# Number of arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) and quantities related to the triangle - standard equilateral 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` in the standard equilateral triangle. It also provides number of vertices (i.e., number of data points inside the standard equilateral triangle $T_e$) and indices of the data points that reside in $T_e$. PE proximity region $N_{PE}(x,r)$ is defined with respect to the standard equilateral triangle $T_e=T(v=1,v=2,v=3)=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ 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 $T_e$; default is $M=(1,1,1)$, i.e., the center of mass of $T_e$. For the number of arcs, loops are not allowed so arcs are only possible for points inside $T_e$ for this function. See also (if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="ceyhan:arc-density-PE;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r num.arcsPEstd.tri(Xp, r, M = c(1, 1, 1)) ``` ## Arguments - `Xp`: A set of 2D points which constitute the vertices of the PE-PCD. - `r`: A positive real number which serves as the expansion parameter for 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 standard equilateral triangle $T_e$; default is $M=(1,1,1)$ i.e. the center of mass of $T_e$. ## Returns A `list` with the elements - **desc**: A short description of the output: number of arcs and quantities related to the standard equilateral 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 standard equilateral triangle $T_e$ - **num.in.tri**: Number of `Xp` points in the standard equilateral triangle, $T_e$ - **ind.in.tri**: The vector of indices of the `Xp` points that reside in $T_e$ - **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 $T_e$. - **vertices**: Vertices of the digraph, `Xp`. ## Examples ```r A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2); n<-10 #try also n<-20 set.seed(1) Xp<-runif.std.tri(n)$gen.points M<-c(.6,.2) #try also M<-c(1,1,1) Narcs = num.arcsPEstd.tri(Xp,r=1.25,M) Narcs summary(Narcs) oldpar <- par(pty="s") plot(Narcs,asp=1) par(oldpar) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `num.arcsPEtri`, `num.arcsPE`, and `num.arcsCSstd.tri` ## Author(s) Elvan Ceyhan

num.arcsPEstd.tri function