IarcPEtetra {pcds}

The indicator for the presence of an arc from one 3D point to another 3D point for Proportional Edge Proximity Catch Digraphs (PE-PCDs)

Description

Returns I(p2 is in N_{PE}(p1,r)) for 3D points p1 and p2, that is, returns 1 if p2 is in N_{PE}(p1,r), returns 0 otherwise, where N_{PE}(x,r) is the PE proximity region for point x with the expansion parameter r \ge 1.

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". rv is the index of the vertex region p1 resides, with default=NULL.

If p1 and p2 are distinct and either of them are outside th, it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

See also (Ceyhan (2005, 2010)).

Usage

IarcPEtetra(p1, p2, th, r, M = "CM", rv = NULL)

Arguments

p1

A 3D point whose PE proximity region is constructed.

p2

A 3D point. The function determines whether p2 is inside the PE proximity region of p1 or not.

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".

rv

Index of the M-vertex region containing the point, either 1,2,3,4 (default is NULL).

Value

I(p2 is in N_{PE}(p1,r)) for p1, that is, returns 1 if p2 is in N_{PE}(p1,r), 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

IarcPEstd.tetra, IarcPEtri and IarcPEint

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<-3  #try also n<-20

Xp<-runif.tetra(n,tetra)$g

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

IarcPEtetra(Xp[1,],Xp[2,],tetra,r)  #uses the default M="CM"
IarcPEtetra(Xp[1,],Xp[2,],tetra,r,M)

IarcPEtetra(c(.4,.4,.4),c(.5,.5,.5),tetra,r,M)

#or try
RV<-rel.vert.tetraCC(Xp[1,],tetra)$rv
IarcPEtetra(Xp[1,],Xp[3,],tetra,r,M,rv=RV)

P1<-c(.1,.1,.1)
P2<-c(.5,.5,.5)
IarcPEtetra(P1,P2,tetra,r,M)