# The indicator for the presence of an arc from a point in set `S` to the point `p` for Arc Slice Proximity Catch Digraphs (AS-PCDs) - one triangle case Returns I($pt \in N_{AS}(x)$ for some $x \in S$), that is, returns 1 if $p$ is in $\cup_{x \in S}N_{AS}(x)$, 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 vertices of `tri` are also labeled as 1,2, and 3, respectively. 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`. If `p` is not in `S` and either `p` or all points in `S` are outside `tri`, it returns 0, but if `p` is in `S`, then it always returns 1 (i.e., loops are allowed). 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 IarcASset2pnt.tri(S, p, tri, M = "CC") ``` ## Arguments - `S`: A set of 2D points whose AS proximity regions are considered. - `p`: A 2D point. The function determines whether `p` is inside the union of AS proximity regions of points in `S` 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`. ## Returns $I(pt \in \cup_{x in S}N_{AS}(x,r))$, that is, returns 1 if `p` is in `S` or inside $N_{AS}(x)$ for at least one $x$ in `S`, returns 0 otherwise, where AS proximity region is constructed in `tri` ## Examples ```r A<-c(1,1); B<-c(2,0); C<-c(1.5,2); Tr<-rbind(A,B,C); n<-10 set.seed(1) Xp<-runif.tri(n,Tr)$gen.points S<-rbind(Xp[1,],Xp[2,]) #try also S<-c(1.5,1) M<-as.numeric(runif.tri(1,Tr)$g) #try also M<-c(1.6,1.2) IarcASset2pnt.tri(S,Xp[3,],Tr,M) S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,]) IarcASset2pnt.tri(S,Xp[3,],Tr,M) IarcASset2pnt.tri(S,Xp[6,],Tr,M) S<-rbind(c(.1,.1),c(.3,.4),c(.5,.3)) IarcASset2pnt.tri(S,Xp[3,],Tr,M) IarcASset2pnt.tri(c(.2,.5),Xp[2,],Tr,M) IarcASset2pnt.tri(Xp,c(.2,.5),Tr,M) IarcASset2pnt.tri(Xp,Xp[2,],Tr,M) IarcASset2pnt.tri(c(.2,.5),c(.2,.5),Tr,M) IarcASset2pnt.tri(Xp[5,],Xp[2,],Tr,M) S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,],c(.2,.5)) IarcASset2pnt.tri(S,Xp[3,],Tr,M) P<-c(.4,.2) S<-Xp[c(1,3,4),] IarcASset2pnt.tri(Xp,P,Tr,M) IarcASset2pnt.tri(S,P,Tr,M) IarcASset2pnt.tri(rbind(S,S),P,Tr,M) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `IarcAStri`, `IarcASset2pnt.tri`, and `IarcCSset2pnt.tri` ## Author(s) Elvan Ceyhan

IarcASset2pnt.tri function