# Number of arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) and quantities related to the triangle - one triangle case An object of class `"NumArcs"`. Returns the number of arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) whose vertices are the given 2D numerical data set, `Xp`. It also provides number of vertices (i.e., number of data points inside the triangle) and indices of the data points that reside in the triangle. PE proximity region $N_{PE}(x,r)$ is defined with respect to the triangle, `tri` with expansion parameter $r \ge 1$ and vertex regions are based on the 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 circumcenter of `tri`; default is $M=(1,1,1)$, i.e., the center of mass of `tri`. For the number of arcs, loops are not allowed so arcs are only possible for points inside the triangle `tri` for this function. See also (if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="ceyhan:Phd-thesis,ceyhan:stamet2016;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r num.arcsPEtri(Xp, tri, r, M = c(1, 1, 1)) ``` ## Arguments - `Xp`: A set of 2D points which constitute the vertices of 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 - **desc**: A short description of the output: number of arcs and quantities related to the triangle - **num.arcs**: Number of arcs of the PE-PCD - **tri.num.arcs**: Number of arcs of the induced subdigraph of the PE-PCD for vertices in the triangle `tri` - **num.in.tri**: Number of `Xp` points in the triangle, `tri` - **ind.in.tri**: The vector of indices of the `Xp` points that reside in the triangle - **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`. - **vertices**: Vertices of the digraph, `Xp`. ## Examples ```r A<-c(1,1); B<-c(2,0); C<-c(1.5,2); Tr<-rbind(A,B,C); n<-10 #try also n<-20 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) Narcs = num.arcsPEtri(Xp,Tr,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.arcsPEstd.tri`, `num.arcsPE`, `num.arcsCStri`, and `num.arcsAStri` ## Author(s) Elvan Ceyhan

num.arcsPEtri function