num.arcsPEstd.tri: Number of arcs of Proportional Edge Proximity Catch Digraphs...

In elvanceyhan/pcds: Proximity Catch Digraphs and Their Applications

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 (\insertCiteceyhan:arc-density-PE;textualpcds).

Usage

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

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.

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
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	Points on which the tessellation of the study region is performed, here, tessellation is the support triangle T_e.
vertices	Vertices of the digraph, Xp.

Author(s)

Elvan Ceyhan

References

\insertAllCited

See Also

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

Examples

## Not run: 
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)
par(pty="s")
plot(Narcs,asp=1)

## End(Not run)

elvanceyhan/pcds documentation built on June 29, 2023, 8:12 a.m.