# The vertices of the Proportional Edge (PE) Proximity Region in a standard basic triangle Returns the vertices of the PE proximity region (which is itself a triangle) for a point in the standard basic triangle $T_b=T((0,0),(1,0),(c_1,c_2))=$`(rv=1,rv=2,rv=3)`. PE proximity region is defined with respect to the standard basic triangle $T_b$ with expansion parameter $r \ge 1$ and vertex 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 basic triangle $T_b$ or based on the circumcenter of $T_b$; default is $M=(1,1,1)$, i.e., the center of mass of $T_b$. Vertex regions are labeled as $1,2,3$ rowwise for the vertices of the triangle $T_b$. `rv` is the index of the vertex 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:mcap2012;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r NPEbasic.tri(p, r, c1, c2, M = c(1, 1, 1), rv = NULL) ``` ## Arguments - `p`: A 2D point whose PE proximity region is to be computed. - `r`: A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$. - `c1, c2`: Positive real numbers representing the top vertex in standard basic triangle $T_b=T((0,0),(1,0),(c_1,c_2))$, $c_1$ must be in $[0,1/2]$, $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$. - `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 basic triangle $T_b$ or the circumcenter of $T_b$ which may be entered as "CC" as well; default is $M=(1,1,1)$, i.e., the center of mass of $T_b$. - `rv`: Index of the `M`-vertex region containing the point `p`, either `1,2,3` or `NULL` (default is `NULL`). ## Returns Vertices of the triangular region which constitutes the PE proximity region with expansion parameter r and center `M` for a point `p` ## Examples ```r c1<-.4; c2<-.6 A<-c(0,0); B<-c(1,0); C<-c(c1,c2); Tb<-rbind(A,B,C); M<-as.numeric(runif.basic.tri(1,c1,c2)$g) #try also M<-c(.6,.2) r<-2 P1<-as.numeric(runif.basic.tri(1,c1,c2)$g) #try also P1<-c(.4,.2) NPEbasic.tri(P1,r,c1,c2,M) #or try Rv<-rel.vert.basic.tri(P1,c1,c2,M)$rv NPEbasic.tri(P1,r,c1,c2,M,Rv) P1<-c(1.4,1.2) P2<-c(1.5,1.26) NPEbasic.tri(P1,r,c1,c2,M) #gives an error if M=c(1.3,1.3) #since center is not the circumcenter or not in the interior of the triangle ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `NPEtri`, `NAStri`, `NCStri`, and `IarcPEbasic.tri` ## Author(s) Elvan Ceyhan

NPEbasic.tri function