# Generation of points segregated (in a Type II fashion) from the vertices of $T_e$ 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.) ```r rsegII.std.tri(n, eps) ``` ## 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 $T_e$). ## Returns 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 ## Examples ```r 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) ``` ## See Also `rseg.circular`, `rassoc.circular`, `rseg.std.tri`, and `rseg.multi.tri` ## Author(s) Elvan Ceyhan

rsegII.std.tri function