# The arcs of Proportional Edge Proximity Catch Digraph (PE-PCD) for 2D data - multiple 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 multiple triangle case. PE proximity regions are defined with respect to the Delaunay triangles based on `Yp` points with expansion parameter $r \ge 1$ and vertex regions in each triangle are based on the center $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of each Delaunay triangle or based on circumcenter of each Delaunay triangle (default for $M=(1,1,1)$ which is the center of mass of the triangle). Each Delaunay triangle is first converted to an (nonscaled) basic triangle so that `M` will be the same type of center for each Delaunay triangle (this conversion is not necessary when `M` is $CM$). Convex hull of `Yp` is partitioned by the Delaunay triangles based on `Yp` points (i.e., multiple triangles are the set of these Delaunay triangles whose union constitutes the convex hull of `Yp` points). For the number of arcs, loops are not allowed so arcs are only possible for points inside the convex hull of `Yp` points. See (if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="ceyhan:Phd-thesis,ceyhan:arc-density-PE,ceyhan:dom-num-NPE-Spat2011;textual",package="pcds",cached_env=.Rdpack.currefs) ) for more on the PE-PCDs. Also, see (if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="okabe:2000,ceyhan:comp-geo-2010,sinclair:2016;textual",package="pcds",cached_env=.Rdpack.currefs) ) for more on Delaunay triangulation and the corresponding algorithm. ```r arcsPE(Xp, Yp, r, M = c(1, 1, 1)) ``` ## Arguments - `Xp`: A set of 2D points which constitute the vertices of the PE-PCD. - `Yp`: A set of 2D points which constitute the vertices of the Delaunay triangles. - `r`: A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$. - `M`: A 3D point in barycentric coordinates which serves as a center in the interior of each Delaunay triangle or circumcenter of each Delaunay triangle (for this, argument should be set as `M="CC"`), default for $M=(1,1,1)$ which is the center of mass of each triangle. ## Returns A `list` with the elements - **type**: A description of the type of the digraph - **parameters**: Parameters of the digraph, the center used to construct the vertex regions and the expansion parameter. - **tess.points**: Tessellation points, i.e., points on which the tessellation of the study region is performed, here, tessellation is the Delaunay triangulation based on `Yp` points. - **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 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 #nx is number of X points (target) and ny is number of Y points (nontarget) nx<-20; ny<-5; #try also nx<-40; ny<-10 or nx<-1000; ny<-10; set.seed(1) Xp<-cbind(runif(nx,0,1),runif(nx,0,1)) Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1)) #try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1)) M<-c(1,1,1) #try also M<-c(1,2,3) r<-1.5 #try also r<-2 Arcs<-arcsPE(Xp,Yp,r,M) #or try with the default center Arcs<-arcsPE(Xp,Yp,r) Arcs summary(Arcs) plot(Arcs) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `arcsPEtri`, `arcsAS`, and `arcsCS` ## Author(s) Elvan Ceyhan

arcsPE function