runif.std.tetra: Generation of Uniform Points in the Standard Regular Tetrahedron \(T_h\)

Description

An object of class "Uniform". Generates n points uniformly in the standard regular tetrahedron \(T_h=T((0,0,0),(1,0,0),(1/2,\sqrt{3}/2,0),(1/2,\sqrt{3}/6,\sqrt{6}/3))\).

Usage

runif.std.tetra(n)

Value

A list with the elements

type: The type of the pattern from which points are to be generated
mtitle: The "main" title for the plot of the point pattern
tess.points: The vertices of the support region of the uniformly generated points, it is the standard regular tetrahedron \(T_h\) for this function
gen.points: The output set of generated points uniformly in the standard regular tetrahedron \(T_h\).
out.region: The outer region which contains the support region, NULL for this function.
desc.pat: Description of the point pattern from which points are to be generated
num.points: The vector of two numbers, which are the number of generated points and the number of vertices of the support points (here it is 4).
txt4pnts: Description of the two numbers in num.points
xlimit,ylimit,zlimit: The ranges of the \(x\)-, \(y\)-, and \(z\)-coordinates of the support, \(T_h\)

Arguments

n: A positive integer representing the number of uniform points to be generated in the standard regular tetrahedron \(T_h\).

Author

Elvan Ceyhan

Examples

Run this code

# \donttest{
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)
n<-100

set.seed(1)
Xdt<-runif.std.tetra(n)
Xdt
summary(Xdt)
plot(Xdt)

Xp<-runif.std.tetra(n)$g

Xlim<-range(tetra[,1])
Ylim<-range(tetra[,2])
Zlim<-range(tetra[,3])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
zd<-Zlim[2]-Zlim[1]

plot3D::scatter3D(Xp[,1],Xp[,2],Xp[,3],
phi =20,theta=15, bty = "g", pch = 20, cex = 1,
ticktype = "detailed",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05),
zlim=Zlim+zd*c(-.05,.05))
#add the vertices of the tetrahedron
plot3D::points3D(tetra[,1],tetra[,2],tetra[,3], add=TRUE)
L<-rbind(A,A,A,B,B,C); R<-rbind(B,C,D,C,D,D)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3],
add=TRUE,lwd=2)

plot3D::text3D(tetra[,1]+c(.05,0,0,0),tetra[,2],tetra[,3],
labels=c("A","B","C","D"), add=TRUE)

s3d<-scatterplot3d::scatterplot3d(Xp, highlight.3d=TRUE,xlab="x",
ylab="y",zlab="z", col.axis="blue", col.grid="lightblue",
               main="3D Scatterplot of the data", pch=20)
s3d$points3d(tetra,pch=20,col="blue")
# }