cp2e.bastri: Projections of a point inside the basic triangle to its edges

Description

Returns the projections from a general center $M = (m_{1}, m_{2})$ in Cartesian coordinates or $M = (α, β, γ)$ in barycentric coordinates in the interior of the basic triangle $T_{b} = T ((0, 0), (1, 0), (c_{1}, c_{2}))$ to the edges on the extension of the lines joining M to the vertices (see the examples for an illustration). In the basic triangle $T_{b}$ , $c_{1}$ is in $[0, 1 / 2]$ , $c_{2} > 0$ and $(1 - c_{1})^{2} + c_{2}^{2} \leq 1$ .

Any given triangle can be mapped to the 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 basic triangle is useful for simulation studies under the uniformity hypothesis.

See also (ceyhan:Phd-thesis,ceyhan:comp-geo-2010;textualpcds).

Usage

cp2e.bastri(c1, c2, M)

Value

Three projection points (stacked row-wise) from a general center $M = (m_{1}, m_{2})$ in Cartesian coordinates or $M = (α, β, γ)$ in barycentric coordinates in the interior of a basic triangle to the edges on the extension of the lines joining M to the vertices; row $i$ is the projection point into edge $i$ , for $i = 1, 2, 3$ .

Arguments

c1, c2: Positive real numbers which constitute the vertex of the basic triangle adjacent to the shorter edges; $c_{1}$ must be in $[0, 1 / 2]$ , $c_{2} > 0$ and $(1 - c_{1})^{2} + c_{2}^{2} \leq 1$ .
M: A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the basic triangle.

Author

Elvan Ceyhan

References

Examples

Run this code

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.bastri(1,c1,c2)$g)  #try also M<-c(.6,.2)

cp2e.bastri(c1,c2,M)

Ds<-cp2e.bastri(c1,c2,M)

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)

cp2e.bastri(c1,c2,M)

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