# The arcs of Proportional Edge Proximity Catch Digraph (PE-PCD) for 2D data - one triangle case An object of class `"PCDs"`. Returns arcs of PE-PCD as tails (or sources) and heads (or arrow ends) and related parameters and the quantities of the digraph. The vertices of the PE-PCD are the data points in `Xp` in the one triangle case. PE proximity regions are constructed with respect to the triangle `tri` with expansion parameter $r \ge 1$, i.e., arcs may exist only for points inside `tri`. It also provides various descriptions and quantities about the arcs of the PE-PCD such as number of arcs, arc density, etc. Vertex regions are 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` or based on the circumcenter of `tri`; default is $M=(1,1,1)$, i.e., the center of mass of `tri`. When the center is the circumcenter, `CC`, the vertex regions are constructed based on the orthogonal projections to the edges, while with any interior center `M`, the vertex regions are constructed using the extensions of the lines combining vertices with `M`. `M`-vertex regions are recommended spatial inference, due to geometry invariance property of the arc density and domination number the PE-PCDs based on uniform data. See also (if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="ceyhan:Phd-thesis,ceyhan:arc-density-PE;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r arcsPEtri(Xp, tri, r, M = c(1, 1, 1)) ``` ## Arguments - `Xp`: A set of 2D points which constitute the vertices of the PE-PCD. - `tri`: A $3 \times 2$ matrix with each row representing a vertex of the triangle. - `r`: A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$. - `M`: A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri` or the circumcenter of `tri` which may be entered as "CC" as well; default is $M=(1,1,1)$, i.e., the center of mass of `tri`. ## Returns A `list` with the elements - **type**: A description of the type of the digraph - **parameters**: Parameters of the digraph, the center `M` used to construct the vertex regions and the expansion parameter `r`. - **tess.points**: Tessellation points, i.e., points on which the tessellation of the study region is performed, here, tessellation points are the vertices of the support triangle `tri`. - **tess.name**: Name of the tessellation points `tess.points` - **vertices**: Vertices of the digraph, `Xp` points - **vert.name**: Name of the data set which constitutes the vertices of the digraph - **S**: Tails (or sources) of the arcs of PE-PCD for 2D data set `Xp` as vertices of the digraph - **E**: Heads (or arrow ends) of the arcs of PE-PCD for 2D data set `Xp` as vertices of the digraph - **mtitle**: Text for `"main"` title in the plot of the digraph - **quant**: Various quantities for the digraph: number of vertices, number of partition points, number of triangles, number of arcs, and arc density. ## Examples ```r A<-c(1,1); B<-c(2,0); C<-c(1.5,2); Tr<-rbind(A,B,C); n<-10 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) r<-1.5 #try also r<-2 Arcs<-arcsPEtri(Xp,Tr,r,M) #or try with the default center Arcs<-arcsPEtri(Xp,Tr,r); M= (Arcs$param)$cent Arcs summary(Arcs) plot(Arcs) #can add vertex regions #but we first need to determine center is the circumcenter or not, #see the description for more detail. CC<-circumcenter.tri(Tr) if (isTRUE(all.equal(M,CC))) {cent<-CC D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2; Ds<-rbind(D1,D2,D3) cent.name<-"CC" } else {cent<-M cent.name<-"M" Ds<-prj.cent2edges(Tr,M) } L<-rbind(cent,cent,cent); R<-Ds segments(L[,1], L[,2], R[,1], R[,2], lty=2) #now we can add the vertex names and annotation txt<-rbind(Tr,cent,Ds) xc<-txt[,1]+c(-.02,.02,.02,.02,.03,-.03,-.01) yc<-txt[,2]+c(.02,.02,.03,.06,.04,.05,-.07) txt.str<-c("A","B","C","M","D1","D2","D3") text(xc,yc,txt.str) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `arcsPE`, `arcsAStri`, and `arcsCStri` ## Author(s) Elvan Ceyhan

arcsPEtri function