num.arcsAStri {pcds}

Number of arcs of Arc Slice Proximity Catch Digraphs (AS-PCDs) and quantities related to the triangle - one triangle case

Description

An object of class "NumArcs". Returns the number of arcs of Arc Slice Proximity Catch Digraphs (AS-PCDs) whose vertices are the 2D 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.

The data points could be inside or outside a general triangle tri=T(A,B,C)=(rv=1,rv=2,rv=3), with vertices of tri stacked row-wise.

AS proximity regions are defined with respect to the triangle tri 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 the triangle tri or based on circumcenter of tri; default is M="CC", i.e., circumcenter of tri. For the number of arcs, loops are not allowed, so arcs are only possible for points inside the triangle, tri.

See also (Ceyhan (2005, 2010)).

Usage

num.arcsAStri(Xp, tri, M = "CC")

Arguments

Xp

A set of 2D points which constitute the vertices of the digraph (i.e., AS-PCD).

tri

Three 2D points, stacked row-wise, each row representing a vertex of the triangle.

M

The center of the triangle. "CC" stands for circumcenter of the triangle tri or a 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of tri; default is M="CC" i.e., the circumcenter of tri.

Value

A list with the elements

desc

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

num.arcs

Number of arcs of the AS-PCD

tri.num.arcs

Number of arcs of the induced subdigraph of the AS-PCD for vertices in the triangle tri

num.in.tri

Number of Xp points in the triangle, tri

ind.in.tri

The vector of indices of the Xp points that reside in the triangle

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 tri.

vertices

Vertices of the digraph, Xp.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2005). An Investigation of Proximity Catch Digraphs in Delaunay Tessellations, also available as technical monograph titled Proximity Catch Digraphs: Auxiliary Tools, Properties, and Applications. Ph.D. thesis, The Johns Hopkins University, Baltimore, MD, 21218.

Ceyhan E (2010). “Extension of One-Dimensional Proximity Regions to Higher Dimensions.” Computational Geometry: Theory and Applications, 43(9), 721-748.

Ceyhan E (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

See Also

num.arcsAS, num.arcsPEtri, and num.arcsCStri

Examples

A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

n<-10  #try also n<-20
set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

Narcs = num.arcsAStri(Xp,Tr,M)
Narcs
summary(Narcs)
plot(Narcs)