rassocII.std.tri: Generation of points associated (in a Type II fashion) with the edges of \(T_e\)

Description

An object of class "Patterns". Generates n points uniformly in the standard equilateral triangle \(T_e=T((0,0),(1,0),(1/2,\sqrt{3}/2))\) under the type II association alternative for eps in \((0,\sqrt{3}/6=0.2886751]\).

In the type II association, the annular allowed regions around the edges are determined by the parameter eps where \(\sqrt{3}/6\)-eps is the distance from the interior triangle (i.e., forbidden region for association) to \(T_e\) (see examples for a sample plot.)

Usage

rassocII.std.tri(n, eps)

Value

A list with the elements

type: The type of the point pattern
mtitle: The "main" title for the plot of the point pattern
parameters: The attraction parameter, eps, of the association pattern, where \(\sqrt{3}/6\)-eps is the distance from the interior triangle to \(T_e\)
ref.points: The input set of points Y; reference points, i.e., points with which generated points are associated (i.e., vertices of \(T_e\)).
gen.points: The output set of generated points associated with Y points (i.e., edges of \(T_e\)).
tri.Y: Logical output for triangulation based on Y points should be implemented or not. if TRUE triangulation based on Y points is to be implemented (default is set to FALSE).
desc.pat: Description of the point pattern
num.points: The vector of two numbers, which are the number of generated points and the number of reference (i.e., Y) points, which is 3 here.
xlimit,ylimit: The ranges of the \(x\)- and \(y\)-coordinates of the reference points, which are the vertices of \(T_e\) here.

Arguments

n: A positive integer representing the number of points to be generated.
eps: A positive real number representing the parameter of type II association (where \(\sqrt{3}/6\)-eps is the distance from the interior triangle distance from the interior triangle to \(T_e\)).

Author

Elvan Ceyhan

Examples

Run this code

# \donttest{
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-100  #try also n<-20 or n<-100 or 1000
eps<-.2  #try also .25, .1

set.seed(1)
Xdt<-rassocII.std.tri(n,eps)
Xdt
summary(Xdt)
plot(Xdt,asp=1)

Xlim<-range(Te[,1])
Ylim<-range(Te[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

Xp<-Xdt$gen.points
plot(Te,pch=".",xlab="",ylab="",
main="Type II association in the \n standard equilateral triangle",
     xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01))
polygon(Te)
points(Xp)

#The support for the Type II association alternative
A1<-c(1/2-eps*sqrt(3),sqrt(3)/6-eps);
B1<-c(1/2+eps*sqrt(3),sqrt(3)/6-eps);
C1<-c(1/2,sqrt(3)/6+2*eps);
supp<-rbind(A1,B1,C1)

plot(Te,asp=1,pch=".",xlab="",ylab="",
main="Support of the Type II Association",
xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01))
polygon(Te,col=5)
polygon(supp,col=0)
points(Xp)
# }