# The closest points among a data set in the vertex regions to the respective edges in a 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 triangle `tri`$=T(A,B,C)$. 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 triangle `tri` or based on the circumcenter of `tri`. Two methods of finding these extrema are provided in the function, which can be chosen in the logical argument `alt`, whose default is `alt=FALSE`. When `alt=FALSE`, the function sequentially finds the vertex region of the data point and then updates the minimum distance to the opposite edge and the relevant extrema objects, and when `alt=TRUE`, it first partitions the data set according which vertex regions they reside, and then finds the minimum distance to the opposite edge and the relevant extrema on each partition. Both options yield equivalent results for the extrema points and indices, with the default being slightly ~ 20 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 cl2edgesMvert.reg(Xp, tri, M, alt = FALSE) ``` ## Arguments - `Xp`: A set of 2D points representing the set of data points. - `tri`: A $3 \times 2$ matrix with each row representing a vertex of the triangle. - `M`: A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri` or the circumcenter of `tri`; which may be entered as "CC" as well; - `alt`: A logical argument for alternative method of finding the closest points to the edges, default `alt=FALSE`. When `alt=FALSE`, the function sequentially finds the vertex region of the data point and then the minimum distance to the opposite edge and the relevant extrema objects, and when `alt=TRUE`, it first partitions the data set according which vertex regions they reside, and then finds the minimum distance to the opposite edge and the relevant extrema on each partition. ## 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 respective vertex region. - **ind.ext**: The data indices of extrema points, `ext`. - **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 `tri` - **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, `tri`, provided as a `list` - **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 `tri` - **dist2ref**: Distances of closest points in the `M`-vertex regions to corresponding edges. ## Examples ```r A<-c(1,1); B<-c(2,0); C<-c(1.5,2); Tr<-rbind(A,B,C); n<-20 #try also n<-100 set.seed(1) Xp<-runif.tri(n,Tr)$g M<-as.numeric(runif.tri(1,Tr)$g) #try also M<-c(1.6,1.0) Ext<-cl2edgesMvert.reg(Xp,Tr,M) Ext summary(Ext) plot(Ext) cl2e<-Ext Ds<-prj.cent2edges(Tr,M) Xlim<-range(Tr[,1],Xp[,1]) Ylim<-range(Tr[,2],Xp[,2]) xd<-Xlim[2]-Xlim[1] yd<-Ylim[2]-Ylim[1] if (dimension(M)==3) {M<-bary2cart(M,Tr)} #need to run this when M is given in barycentric coordinates plot(Tr,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(Tr) 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<-Tr[,1]+c(-.02,.03,.02) yc<-Tr[,2]+c(.02,.02,.04) txt.str<-c("A","B","C") text(xc,yc,txt.str) txt<-rbind(M,Ds) xc<-txt[,1]+c(-.02,.05,-.02,-.01) yc<-txt[,2]+c(-.03,.02,.08,-.07) 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 `cl2edges.vert.reg.basic.tri`, `cl2edgesCMvert.reg`, and `cl2edges.std.tri` ## Author(s) Elvan Ceyhan

cl2edgesMvert.reg function