# The index of the edge region in a triangle that contains the point Returns the index of the edge whose region contains point, `p`, in the triangle `tri`$=T(A,B,C)$ with edge regions 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 `tri`. 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 $T(B,C,M)$, edge region 2 is $T(A,C,M)$, and edge region 3 is $T(A,B,M)$. 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.tri(p, tri, M) ``` ## Arguments - `p`: A 2D point for which `M`-edge 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 three elements - **re**: Index of the `M`-edge region that contains point, `p` in the triangle `tri`. - **tri**: The vertices of the triangle, where row labels are $A$, $B$, and $C$ 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 A<-c(1,1); B<-c(2,0); C<-c(1.5,2); Tr<-rbind(A,B,C); P<-c(1.4,1.2) M<-as.numeric(runif.tri(1,Tr)$g) #try also M<-c(1.6,1.2) rel.edge.tri(P,Tr,M) n<-20 #try also n<-40 Xp<-runif.tri(n,Tr)$g re<-vector() for (i in 1:n) re<-c(re,rel.edge.tri(Xp[i,],Tr,M)$re) re 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)} plot(Tr,xlab="",ylab="",axes=TRUE,pch=".",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05)) polygon(Tr) points(Xp,pch=".") L<-Tr; R<-rbind(M,M,M) segments(L[,1], L[,2], R[,1], R[,2], lty = 2) text(Xp,labels=factor(re)) txt<-rbind(Tr,M) xc<-txt[,1] yc<-txt[,2] txt.str<-c("A","B","C","M") text(xc,yc,txt.str) p1<-(A+B+M)/3 p2<-(B+C+M)/3 p3<-(A+C+M)/3 plot(Tr,xlab="",ylab="", main="Illustration of M-edge regions in a triangle", axes=TRUE,pch=".",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05)) polygon(Tr) L<-Tr; R<-rbind(M,M,M) segments(L[,1], L[,2], R[,1], R[,2], lty = 2) txt<-rbind(Tr,M,p1,p2,p3) xc<-txt[,1]+c(-.02,.02,.02,.02,.02,.02,.02) yc<-txt[,2]+c(.02,.02,.04,.05,.02,.02,.02) txt.str<-c("A","B","C","M","re=3","re=1","re=2") 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.basic.triCM`, `rel.edge.basic.tri`, `rel.edge.std.triCM`, and `edge.reg.triCM` ## Author(s) Elvan Ceyhan

rel.edge.tri function