# The index of the vertex region in the standard equilateral triangle that contains a given point Returns the index of the vertex whose region contains point `p` in standard equilateral triangle $T_e=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ with vertex regions are constructed with center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of $T_e$. (see the plots in the example for illustrations). The vertices of triangle, $T_e$, are labeled as $1,2,3$ according to the row number the vertex is recorded in $T_e$. If the point, `p`, is not inside $T_e$, 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 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.std.tri(p, M) ``` ## Arguments - `p`: A 2D point for which `M`-vertex region it resides in is to be determined in the standard equilateral triangle $T_e$. - `M`: A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the standard equilateral triangle $T_e$. ## Returns A `list` with two elements - **rv**: Index of the vertex whose region contains point, `p`. - **tri**: The vertices of the triangle, $T_e$, where row number corresponds to the vertex index in `rv` with row $1=(0,0)$, row $2=(1,0)$, and row $3=(1/2,\sqrt{3}/2)$. ## Examples ```r A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2); Te<-rbind(A,B,C) n<-20 #try also n<-40 set.seed(1) Xp<-runif.std.tri(n)$gen.points M<-as.numeric(runif.std.tri(1)$g) #try also M<-c(.6,.2) rel.vert.std.tri(Xp[1,],M) Rv<-vector() for (i in 1:n) Rv<-c(Rv,rel.vert.std.tri(Xp[i,],M)$rv) Rv Ds<-prj.cent2edges(Te,M) Xlim<-range(Te[,1],Xp[,1]) Ylim<-range(Te[,2],Xp[,2]) xd<-Xlim[2]-Xlim[1] yd<-Ylim[2]-Ylim[1] if (dimension(M)==3) {M<-bary2cart(M,Te)} #need to run this when M is given in barycentric coordinates plot(Te,asp=1,pch=".",xlab="",ylab="",axes=TRUE, xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05)) polygon(Te) points(Xp,pch=".",col=1) L<-rbind(M,M,M); R<-Ds segments(L[,1], L[,2], R[,1], R[,2], lty = 2) txt<-rbind(Te,M) xc<-txt[,1]+c(-.02,.03,.02,0) yc<-txt[,2]+c(.02,.02,.03,.05) txt.str<-c("A","B","C","M") 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.std.triCM`, `rel.vert.tri`, `rel.vert.triCC`, `rel.vert.basic.triCC`, `rel.vert.triCM`, and `rel.vert.basic.tri` ## Author(s) Elvan Ceyhan

rel.vert.std.tri function