IarcCSbasic.tri {pcds}

The indicator for the presence of an arc from a point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard basic triangle case

Description

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

CS proximity region is defined with respect to 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.

Edge 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; default is M=(1,1,1) i.e., the center of mass of T_b. re is the index of the edge region p1 resides, with default=NULL.

If p1 and p2 are distinct and either of them are outside T_b, it 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); Ceyhan et al. (2007)).

Usage

IarcCSbasic.tri(p1, p2, t, c1, c2, M = c(1, 1, 1), re = NULL)

Arguments

p1

A 2D point whose CS proximity region is constructed.

p2

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

t

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

c1, c2

Positive real numbers which constitute the vertex of the standard basic triangle adjacent to the shorter edges; c_1 must be in [0,1/2], c_2>0 and (1-c_1)^2+c_2^2 \le 1.

M

A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the standard basic triangle or circumcenter of T_b; default is M=(1,1,1) i.e., the center of mass of T_b.

re

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

Value

I(p2 is in N_{CS}(p1,t)) for points p1 and p2, that is, returns 1 if p2 is in N_{CS}(p1,t), 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, Priebe CE, Marchette DJ (2007). “A new family of random graphs for testing spatial segregation.” Canadian Journal of Statistics, 35(1), 27-50.

See Also

IarcCStri and IarcCSstd.tri

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)

tau<-2

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

P1<-c(.4,.2)
P2<-c(.5,.26)
IarcCSbasic.tri(P1,P2,tau,c1,c2,M)
IarcCSbasic.tri(P1,P1,tau,c1,c2,M)

#or try
Re<-rel.edge.basic.tri(P1,c1,c2,M)$re
IarcCSbasic.tri(P1,P2,tau,c1,c2,M,Re)
IarcCSbasic.tri(P1,P1,tau,c1,c2,M,Re)