Idom.num2PEtetra {pcds}

The indicator for two 3D points constituting a dominating set for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one tetrahedron case

Description

Returns I({p1,p2} is a dominating set of the PE-PCD) where the vertices of the PE-PCD are the 3D data set Xp in the tetrahedron th, that is, returns 1 if {p1,p2} is a dominating set of PE-PCD, returns 0 otherwise.

Point, p1, is in the region of vertex rv1 (default is NULL) and point, p2, is in the region of vertex rv2 (default is NULL); vertices (and hence rv1 and rv2) are labeled as 1,2,3,4 in the order they are stacked row-wise in th.

PE proximity region is constructed with respect to the tetrahedron th with expansion parameter r \ge 1 and vertex regions are based on center of mass (M="CM") or circumcenter (M="CC") only.

ch.data.pnts is for checking whether points p1 and p2 are both data points in Xp or not (default is FALSE), so by default this function checks whether the points p1 and p2 would constitute a dominating set if they actually were both in the data set.

See also (Ceyhan (2005, 2010)).

Usage

Idom.num2PEtetra(
  p1,
  p2,
  Xp,
  th,
  r,
  M = "CM",
  rv1 = NULL,
  rv2 = NULL,
  ch.data.pnts = FALSE
)

Arguments

p1, p2

Two 3D points to be tested for constituting a dominating set of the PE-PCD.

Xp

A set of 3D points which constitutes the vertices of the PE-PCD.

th

A 4 \times 3 matrix with each row representing a vertex of the tetrahedron.

r

A positive real number which serves as the expansion parameter in PE proximity region; must be \ge 1.

M

The center to be used in the construction of the vertex regions in the tetrahedron, th. Currently it only takes "CC" for circumcenter and "CM" for center of mass; default="CM".

rv1, rv2

The indices of the vertices whose regions contains p1 and p2, respectively. They take the vertex labels as 1,2,3,4 as in the row order of the vertices in th (default is NULL for both).

ch.data.pnts

A logical argument for checking whether both points p1 and p2 are data points in Xp or not (default is FALSE).

Value

I({p1,p2} is a dominating set of the PE-PCD) where the vertices of the PE-PCD are the 3D data set Xp), that is, returns 1 if {p1,p2} is a dominating set of PE-PCD, returns 0 otherwise

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

Idom.num2PEstd.tetra, Idom.num2PEtri and Idom.num2PEbasic.tri

Examples

A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)
n<-5

set.seed(1)
Xp<-runif.tetra(n,tetra)$g  #try also Xp<-cbind(runif(n),runif(n),runif(n))

M<-"CM";  #try also M<-"CC";
r<-1.5

Idom.num2PEtetra(Xp[1,],Xp[2,],Xp,tetra,r,M)
Idom.num2PEtetra(c(-1,-1,-1),Xp[2,],Xp,tetra,r,M)

ind.gam2<-ind.gamn2<-vector()
for (i in 1:(n-1))
 for (j in (i+1):n)
 {if (Idom.num2PEtetra(Xp[i,],Xp[j,],Xp,tetra,r,M)==1)
 {ind.gam2<-rbind(ind.gam2,c(i,j))
 }
 }
ind.gam2

#or try
rv1<-rel.vert.tetraCC(Xp[1,],tetra)$rv;rv2<-rel.vert.tetraCC(Xp[2,],tetra)$rv
Idom.num2PEtetra(Xp[1,],Xp[2,],Xp,tetra,r,M,rv1,rv2)

#or try
rv1<-rel.vert.tetraCC(Xp[1,],tetra)$rv;
Idom.num2PEtetra(Xp[1,],Xp[2,],Xp,tetra,r,M,rv1)

#or try
rv2<-rel.vert.tetraCC(Xp[2,],tetra)$rv
Idom.num2PEtetra(Xp[1,],Xp[2,],Xp,tetra,r,M,rv2=rv2)

P1<-c(.1,.1,.1)
P2<-c(.4,.1,.2)
Idom.num2PEtetra(P1,P2,Xp,tetra,r,M)

Idom.num2PEtetra(c(-1,-1,-1),Xp[2,],Xp,tetra,r,M,ch.data.pnts = FALSE)
#gives an error message if ch.data.pnts = TRUE
#since not both points, p1 and p2, are data points in Xp