# Projections of a point inside the standard basic triangle form to its edges Returns the projections from a general 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 form $T_b=T((0,0),(1,0),(c_1,c_2))$ to the edges on the extension of the lines joining `M` to the vertices (see the examples for an illustration). In the standard basic triangle form $T_b$, $c_1$ is in $[0,1/2]$, $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$. Any given triangle can be mapped to the standard basic triangle form 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 form 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;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r prj.cent2edges.basic.tri(c1, c2, M) ``` ## Arguments - `c1, c2`: Positive real numbers which constitute the vertex of the standard basic triangle form 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 form. ## Returns Three projection points (stacked row-wise) from a general center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of a standard basic triangle form to the edges on the extension of the lines joining `M` to the vertices; row $i$ is the projection point into edge $i$, for $i=1,2,3$. ## Examples ```r c1<-.4; c2<-.6 A<-c(0,0); B<-c(1,0); C<-c(c1,c2); Tb<-rbind(A,B,C); M<-as.numeric(runif.basic.tri(1,c1,c2)$g) #try also M<-c(.6,.2) Ds<-prj.cent2edges.basic.tri(c1,c2,M) Ds Xlim<-range(Tb[,1]) Ylim<-range(Tb[,2]) xd<-Xlim[2]-Xlim[1] yd<-Ylim[2]-Ylim[1] if (dimension(M)==3) {M<-bary2cart(M,Tb)} #need to run this when M is given in barycentric coordinates plot(Tb,pch=".",xlab="",ylab="",axes=TRUE, xlim=Xlim+xd*c(-.1,.1),ylim=Ylim+yd*c(-.05,.05)) polygon(Tb) L<-rbind(M,M,M); R<-Ds segments(L[,1], L[,2], R[,1], R[,2], lty = 2) L<-rbind(M,M,M); R<-Tb segments(L[,1], L[,2], R[,1], R[,2], lty = 3,col=2) xc<-Tb[,1]+c(-.04,.05,.04) yc<-Tb[,2]+c(.02,.02,.03) txt.str<-c("rv=1","rv=2","rv=3") text(xc,yc,txt.str) txt<-rbind(M,Ds) xc<-txt[,1]+c(-.02,.03,-.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 `prj.cent2edges` and `prj.nondegPEcent2edges` ## Author(s) Elvan Ceyhan

prj.cent2edges.basic.tri function