# The arcs of Central Similarity Proximity Catch Digraphs (CS-PCD) for 2D data - one triangle case An object of class `"PCDs"`. Returns arcs of CS-PCD as tails (or sources) and heads (or arrow ends) and related parameters and the quantities of the digraph. The vertices of the CS-PCD are the data points in `Xp` in the one triangle case. CS proximity regions are constructed with respect to the triangle `tri` with expansion parameter $t>0$, i.e., arcs may exist for points only inside `tri`. It also provides various descriptions and quantities about the arcs of the CS-PCD such as number of arcs, arc density, etc. Edge 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`; default is $M=(1,1,1)$ i.e., the center of mass of `tri`. See also (if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="ceyhan:Phd-thesis,ceyhan:arc-density-CS,ceyhan:test2014;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r arcsCStri(Xp, tri, t, M = c(1, 1, 1)) ``` ## Arguments - `Xp`: A set of 2D points which constitute the vertices of the CS-PCD. - `tri`: A $3 \times 2$ matrix with each row representing a vertex of the triangle. - `t`: A positive real number which serves as the expansion parameter in CS proximity region. - `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`; 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 edge regions and the expansion parameter `t`. - **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 constitute the vertices of the digraph - **S**: Tails (or sources) of the arcs of CS-PCD for 2D data set `Xp` as vertices of the digraph - **E**: Heads (or arrow ends) of the arcs of CS-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) t<-1.5 #try also t<-2 Arcs<-arcsCStri(Xp,Tr,t,M) #or try with the default center Arcs<-arcsCStri(Xp,Tr,t); M= (Arcs$param)$c Arcs summary(Arcs) plot(Arcs) #can add edge regions L<-rbind(M,M,M); R<-Tr segments(L[,1], L[,2], R[,1], R[,2], lty=2) #now we can add the vertex names and annotation txt<-rbind(Tr,M) xc<-txt[,1]+c(-.02,.03,.02,.03) yc<-txt[,2]+c(.02,.02,.03,.06) txt.str<-c("A","B","C","M") text(xc,yc,txt.str) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `arcsCS`, `arcsAStri` and `arcsPEtri` ## Author(s) Elvan Ceyhan

arcsCStri function