pcds (version 0.1.6)

NumArcsCSTe: Number of arcs of Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case

Description

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 triangle) and indices of the data points that reside in the triangle.

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

NumArcsCSTe(Xp, t, M = c(1, 1, 1))

Value

A list with the elements

num.arcs

Number of arcs of the CS-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 \(T_e\)

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

NumArcsCStri, NumArcsCS, and NumArcsPETe,

Examples

if (FALSE) {
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)

NumArcsCSTe(Xp,t=.5,M)
NumArcsCSTe(Xp,t=1.5,M)
}