# The indicator for the presence of an arc from a point to another for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard equilateral triangle case Returns $I($`p2` is in $N_{PE}(p1,r))$ for points `p1` and `p2` in the standard equilateral triangle, that is, returns 1 if `p2` is in $N_{PE}(p1,r)$, and returns 0 otherwise, where $N_{PE}(x,r)$ is the PE proximity region for point $x$ with expansion parameter $r \ge 1$. PE proximity region is defined with respect to the standard equilateral triangle $T_e=T(v=1,v=2,v=3)=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ 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 $T_e$. `rv` is the index of the vertex region `p1` resides, with default=`NULL`. If `p1` and `p2` are distinct and either of them are outside $T_e$, it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops). See also (if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="ceyhan:Phd-thesis,ceyhan:comp-geo-2010,ceyhan:arc-density-CS;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r IarcPEstd.tri(p1, p2, r, M = c(1, 1, 1), rv = NULL) ``` ## Arguments - `p1`: A 2D point whose PE proximity region is constructed. - `p2`: A 2D point. The function determines whether `p2` is inside the PE proximity region of `p1` or not. - `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 $T_e$. - `rv`: The index of the vertex region in $T_e$ containing the point, either `1,2,3` or `NULL` (default is `NULL`). ## Returns $I($`p2` is in $N_{PE}(p1,r))$ for points `p1` and `p2` in the standard equilateral triangle, that is, returns 1 if `p2` is in $N_{PE}(p1,r)$, and returns 0 otherwise. ## Examples ```r A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2); Te<-rbind(A,B,C) n<-3 set.seed(1) Xp<-runif.std.tri(n)$gen.points M<-as.numeric(runif.std.tri(1)$g) #try also M<-c(.6,.2) IarcPEstd.tri(Xp[1,],Xp[3,],r=1.5,M) IarcPEstd.tri(Xp[1,],Xp[3,],r=2,M) #or try Rv<-rel.vert.std.triCM(Xp[1,])$rv IarcPEstd.tri(Xp[1,],Xp[3,],r=2,rv=Rv) P1<-c(.4,.2) P2<-c(.5,.26) r<-2 IarcPEstd.tri(P1,P2,r,M) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `IarcPEtri`, `IarcPEbasic.tri`, and `IarcCSstd.tri` ## Author(s) Elvan Ceyhan

IarcPEstd.tri function