# Each function is for the presence of an arc from a point in one of the edge regions to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case Three indicator functions: `IarcCSstd.triRAB`, `IarcCSstd.triRBC` and `IarcCSstd.triRAC`. The function `IarcCSstd.triRAB` returns I(`p2` is in $N_{CS}(p1,t)$ for `p1` in $RAB$ (edge region for edge $AB$, i.e., edge 3) in the standard equilateral triangle $T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2))$; `IarcCSstd.triRBC` returns I(`p2` is in $N_{CS}(p1,t)$ for `p1` in $RBC$ (edge region for edge $BC$, i.e., edge 1) in $T_e$; and `IarcCSstd.triRAC` returns I(`p2` is in $N_{CS}(p1,t)$ for `p1` in $RAC$ (edge region for edge $AC$, i.e., edge 2) in $T_e$. That is, each function returns 1 if `p2` is in $N_{CS}(p1,t)$, returns 0 otherwise. CS proximity region is defined with respect to $T_e$ whose vertices are also labeled as $T_e=T(v=1,v=2,v=3)$ with 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 $T_e$ If `p1` and `p2` are distinct and `p1` is outside the corresponding edge region and `p2` is outside $T_e$, it returns 0, but if they are identical, then it returns 1 regardless of their location (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;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r IarcCSstd.triRAB(p1, p2, t, M) IarcCSstd.triRBC(p1, p2, t, M) IarcCSstd.triRAC(p1, p2, t, M) ``` ## Arguments - `p1`: A 2D point whose CS proximity region is constructed. - `p2`: A 2D point. The function determines whether `p2` is inside the CS proximity region of `p1` or not. - `t`: A positive real number which serves as the expansion parameter in CS proximity region. - `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 $T_e$. ## Returns Each function returns $I($`p2` is in $N_{CS}(p1,t))$ for `p1`, that is, returns 1 if `p2` is in $N_{CS}(p1,t)$, returns 0 otherwise ## Examples ```r #Examples for IarcCSstd.triRAB A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2); CM<-(A+B+C)/3 T3<-rbind(A,B,CM); set.seed(1) Xp<-runif.std.tri(3)$gen.points M<-as.numeric(runif.std.tri(1)$g) #try also M<-c(.6,.2) t<-1 IarcCSstd.triRAB(Xp[1,],Xp[2,],t,M) IarcCSstd.triRAB(c(.2,.5),Xp[2,],t,M) #Examples for IarcCSstd.triRBC A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2); CM<-(A+B+C)/3 T1<-rbind(B,C,CM); set.seed(1) Xp<-runif.std.tri(3)$gen.points M<-as.numeric(runif.std.tri(1)$g) #try also M<-c(.6,.2) t<-1 IarcCSstd.triRBC(Xp[1,],Xp[2,],t,M) IarcCSstd.triRBC(c(.2,.5),Xp[2,],t,M) #Examples for IarcCSstd.triRAC A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2); CM<-(A+B+C)/3 T2<-rbind(A,C,CM); set.seed(1) Xp<-runif.std.tri(3)$gen.points M<-as.numeric(runif.std.tri(1)$g) #try also M<-c(.6,.2) t<-1 IarcCSstd.triRAC(Xp[1,],Xp[2,],t,M) IarcCSstd.triRAC(c(.2,.5),Xp[2,],t,M) ``` ## See Also `IarcCSt1.std.triRAB`, `IarcCSt1.std.triRBC` and `IarcCSt1.std.triRAC` ## Author(s) Elvan Ceyhan

funsCSEdgeRegs function