inci.matPEtetra {pcds}

Incidence matrix for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one tetrahedron case

Description

Returns the incidence matrix for the PE-PCD whose vertices are the given 3D numerical data set, Xp, in the tetrahedron th=T(v=1,v=2,v=3,v=4).

PE proximity regions are constructed with respect to tetrahedron th with expansion parameter r \ge 1 and vertex regions are based on the center M which is circumcenter ("CC") or center of mass ("CM") of th with default="CM". Loops are allowed, so the diagonal entries are all equal to 1.

See also (Ceyhan (2005, 2010)).

Usage

inci.matPEtetra(Xp, th, r, M = "CM")

Arguments

Xp

A set of 3D points which constitute the vertices of 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".

Value

Incidence matrix for the PE-PCD with vertices being 3D data set, Xp, in the tetrahedron th with vertex regions based on circumcenter or center of mass

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

inci.matPEtri, inci.matPE1D, and inci.matPE

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

Xp<-runif.tetra(n,tetra)$g  #try also Xp<-c(.5,.5,.5)

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

IM<-inci.matPEtetra(Xp,tetra,r=1.25)  #uses the default M="CM"
IM<-inci.matPEtetra(Xp,tetra,r=1.25,M)
IM
dom.num.greedy(IM)
Idom.num.up.bnd(IM,3) #try also dom.num.exact(IM)  #this might take a long time for large n