pcds.ugraph (version 0.1.1)

num.edgesCSstd.tri: Number of edges in the underlying or reflexivity graphs of Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case

Description

An object of class "NumEdges". Returns the number of edges of the underlying or reflexivity graphs of Central Similarity Proximity Catch Digraphs (CS-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 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 edges, loops are not allowed so edges are only possible for points inside \(T_e\) for this function.

See also (ceyhan:stamet2016;textualpcds.ugraph).

Usage

num.edgesCSstd.tri(
  Xp,
  t,
  M = c(1, 1, 1),
  ugraph = c("underlying", "reflexivity")
)

Value

A list with the elements

desc

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

und.graph

Type of the graph as "Underlying" or "Reflexivity" for the CS-PCD

num.edges

Number of edges of the underlying or reflexivity graphs based on the CS-PCD with 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 is the support triangle \(T_e\).

vertices

Vertices of the underlying or reflexivity graph, Xp.

Arguments

Xp

A set of 2D points which constitute the vertices of the underlying or reflexivity graphs based on the CS-PCD.

t

A positive real number which serves as the expansion parameter for 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 \(T_e\).

ugraph

The type of the graph based on CS-PCDs, "underlying" is for the underlying graph, and "reflexivity" is for the reflexivity graph (default is "underlying").

Author

Elvan Ceyhan

References

See Also

num.edgesCStri, num.edgesCS, and num.arcsCSstd.tri

Examples

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

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

M<-c(.6,.2)

Nedges = num.edgesCSstd.tri(Xp,t=1.5,M)
Nedges
summary(Nedges)
plot(Nedges)
#}