# The indicator for the presence of an arc from one 3D point to another 3D point for Proportional Edge Proximity Catch Digraphs (PE-PCDs) Returns $I($`p2` is in $N_{PE}(p1,r))$ for 3D points `p1` and `p2`, that is, returns 1 if `p2` is in $N_{PE}(p1,r)$, returns 0 otherwise, where $N_{PE}(x,r)$ is the PE proximity region for point $x$ with the expansion parameter $r \ge 1$. PE proximity region is constructed with respect to the tetrahedron `th` and vertex regions are based on the center `M` which is circumcenter (`"CC"`) or center of mass (`"CM"`) of `th` with default=`"CM"`. `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 `th`, 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;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r IarcPEtetra(p1, p2, th, r, M = "CM", rv = NULL) ``` ## Arguments - `p1`: A 3D point whose PE proximity region is constructed. - `p2`: A 3D point. The function determines whether `p2` is inside the PE proximity region of `p1` or not. - `th`: A $4 \times 3$ matrix with each row representing a vertex of the tetrahedron. - `r`: A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$. - `M`: The center to be used in the construction of the vertex regions in the tetrahedron, `th`. Currently it only takes `"CC"` for circumcenter and `"CM"` for center of mass; default=`"CM"`. - `rv`: Index of the `M`-vertex region containing the point, either `1,2,3,4` (default is `NULL`). ## Returns $I($`p2` is in $N_{PE}(p1,r))$ for `p1`, that is, returns 1 if `p2` is in $N_{PE}(p1,r)$, returns 0 otherwise ## 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 #try also n<-20 Xp<-runif.tetra(n,tetra)$g M<-"CM" #try also M<-"CC" r<-1.5 IarcPEtetra(Xp[1,],Xp[2,],tetra,r) #uses the default M="CM" IarcPEtetra(Xp[1,],Xp[2,],tetra,r,M) IarcPEtetra(c(.4,.4,.4),c(.5,.5,.5),tetra,r,M) #or try RV<-rel.vert.tetraCC(Xp[1,],tetra)$rv IarcPEtetra(Xp[1,],Xp[3,],tetra,r,M,rv=RV) P1<-c(.1,.1,.1) P2<-c(.5,.5,.5) IarcPEtetra(P1,P2,tetra,r,M) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `IarcPEstd.tetra`, `IarcPEtri` and `IarcPEint` ## Author(s) Elvan Ceyhan

IarcPEtetra function