# Number of arcs of Arc Slice Proximity Catch Digraphs (AS-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 Arc Slice Proximity Catch Digraph (AS-PCD) whose vertices are the data points in `Xp` in the multiple triangle case (with triangulation based on `Yp` points). AS proximity regions are defined with respect to the Delaunay triangles based on `Yp` points and vertex regions in each triangle are based on the center `M="CC"` for circumcenter of each Delaunay triangle or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of each Delaunay triangle; default is `M="CC"` i.e., circumcenter of each triangle. 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:comp-geo-2010,ceyhan:mcap2012;textual",package="pcds",cached_env=.Rdpack.currefs) ) for more on AS-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.arcsAS(Xp, Yp, M = "CC") ``` ## Arguments - `Xp`: A set of 2D points which constitute the vertices of the AS-PCD. - `Yp`: A set of 2D points which constitute the vertices of the Delaunay triangles. - `M`: The center of the triangle. `"CC"` stands for circumcenter of each Delaunay triangle or 3D point in barycentric coordinates which serves as a center in the interior of each Delaunay triangle; default is `M="CC"` i.e., the circumcenter 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 AS-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 AS-PCD in the Delaunay triangles based on `Yp` points - **del.tri.ind**: A matrix of indices of Delaunay triangles based on `Yp` points, each column corresponds to the vector of indices of the vertices of one of the Delaunay 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<-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<-"CC" #try also M<-c(1,1,1) Narcs = num.arcsAS(Xp,Yp,M) Narcs summary(Narcs) plot(Narcs) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `num.arcsAStri`, `num.arcsPE`, and `num.arcsCS` ## Author(s) Elvan Ceyhan

num.arcsAS function