# The indicator for the presence of an arc from a point in set `S` to the point `p` for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one triangle case Returns $I($`p` in $N_{PE}(x,r)$ for some $x$ in `S`$)$, that is, returns 1 if `p` is in $\cup_{x in S}N_{PE}(x,r)$, and returns 0 otherwise. PE proximity region is constructed with respect to the triangle `tri` with the 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 the circumcenter of `tri`; default is $M=(1,1,1)$, i.e., the center of mass of `tri`. Vertices of `tri` are also labeled as 1, 2, and 3, respectively. If `p` is not in `S` and either `p` or all points in `S` are outside `tri`, it returns 0, but if `p` is in `S`, then it always returns 1 regardless of its location (i.e., loops are allowed). ```r IarcPEset2pnt.tri(S, p, tri, r, M = c(1, 1, 1)) ``` ## Arguments - `S`: A set of 2D points. Presence of an arc from a point in `S` to point `p` is checked by the function. - `p`: A 2D point. Presence of an arc from a point in `S` to point `p` is checked by the function. - `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 constructed in the triangle `tri`; 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 $I($`p` is in $U_{x in S} N_{PE}(x,r))$, that is, returns 1 if `p` is in `S` or inside $N_{PE}(x,r)$ for at least one $x$ in `S`, and returns 0 otherwise, where PE proximity region is constructed with respect to the triangle `tri` ## Examples ```r A<-c(1,1); B<-c(2,0); C<-c(1.5,2); Tr<-rbind(A,B,C); n<-10 set.seed(1) Xp<-runif.tri(n,Tr)$gen.points M<-as.numeric(runif.tri(1,Tr)$g) #try also M<-c(1.6,1.0) r<-1.5 S<-rbind(Xp[1,],Xp[2,]) #try also S<-c(1.5,1) IarcPEset2pnt.tri(S,Xp[3,],Tr,r,M) IarcPEset2pnt.tri(S,Xp[3,],r=1,Tr,M) S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,]) IarcPEset2pnt.tri(S,Xp[3,],Tr,r,M) S<-rbind(c(.1,.1),c(.3,.4),c(.5,.3)) IarcPEset2pnt.tri(S,Xp[3,],Tr,r,M) P<-c(.4,.2) S<-Xp[c(1,3,4),] IarcPEset2pnt.tri(Xp,P,Tr,r,M) ``` ## See Also `IarcPEset2pnt.std.tri`, `IarcPEtri`, `IarcPEstd.tri`, `IarcASset2pnt.tri`, and `IarcCSset2pnt.tri` ## Author(s) Elvan Ceyhan

IarcPEset2pnt.tri function