# The indicator for the presence of an arc from a point to another for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one triangle case Returns $I($`p2` is in $N_{PE}(p1,r))$ for points `p1` and `p2`, 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 the expansion parameter $r \ge 1$. PE proximity region is constructed with respect to the triangle `tri` 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 `tri` or based on the circumcenter of `tri`; default is $M=(1,1,1)$, i.e., the center of mass of `tri`. `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 `tri`, 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:arc-density-PE,ceyhan:dom-num-NPE-Spat2011;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r IarcPEtri(p1, p2, tri, 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. - `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`. - `rv`: Index of the `M`-vertex region 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`, that is, returns 1 if `p2` is in $N_{PE}(p1,r)$, and returns 0 otherwise. ## Examples ```r A<-c(1,1); B<-c(2,0); C<-c(1.5,2); Tr<-rbind(A,B,C); M<-as.numeric(runif.tri(1,Tr)$g) #try also M<-c(1.6,1.0); r<-1.5 n<-3 set.seed(1) Xp<-runif.tri(n,Tr)$g IarcPEtri(Xp[1,],Xp[2,],Tr,r,M) P1<-as.numeric(runif.tri(1,Tr)$g) P2<-as.numeric(runif.tri(1,Tr)$g) IarcPEtri(P1,P2,Tr,r,M) P1<-c(.4,.2) P2<-c(1.8,.5) IarcPEtri(P1,P2,Tr,r,M) IarcPEtri(P2,P1,Tr,r,M) M<-c(1.3,1.3) r<-2 #or try Rv<-rel.vert.tri(P1,Tr,M)$rv IarcPEtri(P1,P2,Tr,r,M,Rv) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `IarcPEbasic.tri`, `IarcPEstd.tri`, `IarcAStri`, and `IarcCStri` ## Author(s) Elvan Ceyhan

IarcPEtri function