prj.cent2edges.basic.tri {pcds}

Projections of a point inside the standard basic triangle form to its edges

Description

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 (Ceyhan (2005, 2010)).

Usage

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.

Value

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.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2005). An Investigation of Proximity Catch Digraphs in Delaunay Tessellations, also available as technical monograph titled Proximity Catch Digraphs: Auxiliary Tools, Properties, and Applications. Ph.D. thesis, The Johns Hopkins University, Baltimore, MD, 21218.

Ceyhan E (2010). “Extension of One-Dimensional Proximity Regions to Higher Dimensions.” Computational Geometry: Theory and Applications, 43(9), 721-748.

See Also

prj.cent2edges and prj.nondegPEcent2edges

Examples

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)