# The indicator for the presence of an arc from a point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case Returns $I($`p2` is in $N_{CS}(p1,t))$ for points `p1` and `p2`, that is, returns 1 if `p2` is in $N_{CS}(p1,t)$, returns 0 otherwise, where $N_{CS}(x,t)$ is the CS proximity region for point $x$ with expansion parameter $t >0$. CS proximity region is defined with respect to the standard equilateral triangle $T_e=T(v=1,v=2,v=3)=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ 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 $T_e$; default is $M=(1,1,1)$ i.e., the center of mass of $T_e$. `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 $T_e$, 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:comp-geo-2010,ceyhan:arc-density-CS;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r IarcCSstd.tri(p1, p2, t, M = c(1, 1, 1), re = NULL) ``` ## 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 $T_e$; default is $M=(1,1,1)$ i.e. the center of mass of $T_e$. - `re`: The index of the edge region in $T_e$ containing the point, either `1,2,3` or `NULL` (default is `NULL`). ## Returns $I($`p2` is in $N_{CS}(p1,t))$ for points `p1` and `p2`, that is, returns 1 if `p2` is in $N_{CS}(p1,t)$, returns 0 otherwise ## Examples ```r A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2); Te<-rbind(A,B,C) n<-3 set.seed(1) Xp<-runif.std.tri(n)$gen.points M<-as.numeric(runif.std.tri(1)$g) #try also M<-c(.6,.2) or M=(A+B+C)/3 IarcCSstd.tri(Xp[1,],Xp[3,],t=2,M) IarcCSstd.tri(c(0,1),Xp[3,],t=2,M) #or try Re<-rel.edge.tri(Xp[1,],Te,M) $re IarcCSstd.tri(Xp[1,],Xp[3,],t=2,M,Re) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `IarcCStri`, `IarcCSbasic.tri`, and `IarcPEstd.tri` ## Author(s) Elvan Ceyhan

IarcCSstd.tri function