# The closest points among a data set in the vertex regions to the corresponding edges in a standard basic triangle An object of class `"Extrema"`. Returns the closest data points among the data set, `Xp`, to edge $i$ in `M`-vertex region $i$ for $i=1,2,3$ in the standard basic triangle $T_b=T(A=(0,0),B=(1,0),C=(c_1,c_2))$ where $c_1$ is in $[0,1/2]$, $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$. Vertex labels are $A=1$, $B=2$, and $C=3$, and corresponding edge labels are $BC=1$, $AC=2$, and $AB=3$. Vertex regions are based on 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 the circumcenter of $T_b$. 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:dom-num-NPE-Spat2011;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r cl2edges.vert.reg.basic.tri(Xp, c1, c2, M) ``` ## Arguments - `Xp`: A set of 2D points representing the set of data points. - `c1, c2`: Positive real numbers which constitute the vertex of the standard basic triangle adjacent to the shorter edges; $c_1$ must be in $[0,1/2]$, $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$. - `M`: A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the standard basic triangle $T_b$ or the circumcenter of $T_b$. ## Returns A `list` with the elements - **txt1**: Vertex labels are $A=1$, $B=2$, and $C=3$ (correspond to row number in Extremum Points). - **txt2**: A short description of the distances as `"Distances to Edges in the Respective \eqn{M}-Vertex Regions"`. - **type**: Type of the extrema points - **desc**: A short description of the extrema points - **mtitle**: The `"main"` title for the plot of the extrema - **ext**: The extrema points, here, closest points to edges in the corresponding vertex region. - **X**: The input data, `Xp`, can be a `matrix` or `data frame` - **num.points**: The number of data points, i.e., size of `Xp` - **supp**: Support of the data points, here, it is $T_b$. - **cent**: The center point used for construction of vertex regions - **ncent**: Name of the center, `cent`, it is `"M"` or `"CC"` for this function - **regions**: Vertex regions inside the triangle, $T_b$. - **region.names**: Names of the vertex regions as `"vr=1"`, `"vr=2"`, and `"vr=3"` - **region.centers**: Centers of mass of the vertex regions inside $T_b$. - **dist2ref**: Distances of closest points in the vertex regions to corresponding edges. ## Examples ```r c1<-.4; c2<-.6 A<-c(0,0); B<-c(1,0); C<-c(c1,c2); Tb<-rbind(A,B,C); set.seed(1) n<-20 Xp<-runif.basic.tri(n,c1,c2)$g M<-as.numeric(runif.basic.tri(1,c1,c2)$g) #try also M<-c(.6,.3) Ext<-cl2edges.vert.reg.basic.tri(Xp,c1,c2,M) Ext summary(Ext) plot(Ext) cl2e<-Ext Ds<-prj.cent2edges.basic.tri(c1,c2,M) Xlim<-range(Tb[,1],Xp[,1]) Ylim<-range(Tb[,2],Xp[,2]) xd<-Xlim[2]-Xlim[1] yd<-Ylim[2]-Ylim[1] plot(Tb,pch=".",xlab="",ylab="", main="Closest Points in M-Vertex Regions \n to the Opposite Edges", axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05)) polygon(Tb) points(Xp,pch=1,col=1) L<-rbind(M,M,M); R<-Ds segments(L[,1], L[,2], R[,1], R[,2], lty=2) points(cl2e$ext,pch=3,col=2) xc<-Tb[,1]+c(-.02,.02,0.02) yc<-Tb[,2]+c(.02,.02,.02) txt.str<-c("A","B","C") text(xc,yc,txt.str) txt<-rbind(M,Ds) xc<-txt[,1]+c(-.02,.04,-.03,0) yc<-txt[,2]+c(-.02,.02,.02,-.03) txt.str<-c("M","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 `cl2edgesCMvert.reg`, `cl2edgesMvert.reg`, and `cl2edges.std.tri` ## Author(s) Elvan Ceyhan

cl2edges.vert.reg.basic.tri function