num.arcsPEtetra {pcds}

Number of arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) and quantities related to the tetrahedron - one tetrahedron case

Description

An object of class "NumArcs". Returns the number of arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) whose vertices are the given 3D numerical data set, Xp. It also provides number of vertices (i.e., number of data points inside the tetrahedron) and indices of the data points that reside in the tetrahedron.

PE proximity region is constructed with respect to the tetrahedron th and vertex regions are based on the center M which is circumcenter ("CC") or center of mass ("CM") of th with default="CM". For the number of arcs, loops are not allowed so arcs are only possible for points inside the tetrahedron th for this function.

See also (Ceyhan (2005, 2010)).

Usage

num.arcsPEtetra(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

A list with the elements

desc

A short description of the output: number of arcs and quantities related to the tetrahedron

num.arcs

Number of arcs of the PE-PCD

tri.num.arcs

Number of arcs of the induced subdigraph of the PE-PCD for vertices in the tetrahedron th

num.in.tetra

Number of Xp points in the tetrahedron, th

ind.in.tetra

The vector of indices of the Xp points that reside in the tetrahedron

tess.points

Tessellation points, i.e., points on which the tessellation of the study region is performed, here, tessellation points are the vertices of the support tetrahedron th.

vertices

Vertices of the digraph, Xp.

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

num.arcsPEtri, num.arcsCStri, and num.arcsAStri

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<-10  #try also n<-20
set.seed(1)
Xp<-runif.tetra(n,tetra)$g

M<-"CM"  #try also M<-"CC"
r<-1.25

Narcs = num.arcsPEtetra(Xp,tetra,r,M)
Narcs
summary(Narcs)
#plot(Narcs)