# The indicator for the presence of an arc from a point to another for Arc Slice Proximity Catch Digraphs (AS-PCDs) - standard basic 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 region is constructed in the standard basic triangle $T_b=T((0,0),(1,0),(c_1,c_2))$ where $c_1$ is in $[0,1/2]$, $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$. 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 standard basic triangle $T_b$ or based on circumcenter of $T_b$; default is `M="CC"`, i.e., circumcenter of $T_b$. `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 $T_b$, the function returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops). Any given triangle can be mapped to the standard basic triangle by a combination of rigid body motions (i.e., translation, rotation and reflection) and scaling, preserving uniformity of the points in the original triangle. Hence standard basic triangle is useful for simulation studies under the uniformity hypothesis. 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 IarcASbasic.tri(p1, p2, c1, c2, 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. - `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`: The center of the triangle. `"CC"` stands for circumcenter or a 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle $T_b$; default is `M="CC"` i.e., the circumcenter of $T_b$. - `rv`: The index of the `M`-vertex region in $T_b$ containing the point, either `1,2,3` or `NULL` (default is `NULL`). ## Returns $I(p2 \in N_{AS}(p1))$ for points `p1` and `p2`, that is, returns 1 if $p2$ is in $N_{AS}(p1)$ (i.e., if there is an arc from `p1` to `p2`), returns 0 otherwise. ## 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)$g) #try also M<-c(.6,.2) P1<-as.numeric(runif.basic.tri(1,c1,c2)$g) P2<-as.numeric(runif.basic.tri(1,c1,c2)$g) IarcASbasic.tri(P1,P2,c1,c2,M) P1<-c(.3,.2) P2<-c(.6,.2) IarcASbasic.tri(P1,P2,c1,c2,M) #or try Rv<-rel.vert.basic.triCC(P1,c1,c2)$rv IarcASbasic.tri(P1,P2,c1,c2,M,Rv) P1<-c(.3,.2) P2<-c(.8,.2) IarcASbasic.tri(P1,P2,c1,c2,M) P3<-c(.5,.4) IarcASbasic.tri(P1,P3,c1,c2,M) P4<-c(1.5,.4) IarcASbasic.tri(P1,P4,c1,c2,M) IarcASbasic.tri(P4,P4,c1,c2,M) c1<-.4; c2<-.6; P1<-c(.3,.2) P2<-c(.6,.2) IarcASbasic.tri(P1,P2,c1,c2,M) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `IarcAStri` and `NAStri` ## Author(s) Elvan Ceyhan

IarcASbasic.tri function