# 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`$=T(A,B,C)$. 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 to the edges on the extension of the lines joining `M` to the vertices or based on the circumcenter of `tri`. 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, `M`, and projection points from `M` to the edges crossing the vertex (as the output of `prj.cent2edges(Tr,M)`) or $CC$-vertex region (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(Xp, tri, M) ``` ## 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. - `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`. ## Returns A `list` with two elements - **rv**: Indices of the vertices whose regions contains points in `Xp`. - **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); M<-c(1.6,1.0) P<-c(.4,.2) rel.verts.tri(P,Tr,M) n<-20 #try also n<-40 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) rel.verts.tri(Xp,Tr,M) rel.verts.tri(rbind(Xp,c(2,2)),Tr,M) rv<-rel.verts.tri(Xp,Tr,M) rv ifelse(identical(M,circumcenter.tri(Tr)), Ds<-rbind((B+C)/2,(A+C)/2,(A+B)/2),Ds<-prj.cent2edges(Tr,M)) Xlim<-range(Tr[,1],M[1],Xp[,1]) Ylim<-range(Tr[,2],M[2],Xp[,2]) xd<-Xlim[2]-Xlim[1] yd<-Ylim[2]-Ylim[1] if (dimension(M)==3) {M<-bary2cart(M,Tr)} #need to run this when M is given in barycentric coordinates plot(Tr,pch=".",xlab="",ylab="", main="Scatterplot of data points \n and M-vertex regions in a triangle", 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] yc<-Tr[,2] 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,.04,.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.nondegPE` ## Author(s) Elvan Ceyhan

rel.verts.tri function