# The indicator for the presence of an arc from a point to another for Arc Slice Proximity Catch Digraphs (AS-PCDs) - one triangle case Returns $I(p2 \in N_{AS}(p1))$ for points `p1` and `p2`, that is, returns 1 if $p2$ is in $N_{AS}(p1)$, returns 0 otherwise, where $N_{AS}(x)$ is the AS proximity region for point $x$. AS proximity regions are constructed with respect to the triangle, `tri`$=T(A,B,C)=$`(rv=1,rv=2,rv=3)`, 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 the triangle `tri` or based on circumcenter of `tri`; default is `M="CC"`, i.e., circumcenter of `tri`. `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 `tri`, the function 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:mcap2012;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r IarcAStri(p1, p2, tri, M = "CC", rv = NULL) ``` ## Arguments - `p1`: A 2D point whose AS proximity region is constructed. - `p2`: A 2D point. The function determines whether `p2` is inside the AS proximity region of `p1` or not. - `tri`: Three 2D points, stacked row-wise, each row representing a vertex of the triangle. - `M`: The center of the triangle. `"CC"` stands for circumcenter of the triangle `tri` or a 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of `tri`; default is `M="CC"` i.e., the circumcenter of `tri`. - `rv`: The index of the `M`-vertex region in `tri` containing the point, either `1,2,3` or `NULL` (default is `NULL`). ## Returns $I(p2 \in N_{AS}(p1))$ for `p1`, that is, returns 1 if $p2$ is in $N_{AS}(p1)$, returns 0 otherwise ## Examples ```r A<-c(1,1); B<-c(2,0); C<-c(1.5,2); Tr<-rbind(A,B,C); M<-as.numeric(runif.tri(1,Tr)$g) #try also M<-c(1.6,1.2) P1<-as.numeric(runif.tri(1,Tr)$g) P2<-as.numeric(runif.tri(1,Tr)$g) IarcAStri(P1,P2,Tr,M) P1<-c(1.3,1.2) P2<-c(1.8,.5) IarcAStri(P1,P2,Tr,M) IarcAStri(P1,P1,Tr,M) #or try Rv<-rel.vert.triCC(P1,Tr)$rv IarcAStri(P1,P2,Tr,M,Rv) P3<-c(1.6,1.4) IarcAStri(P1,P3,Tr,M) P4<-c(1.5,1.0) IarcAStri(P1,P4,Tr,M) P5<-c(.5,1.0) IarcAStri(P1,P5,Tr,M) IarcAStri(P5,P5,Tr,M) #or try Rv<-rel.vert.triCC(P5,Tr)$rv IarcAStri(P5,P5,Tr,M,Rv) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `IarcASbasic.tri`, `IarcPEtri`, and `IarcCStri` ## Author(s) Elvan Ceyhan

IarcAStri function