# Incidence matrix for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard equilateral triangle case Returns the incidence matrix for the PE-PCD whose vertices are the given 2D numerical data set, `Xp`, in the standard equilateral triangle $T_e=T(v=1,v=2,v=3)=T((0,0),(1,0),(1/2,\sqrt{3}/2))$. PE proximity region is constructed with respect to the standard equilateral triangle $T_e$ 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 $T_e$. Loops are allowed, so the diagonal entries are all equal to 1. See also (if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="ceyhan:Phd-thesis,ceyhan:comp-geo-2010,ceyhan:dom-num-NPE-Spat2011;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r inci.matPEstd.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 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 standard equilateral triangle $T_e$; default is $M=(1,1,1)$ i.e. the center of mass of $T_e$. ## Returns Incidence matrix for the PE-PCD with vertices being 2D data set, `Xp` in the standard equilateral triangle where PE proximity regions are defined with `M`-vertex regions. ## Examples ```r A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2); Te<-rbind(A,B,C) n<-10 set.seed(1) Xp<-runif.std.tri(n)$gen.points M<-as.numeric(runif.std.tri(1)$g) #try also M<-c(.6,.2) inc.mat<-inci.matPEstd.tri(Xp,r=1.25,M) inc.mat sum(inc.mat)-n num.arcsPEstd.tri(Xp,r=1.25) dom.num.greedy(inc.mat) Idom.num.up.bnd(inc.mat,2) #try also dom.num.exact(inc.mat) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `inci.matPEtri`, `inci.matPE`, and `inci.matCSstd.tri` ## Author(s) Elvan Ceyhan

inci.matPEstd.tri function