pcds (version 0.1.8)

Idom.setPEstd.tri: The indicator for the set of points S being a dominating set or not for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard equilateral triangle case

Description

Returns \(I(\)S a dominating set of PE-PCD whose vertices are the data points Xp\()\) for S in the standard equilateral triangle, that is, returns 1 if S is a dominating set of PE-PCD, and returns 0 otherwise.

PE proximity region is constructed with respect to the standard equilateral triangle \(T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2))\) with expansion parameter \(r \ge 1\) 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 \(T_e\); default is \(M=(1,1,1)\), i.e., the center of mass of \(T_e\) (which is also equivalent to the circumcenter of \(T_e\)). Vertices of \(T_e\) are also labeled as 1, 2, and 3, respectively.

See also (ceyhan:Phd-thesis,ceyhan:masa-2007,ceyhan:dom-num-NPE-Spat2011,ceyhan:mcap2012;textualpcds).

Usage

Idom.setPEstd.tri(S, Xp, r, M = c(1, 1, 1))

Value

\(I(\)S a dominating set of PE-PCD\()\) for S

in the standard equilateral triangle, that is, returns 1 if S is a dominating set of PE-PCD, and returns 0 otherwise, where PE proximity region is constructed in the standard equilateral triangle \(T_e\).

Arguments

S

A set of 2D points whose PE proximity regions are considered.

Xp

A set of 2D points which constitutes the vertices of the PE-PCD.

r

A positive real number which serves as the expansion parameter in PE proximity region in the standard equilateral triangle \(T_e=T((0,0),(1,0),(1/2,\sqrt{3}/2))\); must be \(\ge 1\).

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 equilateral triangle \(T_e\); default is \(M=(1,1,1)\) i.e. the center of mass of \(T_e\).

Author

Elvan Ceyhan

References

See Also

Idom.setPEtri and Idom.setCSstd.tri

Examples

# \donttest{
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

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

r<-1.5

S<-rbind(Xp[1,],Xp[2,])
Idom.setPEstd.tri(S,Xp,r,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,],c(.2,.5))
Idom.setPEstd.tri(S,Xp[3,],r,M)
# }