# The vertices of the Arc Slice (AS) Proximity Region in the standard basic triangle Returns the end points of the line segments and arc-slices that constitute the boundary of AS proximity region for a point 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 `p` resides, with default=`NULL`. If `p` is outside $T_b$, it returns `NULL` for the proximity region. `dec` is the number of decimals (default is 4) to round the barycentric coordinates when checking whether the end points fall on the boundary of the triangle $T_b$ or not (so as not to miss the intersection points due to precision in the decimals). 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 NASbasic.tri(p, c1, c2, M = "CC", rv = NULL, dec = 4) ``` ## Arguments - `p`: A 2D point whose AS proximity region is to be computed. - `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 of the triangle $T_b$ 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 containing the point, either `1,2,3` or `NULL` (default is `NULL`). - `dec`: a positive integer the number of decimals (default is 4) to round the barycentric coordinates when checking whether the end points fall on the boundary of the triangle $T_b$ or not. ## Returns A `list` with the elements - **L,R**: The end points of the line segments on the boundary of the AS proximity region. Each row in `L` and `R` constitute a line segment on the boundary. - **Arc.Slices**: The end points of the arc-slices on the circular parts of the AS proximity region. Here points in row 1 and row 2 constitute the end points of one arc-slice, points on row 3 and row 4 constitute the end points for the next arc-slice and so on. ## Examples ```r c1<-.4; c2<-.6 #try also c1<-.2; c2<-.2; A<-c(0,0); B<-c(1,0); C<-c(c1,c2); Tb<-rbind(A,B,C) set.seed(1) 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); #try also P1<-c(.3,.2) NASbasic.tri(P1,c1,c2) #default with M="CC" NASbasic.tri(P1,c1,c2,M) #or try Rv<-rel.vert.basic.triCC(P1,c1,c2)$rv NASbasic.tri(P1,c1,c2,M,Rv) NASbasic.tri(c(3,5),c1,c2,M) P2<-c(.5,.4) NASbasic.tri(P2,c1,c2,M) P3<-c(1.5,.4) NASbasic.tri(P3,c1,c2,M) if (dimension(M)==3) {M<-bary2cart(M,Tr)} #need to run this when M is given in barycentric coordinates #plot of the NAS region P1<-as.numeric(runif.basic.tri(1,c1,c2)$g); CC<-circumcenter.basic.tri(c1,c2) 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" rv<-rel.vert.basic.triCC(P1,c1,c2)$rv } else {cent<-M cent.name<-"M" Ds<-prj.cent2edges.basic.tri(c1,c2,M) rv<-rel.vert.basic.tri(P1,c1,c2,M)$rv } RV<-Tb[rv,] rad<-Dist(P1,RV) Int.Pts<-NASbasic.tri(P1,c1,c2,M) Xlim<-range(Tb[,1],P1[1]+rad,P1[1]-rad) Ylim<-range(Tb[,2],P1[2]+rad,P1[2]-rad) xd<-Xlim[2]-Xlim[1] yd<-Ylim[2]-Ylim[1] plot(A,pch=".",asp=1,xlab="",ylab="",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05)) polygon(Tb) points(rbind(Tb,P1,rbind(Int.Pts$L,Int.Pts$R))) L<-rbind(cent,cent,cent); R<-Ds segments(L[,1], L[,2], R[,1], R[,2], lty=2) interp::circles(P1[1],P1[2],rad,lty=2) L<-Int.Pts$L; R<-Int.Pts$R segments(L[,1], L[,2], R[,1], R[,2], lty=1,col=2) Arcs<-Int.Pts$a; if (!is.null(Arcs)) { K<-nrow(Arcs)/2 for (i in 1:K) {A1<-Arcs[2*i-1,]; A2<-Arcs[2*i,]; angles<-angle.str2end(A1,P1,A2)$c plotrix::draw.arc(P1[1],P1[2],rad,angle1=angles[1],angle2=angles[2],col=2) } } #proximity region with the triangle (i.e., for labeling the vertices of the NAS) IP.txt<-intpts<-c() if (!is.null(Int.Pts$a)) { intpts<-unique(round(Int.Pts$a,7)) #this part is for labeling the intersection points of the spherical for (i in 1:(length(intpts)/2)) IP.txt<-c(IP.txt,paste("I",i+1, sep = "")) } txt<-rbind(Tb,P1,cent,intpts) txt.str<-c("A","B","C","P1",cent.name,IP.txt) text(txt+cbind(rep(xd*.02,nrow(txt)),rep(-xd*.03,nrow(txt))),txt.str) c1<-.4; c2<-.6; P1<-c(.3,.2) NASbasic.tri(P1,c1,c2,M) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `NAStri` and `IarcASbasic.tri` ## Author(s) Elvan Ceyhan

NASbasic.tri function