# The indicator for the presence of an arc from a point in set `S` to the point `p` for Central Similarity Proximity Catch Digraphs (CS-PCDs) - one triangle case Returns I(`p` in $N_{CS}(x,t)$ for some $x$ in `S`), that is, returns 1 if `p` in $\cup_{x in S} N_{CS}(x,t)$, returns 0 otherwise. CS proximity region is constructed with respect to the triangle `tri` with the expansion parameter $t>0$ and edge 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`; default is $M=(1,1,1)$ i.e., the center of mass of `tri`. Edges of `tri`$=T(A,B,C)$, $AB$, $BC$, $AC$, are also labeled as edges 3, 1, and 2, 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 IarcCSset2pnt.tri(S, p, tri, t, 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. - `t`: A positive real number which serves as the expansion parameter in CS proximity region constructed in the triangle `tri`. - `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`; default is $M=(1,1,1)$ i.e., the center of mass of `tri`. ## Returns I(`p` is in $\cup_{x in S} N_{CS}(x,t)$), that is, returns 1 if `p` is in `S` or inside $N_{CS}(x,t)$ for at least one $x$ in `S`, returns 0 otherwise where CS 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)$gen.points S<-rbind(Xp[1,],Xp[2,]) #try also S<-c(1.5,1) M<-as.numeric(runif.tri(1,Tr)$g) #try also M<-c(1.6,1.0) tau<-.5 IarcCSset2pnt.tri(S,Xp[3,],Tr,tau,M) IarcCSset2pnt.tri(S,Xp[3,],Tr,t=1,M) IarcCSset2pnt.tri(S,Xp[3,],Tr,t=1.5,M) S<-rbind(c(.1,.1),c(.3,.4),c(.5,.3)) IarcCSset2pnt.tri(S,Xp[3,],Tr,tau,M) ``` ## See Also `IarcCSset2pnt.std.tri`, `IarcCStri`, `IarcCSstd.tri`, `IarcASset2pnt.tri`, and `IarcPEset2pnt.tri` ## Author(s) Elvan Ceyhan

IarcCSset2pnt.tri function