IarcAStri {pcds}

The indicator for the presence of an arc from a point to another for Arc Slice Proximity Catch Digraphs (AS-PCDs) - one triangle case

Description

Returns I(p2 \in N_{AS}(p1)) for points p1 and p2, that is, returns 1 if p2 is in N_{AS}(p1), returns 0 otherwise, where N_{AS}(x) is the AS proximity region for point x.

AS proximity regions are constructed with respect to the triangle, tri=T(A,B,C)=(rv=1,rv=2,rv=3), and vertex regions are based on the center, M=(m_1,m_2) in Cartesian coordinates or M=(\alpha,\beta,\gamma) in barycentric coordinates in the interior of the triangle tri or based on circumcenter of tri; default is M="CC", i.e., circumcenter of tri. 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 tri, the function 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

IarcAStri(p1, p2, tri, M = "CC", rv = NULL)

Arguments

p1

A 2D point whose AS proximity region is constructed.

p2

A 2D point. The function determines whether p2 is inside the AS proximity region of p1 or not.

tri

Three 2D points, stacked row-wise, each row representing a vertex of the triangle.

M

The center of the triangle. "CC" stands for circumcenter of the triangle tri or a 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of tri; default is M="CC" i.e., the circumcenter of tri.

rv

The index of the M-vertex region in tri containing the point, either 1,2,3 or NULL (default is NULL).

Value

I(p2 \in N_{AS}(p1)) for p1, that is, returns 1 if p2 is in N_{AS}(p1), 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.

Ceyhan E (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

See Also

IarcASbasic.tri, IarcPEtri, and IarcCStri

Examples

A<-c(1,1); B<-c(2,0); C<-c(1.5,2);

Tr<-rbind(A,B,C);

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

P1<-as.numeric(runif.tri(1,Tr)$g)
P2<-as.numeric(runif.tri(1,Tr)$g)
IarcAStri(P1,P2,Tr,M)

P1<-c(1.3,1.2)
P2<-c(1.8,.5)
IarcAStri(P1,P2,Tr,M)
IarcAStri(P1,P1,Tr,M)

#or try
Rv<-rel.vert.triCC(P1,Tr)$rv
IarcAStri(P1,P2,Tr,M,Rv)

P3<-c(1.6,1.4)
IarcAStri(P1,P3,Tr,M)

P4<-c(1.5,1.0)
IarcAStri(P1,P4,Tr,M)

P5<-c(.5,1.0)
IarcAStri(P1,P5,Tr,M)
IarcAStri(P5,P5,Tr,M)

#or try
Rv<-rel.vert.triCC(P5,Tr)$rv
IarcAStri(P5,P5,Tr,M,Rv)