pcds (version 0.1.8)

num.arcsPEstd.tri: Number of arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) and quantities related to the triangle - standard equilateral triangle case

Description

An object of class "NumArcs". Returns the number of arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) whose vertices are the given 2D numerical data set, Xp in the standard equilateral triangle. It also provides number of vertices (i.e., number of data points inside the standard equilateral triangle \(T_e\)) and indices of the data points that reside in \(T_e\).

PE proximity region \(N_{PE}(x,r)\) is defined with respect to the standard equilateral triangle \(T_e=T(v=1,v=2,v=3)=T((0,0),(1,0),(1/2,\sqrt{3}/2))\) with expansion parameter \(r \ge 1\) and vertex regions are based on the center \(M=(m_1,m_2)\) in Cartesian coordinates or \(M=(\alpha,\beta,\gamma)\) in barycentric coordinates in the interior of \(T_e\); default is \(M=(1,1,1)\), i.e., the center of mass of \(T_e\). For the number of arcs, loops are not allowed so arcs are only possible for points inside \(T_e\) for this function.

See also (ceyhan:arc-density-PE;textualpcds).

Usage

num.arcsPEstd.tri(Xp, r, M = c(1, 1, 1))

Value

A list with the elements

desc

A short description of the output: number of arcs and quantities related to the standard equilateral triangle

num.arcs

Number of arcs of the PE-PCD

tri.num.arcs

Number of arcs of the induced subdigraph of the PE-PCD for vertices in the standard equilateral triangle \(T_e\)

num.in.tri

Number of Xp points in the standard equilateral triangle, \(T_e\)

ind.in.tri

The vector of indices of the Xp points that reside in \(T_e\)

tess.points

Tessellation points, i.e., points on which the tessellation of the study region is performed, here, tessellation points are the vertices of the support triangle \(T_e\).

vertices

Vertices of the digraph, Xp.

Arguments

Xp

A set of 2D points which constitute the vertices of the PE-PCD.

r

A positive real number which serves as the expansion parameter for PE proximity region; must be \(\ge 1\).

M

A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the standard equilateral triangle \(T_e\); default is \(M=(1,1,1)\) i.e. the center of mass of \(T_e\).

Author

Elvan Ceyhan

References

See Also

num.arcsPEtri, num.arcsPE, and num.arcsCSstd.tri

Examples

# \donttest{
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-c(.6,.2)  #try also M<-c(1,1,1)

Narcs = num.arcsPEstd.tri(Xp,r=1.25,M)
Narcs
summary(Narcs)
oldpar <- par(pty="s")
plot(Narcs,asp=1)
par(oldpar)
# }