# The vertices of the Central Similarity (CS) Proximity Region in a general triangle Returns the vertices of the CS proximity region (which is itself a triangle) for a point in the triangle `tri`$=T(A,B,C)=$`(rv=1,rv=2,rv=3)`. CS proximity region is defined with respect to the triangle `tri` with expansion parameter $t>0$ and edge regions based on 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`. Edge regions are labeled as `1,2,3` rowwise for the corresponding vertices of the triangle `tri`. `re` is the index of the edge region `p` resides, with default=`NULL`. If `p` is outside of `tri`, it returns `NULL` for the proximity region. 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 NCStri(p, tri, t, M = c(1, 1, 1), re = NULL) ``` ## Arguments - `p`: A 2D point whose CS proximity region is to be computed. - `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. - `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`. - `re`: Index of the `M`-edge region containing the point `p`, either `1,2,3` or `NULL` (default is `NULL`). ## Returns Vertices of the triangular region which constitutes the CS proximity region with expansion parameter $t>0$ and center `M` for a point `p` ## Examples ```r A<-c(1,1); B<-c(2,0); C<-c(1.5,2); Tr<-rbind(A,B,C); tau<-1.5 M<-as.numeric(runif.tri(1,Tr)$g) #try also M<-c(1.6,1.2) n<-3 set.seed(1) Xp<-runif.tri(n,Tr)$g NCStri(Xp[1,],Tr,tau,M) P1<-as.numeric(runif.tri(1,Tr)$g) #try also P1<-c(.4,.2) NCStri(P1,Tr,tau,M) #or try re<-rel.edges.tri(P1,Tr,M)$re NCStri(P1,Tr,tau,M,re) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `NPEtri`, `NAStri`, and `IarcCStri` ## Author(s) Elvan Ceyhan

NCStri function