ASarc.dens.tri {pcds}

Arc density of Arc Slice Proximity Catch Digraphs (AS-PCDs) - one triangle case

Description

Returns the arc density of AS-PCD whose vertex set is the given 2D numerical data set, Xp, (some of its members are) in the triangle tri.

AS proximity regions are defined with respect to tri and vertex regions are defined with the center M="CC" for circumcenter of tri; or M=(m_1,m_2) in Cartesian coordinates or M=(\alpha,\beta,\gamma) in barycentric coordinates in the interior of the triangle 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 tri for this function.

in.tri.only is a logical argument (default is FALSE) for considering only the points inside the triangle or all the points as the vertices of the digraph. if in.tri.only=TRUE, arc density is computed only for the points inside the triangle (i.e., arc density of the subdigraph induced by the vertices in the triangle is computed), otherwise arc density of the entire digraph (i.e., digraph with all the vertices) is computed.

See also (Ceyhan (2005, 2010)).

Usage

ASarc.dens.tri(Xp, tri, M = "CC", in.tri.only = FALSE)

Arguments

Xp

A set of 2D points which constitute the vertices of the 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.

in.tri.only

A logical argument (default is in.tri.only=FALSE) for computing the arc density for only the points inside the triangle, tri. That is, if in.tri.only=TRUE arc density of the induced subdigraph with the vertices inside tri is computed, otherwise otherwise arc density of the entire digraph (i.e., digraph with all the vertices) is computed.

Value

Arc density of AS-PCD whose vertices are the 2D numerical data set, Xp; AS proximity regions are defined with respect to the triangle tri and CC-vertex regions.

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

ASarc.dens.tri, CSarc.dens.tri, and num.arcsAStri

Examples

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

set.seed(1)
n<-10  #try also n<-20

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)$num.arcs; narcs/(n*(n-1))
ASarc.dens.tri(Xp,Tr,M)
ASarc.dens.tri(Xp,Tr,M,in.tri.only = FALSE)

ASarc.dens.tri(Xp,Tr,M)