pcds (version 0.1.8)

num.arcsCSstd.tri: Number of arcs of Central Similarity Proximity Catch Digraphs (CS-PCDs) and quantities related to the triangle - standard equilateral triangle case

Description

An object of class "NumArcs". Returns the number of arcs of Central Similarity Proximity Catch Digraphs (CS-PCDs) whose vertices are the given 2D numerical data set, Xp. 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 \(T_e\).

CS proximity region \(N_{CS}(x,t)\) 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 \(t>0\) and edge 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:Phd-thesis,ceyhan:arc-density-CS,ceyhan:test2014;textualpcds).

Usage

num.arcsCSstd.tri(Xp, t, 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 CS-PCD

tri.num.arcs

Number of arcs of the induced subdigraph of the CS-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 digraph.

t

A positive real number which serves as the expansion parameter in CS proximity region.

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.arcsCStri, num.arcsCS, and num.arcsPEstd.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<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

Narcs = num.arcsCSstd.tri(Xp,t=.5,M)
Narcs
summary(Narcs)
oldpar <- par(pty="s")
plot(Narcs,asp=1)
par(oldpar)
# }