# The index of the $CM$-edge region in a standard basic triangle form that contains a point Returns the index of the edge whose region contains point, `p`, in the standard basic triangle form $T_b=T(A=(0,0),B=(1,0),C=(c_1,c_2)$ where $c_1$ is in $[0,1/2]$, $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ with edge regions based on center of mass $CM=(A+B+C)/3$. Edges are labeled as 3 for edge $AB$, 1 for edge $BC$, and 2 for edge $AC$. If the point, `p`, is not inside `tri`, then the function yields `NA` as output. Edge region 1 is the triangle $T(B,C,CM)$, edge region 2 is $T(A,C,CM)$, and edge region 3 is $T(A,B,CM)$. 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,ceyhan:arc-density-CS;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r rel.edge.basic.triCM(p, c1, c2) ``` ## Arguments - `p`: A 2D point for which $CM$-edge region it resides in is to be determined in the standard basic triangle form $T_b$. - `c1, c2`: Positive real numbers which constitute the upper vertex of the standard basic triangle form (i.e., the vertex adjacent to the shorter edges of $T_b$); $c_1$ must be in $[0,1/2]$, $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$. ## Returns A `list` with three elements - **re**: Index of the $CM$-edge region that contains point, `p` in the standard basic triangle form $T_b$ - **tri**: The vertices of the triangle, where row labels are $A=(0,0)$, $B=(1,0)$, and $C=(c_1,c_2)$ with edges are labeled as 3 for edge $AB$, 1 for edge $BC$, and 2 for edge $AC$. - **desc**: Description of the edge labels ## Examples ```r c1<-.4; c2<-.6 P<-c(.4,.2) rel.edge.basic.triCM(P,c1,c2) A<-c(0,0);B<-c(1,0);C<-c(c1,c2); Tb<-rbind(A,B,C) CM<-(A+B+C)/3 rel.edge.basic.triCM(A,c1,c2) rel.edge.basic.triCM(B,c1,c2) rel.edge.basic.triCM(C,c1,c2) rel.edge.basic.triCM(CM,c1,c2) n<-20 #try also n<-40 Xp<-runif.basic.tri(n,c1,c2)$g re<-vector() for (i in 1:n) re<-c(re,rel.edge.basic.triCM(Xp[i,],c1,c2)$re) re Xlim<-range(Tb[,1],Xp[,1]) Ylim<-range(Tb[,2],Xp[,2]) xd<-Xlim[2]-Xlim[1] yd<-Ylim[2]-Ylim[1] plot(Tb,xlab="",ylab="",axes=TRUE,pch=".",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05)) points(Xp,pch=".") polygon(Tb) L<-Tb; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE) segments(L[,1], L[,2], R[,1], R[,2], lty = 2) text(Xp,labels=factor(re)) txt<-rbind(Tb,CM) xc<-txt[,1]+c(-.03,.03,.02,0) yc<-txt[,2]+c(.02,.02,.02,-.04) txt.str<-c("A","B","C","CM") text(xc,yc,txt.str) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `rel.edge.triCM`, `rel.edge.tri`, `rel.edge.basic.tri`, `rel.edge.std.triCM`, and `edge.reg.triCM` ## Author(s) Elvan Ceyhan

rel.edge.basic.triCM function