# The indicator for the presence of an arc from a point in set `S` to the point `p` or Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard equilateral triangle case Returns $I($`p` in $N_{PE}(x,r)$ for some $x$ in `S`$)$ for `S`, in the standard equilateral triangle, 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 standard equilateral triangle $T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ with the expansion parameter $r \ge 1$ and vertex regions are based on 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 $T_e$ (which is equivalent to the circumcenter for $T_e$). Vertices of $T_e$ 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 $T_e$, 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.std.tri(S, p, 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. - `r`: A positive real number which serves as the expansion parameter in PE proximity region in the standard equilateral triangle $T_e=T((0,0),(1,0),(1/2,\sqrt{3}/2))$; 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 $I($`p` is in $U_{x in S} N_{PE}(x,r))$ for `S` in the standard equilateral triangle, 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. PE proximity region is constructed with respect to the standard equilateral triangle $T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ 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) r<-1.5 S<-rbind(Xp[1,],Xp[2,]) #try also S<-c(.5,.5) IarcPEset2pnt.std.tri(S,Xp[3,],r,M) IarcPEset2pnt.std.tri(S,Xp[3,],r=1,M) S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,]) IarcPEset2pnt.std.tri(S,Xp[3,],r,M) IarcPEset2pnt.std.tri(S,Xp[6,],r,M) IarcPEset2pnt.std.tri(S,Xp[6,],r=1.25,M) P<-c(.4,.2) S<-Xp[c(1,3,4),] IarcPEset2pnt.std.tri(Xp,P,r,M) ``` ## See Also `IarcPEset2pnt.tri`, `IarcPEstd.tri`, `IarcPEtri`, and `IarcCSset2pnt.std.tri` ## Author(s) Elvan Ceyhan

IarcPEset2pnt.std.tri function