# The index of the vertex region in a triangle that contains a given point Returns the index of the related vertex in the triangle, `tri`, whose region contains point `p`. Vertex regions are based on the general center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle `tri`. Vertices of the triangle `tri` are labeled according to the row number the vertex is recorded. If the point, `p`, is not inside `tri`, then the function yields `NA` as output. The corresponding vertex region is the polygon with the vertex, `M`, and projections from `M` to the edges on the lines joining vertices and `M` (see the illustration in the examples). 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 rel.vert.tri(p, tri, M) ``` ## Arguments - `p`: A 2D point for which `M`-vertex region it resides in is to be determined in the triangle `tri`. - `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`. ## Returns A `list` with two elements - **rv**: Index of the vertex whose region contains point, `p`; index of the vertex is the same as the row number in the triangle, `tri` - **tri**: The vertices of the triangle, `tri`, where row number corresponds to the vertex index `rv` with `rv=1` is row 1, `rv=2` is row 2, and $rv=3$ is is row 3. ## 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(1.5,1.6) rel.vert.tri(P,Tr,M) #try also rel.vert.tri(P,Tr,M=c(2,2)) #center is not in the interior of the triangle 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) Rv<-vector() for (i in 1:n) {Rv<-c(Rv,rel.vert.tri(Xp[i,],Tr,M)$rv)} Rv 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] 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="Illustration of M-Vertex Regions\n 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,-.04,0) yc<-txt[,2]+c(-.02,.04,.05,-.08) txt.str<-c("M","D1","D2","D3") text(xc,yc,txt.str) text(Xp,labels=factor(Rv)) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `rel.vert.triCM`, `rel.vert.triCC`, `rel.vert.basic.triCC`, `rel.vert.basic.triCM`, `rel.vert.basic.tri`, and `rel.vert.std.triCM` ## Author(s) Elvan Ceyhan

rel.vert.tri function