# Number of arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) and related quantities of the induced subdigraphs for points in the Delaunay triangles - multiple triangle case An object of class `"NumArcs"`. Returns the number of arcs and various other quantities related to the Delaunay triangles for Proportional Edge Proximity Catch Digraph (PE-PCD) whose vertices 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 is 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;textual",package="pcds",cached_env=.Rdpack.currefs) ) for more on 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 num.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 - **desc**: A short description of the output: number of arcs and related quantities for the induced subdigraphs in the Delaunay triangles - **num.arcs**: Total number of arcs in all triangles, i.e., the number of arcs for the entire PE-PCD - **num.in.conv.hull**: Number of `Xp` points in the convex hull of `Yp` points - **num.in.tris**: The vector of number of `Xp` points in the Delaunay triangles based on `Yp` points - **weight.vec**: The `vector` of the areas of Delaunay triangles based on `Yp` points - **tri.num.arcs**: The `vector` of the number of arcs of the components of the PE-PCD in the Delaunay triangles based on `Yp` points - **del.tri.ind**: A matrix of indices of vertices of the Delaunay triangles based on `Yp` points, each column corresponds to the vector of indices of the vertices of one triangle. - **data.tri.ind**: A `vector` of indices of vertices of the Delaunay triangles in which data points reside, i.e., column number of `del.tri.ind` for each `Xp` point. - **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. - **vertices**: Vertices of the digraph, `Xp`. ## Examples ```r #nx is number of X points (target) and ny is number of Y points (nontarget) nx<-15; ny<-5; #try also nx<-40; ny<-10 or nx<-1000; ny<-10; set.seed(1) Xp<-cbind(runif(nx),runif(nx)) 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) Narcs = num.arcsPE(Xp,Yp,r=1.25,M) Narcs summary(Narcs) plot(Narcs) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `num.arcsPEtri`, `num.arcsPEstd.tri`, `num.arcsCS`, and `num.arcsAS` ## Author(s) Elvan Ceyhan

num.arcsPE function