rsegII.std.tri: Generation of points segregated (in a Type II fashion) from the vertices 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 segregation alternative for eps in \((0,\sqrt{3}/6=0.2886751]\).

In the type II segregation, the annular forbidden regions around the edges are determined by the parameter eps which is the distance from the interior triangle (i.e., support for the segregation) to \(T_e\) (see examples for a sample plot.)

Usage

rsegII.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 exclusion parameter, eps, of the segregation pattern, which is the distance from the interior triangle to \(T_e\)
ref.points: The input set of points Y; reference points, i.e., points from which generated points are segregated (i.e., vertices of \(T_e\)).
gen.points: The output set of generated points segregated from Y points (i.e., vertices 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 segregation (which is the distance from the interior triangle points to the boundary of \(T_e\)).

Author

Elvan Ceyhan

Examples

Run this code

A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-10  #try also n<-20 or n<-100 or 1000
eps<-.15  #try also .2

set.seed(1)
Xdt<-rsegII.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 segregation 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 segregation alternative
C1<-c(1/2,sqrt(3)/2-2*eps);
A1<-c(eps*sqrt(3),eps); B1<-c(1-eps*sqrt(3),eps);
supp<-rbind(A1,B1,C1)

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