# Projections of Centers for non-degenerate asymptotic distribution of domination number of Proportional Edge Proximity Catch Digraphs (PE-PCDs) to its edges Returns the projections from center `cent` to the edges on the extension of the lines joining `cent` to the vertices in the triangle, `tri`. Here M is one of the three centers which gives nondegenerate asymptotic distribution of the domination number of PE-PCD for uniform data in `tri` for a given expansion parameter `r` in $(1,1.5]$. The center label `cent` values `1,2,3` correspond to the vertices $M_1$, $M_2$, and $M_3$ (i.e., row numbers in the output of `center.nondegPE(tri,r)`); default for `cent` is 1. `cent` becomes center of mass $CM$ for $r=1.5$. See also (if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="ceyhan:Phd-thesis,ceyhan:masa-2007,ceyhan:dom-num-NPE-Spat2011;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r prj.nondegPEcent2edges(tri, r, cent = 1) ``` ## Arguments - `tri`: A $3 \times 2$ matrix with each row representing a vertex of the triangle. - `r`: A positive real number which serves as the expansion parameter in PE proximity region; must be in $(1,1.5]$ for this function. - `cent`: Index of the center (as $1,2,3$ corresponding to $M_1,\,M_2,\,M_3$) which gives nondegenerate asymptotic distribution of the domination number of PE-PCD for uniform data in `tri` for expansion parameter `r` in $(1,1.5]$; default `cent=1`. ## Returns Three projection points (stacked row-wise) from one of the centers (as $1,2,3$ corresponding to $M_1,\,M_2,\,M_3$) which gives nondegenerate asymptotic distribution of the domination number of PE-PCD for uniform data in `tri` for expansion parameter `r` in $(1,1.5]$. ## Examples ```r A<-c(1,1); B<-c(2,0); C<-c(1.5,2); Tr<-rbind(A,B,C); r<-1.35 prj.nondegPEcent2edges(Tr,r,cent=2) Ms<-center.nondegPE(Tr,r) M1=Ms[1,] Ds<-prj.nondegPEcent2edges(Tr,r,cent=1) Xlim<-range(Tr[,1]) Ylim<-range(Tr[,2]) xd<-Xlim[2]-Xlim[1] yd<-Ylim[2]-Ylim[1] plot(Tr,pch=".",xlab="",ylab="", main="Projections from a non-degeneracy center\n to the edges of the triangle", axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05)) polygon(Tr) points(Ms,pch=".",col=1) polygon(Ms,lty = 2) xc<-Tr[,1]+c(-.02,.03,.02) yc<-Tr[,2]+c(-.02,.04,.04) txt.str<-c("A","B","C") text(xc,yc,txt.str) txt<-Ms xc<-txt[,1]+c(-.02,.04,-.04) yc<-txt[,2]+c(-.02,.04,.04) txt.str<-c("M1","M2","M3") text(xc,yc,txt.str) points(Ds,pch=4,col=2) L<-rbind(M1,M1,M1); R<-Ds segments(L[,1], L[,2], R[,1], R[,2], lty = 2,lwd=2,col=4) txt<-Ds xc<-txt[,1]+c(-.02,.04,-.04) yc<-txt[,2]+c(-.02,.04,.04) txt.str<-c("D1","D2","D3") text(xc,yc,txt.str) prj.nondegPEcent2edges(Tr,r,cent=3) #gives an error message if center index, cent, is different from 1, 2 or 3 prj.nondegPEcent2edges(Tr,r=1.49,cent=2) #gives an error message if r>1.5 ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `prj.cent2edges.basic.tri` and `prj.cent2edges` ## Author(s) Elvan Ceyhan

prj.nondegPEcent2edges function