# The arcs of Arc Slice Proximity Catch Digraph (AS-PCD) for 2D data - one triangle case An object of class `"PCDs"`. Returns arcs of AS-PCD as tails (or sources) and heads (or arrow ends) and related parameters and the quantities of the digraph. The vertices of the AS-PCD are the data points in `Xp` in the one triangle case. AS proximity regions are constructed with respect to the triangle `tri`, i.e., arcs may exist for points only inside `tri`. It also provides various descriptions and quantities about the arcs of the AS-PCD such as number of arcs, arc density, etc. 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`. The different consideration of circumcenter vs any other interior center of the triangle is because the projections from circumcenter are orthogonal to the edges, while projections of `M` on the edges are on the extensions of the lines connecting `M` and the vertices. 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 arcsAStri(Xp, tri, M = "CC") ``` ## Arguments - `Xp`: A set of 2D points which constitute the vertices of the AS-PCD. - `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 the triangle $T_b$; default is `M="CC"` i.e., the circumcenter of `tri`. ## Returns A `list` with the elements - **type**: A description of the type of the digraph - **parameters**: Parameters of the digraph, here, it is the center used to construct the vertex regions. - **tess.points**: Tessellation points, i.e., points on which the tessellation of the study region is performed, here, tessellation points are the vertices of the support triangle `tri`. - **tess.name**: Name of the tessellation points `tess.points` - **vertices**: Vertices of the digraph, `Xp`. - **vert.name**: Name of the data set which constitute the vertices of the digraph - **S**: Tails (or sources) of the arcs of AS-PCD for 2D data set `Xp` as vertices of the digraph - **E**: Heads (or arrow ends) of the arcs of AS-PCD for 2D data set `Xp` as vertices of the digraph - **mtitle**: Text for `"main"` title in the plot of the digraph - **quant**: Various quantities for the digraph: number of vertices, number of partition points, number of intervals, number of arcs, and arc density. ## 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)$g M<-as.numeric(runif.tri(1,Tr)$g) #try also M<-c(1.6,1.2) or M<-circumcenter.tri(Tr) Arcs<-arcsAStri(Xp,Tr,M) #try also Arcs<-arcsAStri(Xp,Tr) #uses the default center, namely circumcenter for M Arcs summary(Arcs) plot(Arcs) #use plot(Arcs,asp=1) if M=CC #can add vertex regions #but we first need to determine center is the circumcenter or not, #see the description for more detail. CC<-circumcenter.tri(Tr) M = as.numeric(Arcs$parameters[[1]]) if (isTRUE(all.equal(M,CC)) || identical(M,"CC")) {cent<-CC D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2; Ds<-rbind(D1,D2,D3) cent.name<-"CC" } else {cent<-M cent.name<-"M" Ds<-prj.cent2edges(Tr,M) } L<-rbind(cent,cent,cent); R<-Ds segments(L[,1], L[,2], R[,1], R[,2], lty=2) #now we add the vertex names and annotation txt<-rbind(Tr,cent,Ds) xc<-txt[,1]+c(-.02,.03,.02,.03,.04,-.03,-.01) yc<-txt[,2]+c(.02,.02,.03,.06,.04,.05,-.07) txt.str<-c("A","B","C",cent.name,"D1","D2","D3") text(xc,yc,txt.str) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `arcsAS`, `arcsPEtri`, `arcsCStri`, `arcsPE`, and `arcsCS` ## Author(s) Elvan Ceyhan

arcsAStri function