rel.verts.tri {pcds}

The indices of the vertex regions in a triangle that contains the points in a give data set

Description

Returns the indices of the vertices whose regions contain the points in data set Xp in a triangle tri=T(A,B,C).

Vertex regions are based on center M=(m_1,m_2) in Cartesian coordinates or M=(\alpha,\beta,\gamma) in barycentric coordinates in the interior of the triangle to the edges on the extension of the lines joining M to the vertices or based on the circumcenter of tri. Vertices of triangle tri are labeled as 1,2,3 according to the row number the vertex is recorded.

If a point in Xp is not inside tri, then the function yields NA as output for that entry. The corresponding vertex region is the polygon with the vertex, M, and projection points from M to the edges crossing the vertex (as the output of prj.cent2edges(Tr,M)) or CC-vertex region (see the examples for an illustration).

See also (Ceyhan (2005, 2011)).

Usage

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

Arguments

Xp

A set of 2D points representing the set of data points for which indices of the vertex 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 or the circumcenter of tri.

Value

A list with two elements

rv

Indices of the vertices whose regions contains points in Xp.

tri

The vertices of the triangle, where row number corresponds to the vertex index in rv.

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 (2011). “Spatial Clustering Tests Based on Domination Number of a New Random Digraph Family.” Communications in Statistics - Theory and Methods, 40(8), 1363-1395.

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.

See Also

rel.verts.triCM, rel.verts.triCC, 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(1.6,1.0)

P<-c(.4,.2)
rel.verts.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(1.6,1.0)

rel.verts.tri(Xp,Tr,M)
rel.verts.tri(rbind(Xp,c(2,2)),Tr,M)

rv<-rel.verts.tri(Xp,Tr,M)
rv

ifelse(identical(M,circumcenter.tri(Tr)),
Ds<-rbind((B+C)/2,(A+C)/2,(A+B)/2),Ds<-prj.cent2edges(Tr,M))

Xlim<-range(Tr[,1],M[1],Xp[,1])
Ylim<-range(Tr[,2],M[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 M-vertex regions in a triangle",
axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp,pch=".",col=1)
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

xc<-Tr[,1]
yc<-Tr[,2]
txt.str<-c("rv=1","rv=2","rv=3")
text(xc,yc,txt.str)

txt<-rbind(M,Ds)
xc<-txt[,1]+c(.02,.04,-.03,0)
yc<-txt[,2]+c(.07,.04,.05,-.07)
txt.str<-c("M","D1","D2","D3")
text(xc,yc,txt.str)
text(Xp,labels=factor(rv$rv))