1.35 cent<-2 P<-c(1.4,1.0) rel.verts.tri.nondegPE(P,Tr,r,cent) n<-20 #try also n<-40 set.seed(1) Xp<-runif.tri(n,Tr)$g rel.verts.tri.nondegPE(Xp,Tr,r,cent) rel.verts.tri.nondegPE(rbind(Xp,c(2,2)),Tr,r,cent) rv<-rel.verts.tri.nondegPE(Xp,Tr,r,cent) M<-center.nondegPE(Tr,r)[cent,]; Ds<-prj.nondegPEcent2edges(Tr,r,cent) Xlim<-range(Tr[,1],Xp[,1]) Ylim<-range(Tr[,2],Xp[,2]) xd<-Xlim[2]-Xlim[1] yd<-Ylim[2]-Ylim[1] plot(Tr,pch=".",xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05)) polygon(Tr) points(Xp,pch=".",col=1) L<-rbind(M,M,M); R<-Ds segments(L[,1], L[,2], R[,1], R[,2], lty = 2) xc<-Tr[,1]+c(-.03,.05,.05) yc<-Tr[,2]+c(-.06,.02,.05) txt.str<-c("rv=1","rv=2","rv=3") text(xc,yc,txt.str) txt<-rbind(M,Ds) xc<-txt[,1]+c(.02,.04,-.03,0) yc<-txt[,2]+c(.07,.03,.05,-.07) txt.str<-c("M","D1","D2","D3") text(xc,yc,txt.str) text(Xp,labels=factor(rv$rv)) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `rel.verts.triCM`, `rel.verts.triCC`, and `rel.verts.tri` ## Author(s) Elvan Ceyhan

rel.verts.tri.nondegPE function