# The index of the vertex region in a standard basic triangle form that contains a given point Returns the index of the related vertex in the standard basic triangle form whose region contains point `p`. The standard basic triangle form is $T_b=T((0,0),(1,0),(c_1,c_2))$ where $c_1$ is in $[0,1/2]$, $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$.. 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 standard basic triangle form $T_b$. Vertices of the standard basic triangle form $T_b$ are labeled according to the row number the vertex is recorded, i.e., as 1=(0,0), 2=(1,0),and $3=(c_1,c_2)$. If the point, `p`, is not inside $T_b$, 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`. That is, `rv=1` has vertices $(0,0),D_3,M,D_2$; `rv=2` has vertices $(1,0),D_1,M,D_3$; and $rv=3$ has vertices $(c_1,c_2),D_2,M,D_1$ (see the illustration in the examples). Any given triangle can be mapped to the standard basic triangle form by a combination of rigid body motions (i.e., translation, rotation and reflection) and scaling, preserving uniformity of the points in the original triangle. Hence, standard basic triangle form is useful for simulation studies under the uniformity hypothesis. 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.basic.tri(p, c1, c2, M) ``` ## Arguments - `p`: A 2D point for which `M`-vertex region it resides in is to be determined in the standard basic triangle form $T_b$. - `c1, c2`: Positive real numbers which constitute the vertex of the standard basic triangle form adjacent to the shorter edges; $c_1$ must be in $[0,1/2]$, $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$. - `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 basic triangle form. ## 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 standard basic triangle form, $T_b$ - **tri**: The vertices of the standard basic triangle form, $T_b$, where row number corresponds to the vertex index `rv` with `rv=1` is row $1=(0,0)$, `rv=2` is row $2=(1,0)$, and $rv=3$ is row $3=(c_1,c_2)$. ## Examples ```r c1<-.4; c2<-.6 A<-c(0,0); B<-c(1,0); C<-c(c1,c2); Tb<-rbind(A,B,C); M<-c(.6,.2) P<-c(.4,.2) rel.vert.basic.tri(P,c1,c2,M) n<-20 #try also n<-40 set.seed(1) Xp<-runif.basic.tri(n,c1,c2)$g M<-as.numeric(runif.basic.tri(1,c1,c2)$g) #try also M<-c(.6,.2) Rv<-vector() for (i in 1:n) { Rv<-c(Rv,rel.vert.basic.tri(Xp[i,],c1,c2,M)$rv)} Rv Ds<-prj.cent2edges.basic.tri(c1,c2,M) Xlim<-range(Tb[,1],Xp[,1]) Ylim<-range(Tb[,2],Xp[,2]) xd<-Xlim[2]-Xlim[1] yd<-Ylim[2]-Ylim[1] if (dimension(M)==3) {M<-bary2cart(M,Tb)} #need to run this when M is given in barycentric coordinates plot(Tb,pch=".",xlab="",ylab="",axes=TRUE, xlim=Xlim+xd*c(-.1,.1),ylim=Ylim+yd*c(-.05,.05)) polygon(Tb) points(Xp,pch=".",col=1) L<-rbind(M,M,M); R<-Ds segments(L[,1], L[,2], R[,1], R[,2], lty = 2) xc<-Tb[,1]+c(-.04,.05,.04) yc<-Tb[,2]+c(.02,.02,.03) 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(-.02,.02,.02,-.03) 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.basic.triCM`, `rel.vert.tri`, `rel.vert.triCC`, `rel.vert.basic.triCC`, `rel.vert.triCM`, and `rel.vert.std.triCM` ## Author(s) Elvan Ceyhan

rel.vert.basic.tri function