# Generation of points associated (in a Type I fashion) with 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 I association alternative for `eps` in $(0,\sqrt{3}/3=0.5773503]$. The allowed triangular regions around the vertices are determined by the parameter `eps`. In the type I association, the triangular support regions around the vertices are determined by the parameter `eps` where $\sqrt{3}/3$-`eps` serves as the height of these triangles (see examples for a sample plot.) See also (if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="ceyhan:arc-density-PE,ceyhan:arc-density-CS,ceyhan:dom-num-NPE-Spat2011;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r rassoc.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 I association (where $\sqrt{3}/3$-`eps` serves as the height of the triangular support regions around the vertices). ## 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 attraction parameter of the association pattern, `eps`, where $\sqrt{3}/3$-`eps` serves as the height of the triangular support regions around the vertices - **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., 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. - **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<-100 #try also n<-20 or n<-100 or 1000 eps<-.25 #try also .15, .5, .75 set.seed(1) Xdt<-rassoc.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 I 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 I association alternative sr<-(sqrt(3)/3-eps)/(sqrt(3)/2) C1<-C+sr*(A-C); C2<-C+sr*(B-C) A1<-A+sr*(B-A); A2<-A+sr*(C-A) B1<-B+sr*(A-B); B2<-B+sr*(C-B) supp<-rbind(A1,B1,B2,C2,C1,A2) plot(Te,asp=1,pch=".",xlab="",ylab="", main="Support of the Type I Association", xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01)) if (sr<=.5) { polygon(Te,col=5) polygon(supp,col=0) } else { polygon(Te,col=0,lwd=2.5) polygon(rbind(A,A1,A2),col=5,border=NA) polygon(rbind(B,B1,B2),col=5,border=NA) polygon(rbind(C,C1,C2),col=5,border=NA) } points(Xp) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `rseg.circular`, `rassoc.circular`, `rsegII.std.tri`, and `rseg.multi.tri` ## Author(s) Elvan Ceyhan

rassoc.std.tri function