rel.edges.tri {pcds}

The indices of the M-edge regions in a triangle that contains the points in a give data set

Description

Returns the indices of the edges whose regions contain the points in data set Xp in a triangle tri=T(A,B,C) and edge regions are based on the center M=(m_1,m_2) in Cartesian coordinates or M=(\alpha,\beta,\gamma) in barycentric coordinates in the interior of the triangle tri (see the plots in the example for illustrations).

The vertices of the triangle tri are labeled as 1=A, 2=B, and 3=C also according to the row number the vertex is recorded in tri and the corresponding edges are 1=BC, 2=AC, and 3=AB.

If a point in Xp is not inside tri, then the function yields NA as output for that entry. The corresponding edge region is the polygon with the vertex, M, and vertices other than the non-adjacent vertex, i.e., edge region 1 is the triangle T(B,M,C), edge region 2 is T(A,M,C) and edge region 3 is T(A,B,M).

See also (Ceyhan (2005, 2010); Ceyhan et al. (2007)).

Usage

rel.edges.tri(Xp, tri, M)

Arguments

Xp

A set of 2D points representing the set of data points for which indices of the edge regions containing them are to be determined.

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.

Value

A list with the elements

re

Indices (i.e., a vector of indices) of the edges whose region contains points in Xp in the triangle tri

tri

The vertices of the triangle, where row number corresponds to the vertex index opposite to edge whose index is given in re.

desc

Description of the edge labels as "Edge labels are AB=3, BC=1, and AC=2".

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.

Ceyhan E (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

Ceyhan E, Priebe CE, Marchette DJ (2007). “A new family of random graphs for testing spatial segregation.” Canadian Journal of Statistics, 35(1), 27-50.

See Also

rel.edges.triCM, rel.verts.tri, and rel.verts.tri.nondegPE

Examples

A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

M<-c(1.6,1.2)

P<-c(.4,.2)
rel.edges.tri(P,Tr,M)

n<-20  #try also n<-40
set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

(re<-rel.edges.tri(Xp,Tr,M))

D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

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="Scatterplot of data points \n and the M-edge regions",axes=TRUE,
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp,pch=".",col=1)
L<-Tr; R<-rbind(M,M,M)
segments(L[,1], L[,2], R[,1], R[,2], lty = 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(.05,.06,-.05,-.02)
yc<-txt[,2]+c(.03,.03,.05,-.08)
txt.str<-c("M","re=2","re=3","re=1")
text(xc,yc,txt.str)
text(Xp,labels=factor(re$re))