NPEstd.tetra {pcds}

The vertices of the Proportional Edge (PE) Proximity Region in the standard regular tetrahedron

Description

Returns the vertices of the PE proximity region (which is itself a tetrahedron) for a point in the standard regular tetrahedron T_h=T((0,0,0),(1,0,0),(1/2,\sqrt{3}/2,0),(1/2,\sqrt{3}/6,\sqrt{6}/3))= (rv=1,rv=2,rv=3,rv=4).

PE proximity region is defined with respect to the tetrahedron T_h with expansion parameter r \ge 1 and vertex regions based on the circumcenter of T_h (which is equivalent to the center of mass in the standard regular tetrahedron).

Vertex regions are labeled as 1,2,3,4 rowwise for the vertices of the tetrahedron T_h. rv is the index of the vertex region p resides, with default=NULL. If p is outside of T_h, it returns NULL for the proximity region.

See also (Ceyhan (2005, 2010)).

Usage

NPEstd.tetra(p, r, rv = NULL)

Arguments

p

A 3D point whose PE proximity region is to be computed.

r

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

rv

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

Value

Vertices of the tetrahedron which constitutes the PE proximity region with expansion parameter r and circumcenter (or center of mass) for a point p in the standard regular tetrahedron

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

NPEtetra, NPEtri and NPEint

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
Xp<-runif.std.tetra(n)$g
r<-1.5
NPEstd.tetra(Xp[1,],r)

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

NPEstd.tetra(c(-1,-1,-1),r,rv=NULL)