1d:T7f3,# The vertices of the Proportional Edge (PE) Proximity Region in the standard regular tetrahedron Returns the vertices of the PE proximity region (which is itself a tetrahedron) for a point in the standard regular tetrahedron $T_h=T((0,0,0),(1,0,0),(1/2,\sqrt{3}/2,0),(1/2,\sqrt{3}/6,\sqrt{6}/3))=$ `(rv=1,rv=2,rv=3,rv=4)`. PE proximity region is defined with respect to the tetrahedron $T_h$ with expansion parameter $r \ge 1$ and vertex regions based on the circumcenter of $T_h$ (which is equivalent to the center of mass in the standard regular tetrahedron). Vertex regions are labeled as `1,2,3,4` rowwise for the vertices of the tetrahedron $T_h$. `rv` is the index of the vertex region `p` resides, with default=`NULL`. If `p` is outside of $T_h$, 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;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r NPEstd.tetra(p, r, rv = NULL) ``` ## Arguments - `p`: A 3D 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 $\ge 1$. - `rv`: Index of the vertex region containing the point, either `1,2,3,4` or `NULL` (default is `NULL`). ## Returns Vertices of the tetrahedron which constitutes the PE proximity region with expansion parameter `r` and circumcenter (or center of mass) for a point `p` in the standard regular tetrahedron ## Examples ```r A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3) tetra<-rbind(A,B,C,D) n<-3 Xp<-runif.std.tetra(n)$g r<-1.5 NPEstd.tetra(Xp[1,],r) #or try RV<-rel.vert.tetraCC(Xp[1,],tetra)$rv NPEstd.tetra(Xp[1,],r,rv=RV) NPEstd.tetra(c(-1,-1,-1),r,rv=NULL) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `NPEtetra`, `NPEtri` and `NPEint` ## Author(s) Elvan Ceyhan

NPEstd.tetra function