IarcASbasic.tri {pcds}

The indicator for the presence of an arc from a point to another for Arc Slice Proximity Catch Digraphs (AS-PCDs) - standard basic 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 region is constructed in the standard basic triangle T_b=T((0,0),(1,0),(c_1,c_2)) where c_1 is in [0,1/2], c_2>0 and (1-c_1)^2+c_2^2 \le 1.

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 standard basic triangle T_b or based on circumcenter of T_b; default is M="CC", i.e., circumcenter of T_b. 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 T_b, the function returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

Any given triangle can be mapped to the standard basic triangle by a combination of rigid body motions (i.e., translation, rotation and reflection) and scaling, preserving uniformity of the points in the original triangle. Hence standard basic triangle is useful for simulation studies under the uniformity hypothesis.

See also (Ceyhan (2005, 2010)).

Usage

IarcASbasic.tri(p1, p2, c1, c2, 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.

c1, c2

Positive real numbers representing the top vertex in standard basic triangle T_b=T((0,0),(1,0),(c_1,c_2)), c_1 must be in [0,1/2], c_2>0 and (1-c_1)^2+c_2^2 \le 1.

M

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

rv

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

Value

I(p2 \in N_{AS}(p1)) for points p1 and p2, that is, returns 1 if p2 is in N_{AS}(p1) (i.e., if there is an arc from p1 to p2), 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

IarcAStri and NAStri

Examples

c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C)

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.2)

P1<-as.numeric(runif.basic.tri(1,c1,c2)$g)
P2<-as.numeric(runif.basic.tri(1,c1,c2)$g)
IarcASbasic.tri(P1,P2,c1,c2,M)

P1<-c(.3,.2)
P2<-c(.6,.2)
IarcASbasic.tri(P1,P2,c1,c2,M)

#or try
Rv<-rel.vert.basic.triCC(P1,c1,c2)$rv
IarcASbasic.tri(P1,P2,c1,c2,M,Rv)

P1<-c(.3,.2)
P2<-c(.8,.2)
IarcASbasic.tri(P1,P2,c1,c2,M)

P3<-c(.5,.4)
IarcASbasic.tri(P1,P3,c1,c2,M)

P4<-c(1.5,.4)
IarcASbasic.tri(P1,P4,c1,c2,M)
IarcASbasic.tri(P4,P4,c1,c2,M)

c1<-.4; c2<-.6;
P1<-c(.3,.2)
P2<-c(.6,.2)
IarcASbasic.tri(P1,P2,c1,c2,M)