IarcCSstd.tri: The indicator for the presence of an arc from a point to...

In pcds: Proximity Catch Digraphs and Their Applications

The indicator for the presence of an arc from a point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case

Description

Returns I(p2 is in N_{CS}(p1,t)) for points p1 and p2, that is, returns 1 if p2 is in N_{CS}(p1,t), returns 0 otherwise, where N_{CS}(x,t) is the CS proximity region for point x with expansion parameter t >0.

CS proximity region 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)) 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 T_e; default is M=(1,1,1) i.e., the center of mass of T_e. rv is the index of the vertex region p1 resides, with default=NULL.

If p1 and p2 are distinct and either of them are outside T_e, it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

See also (\insertCiteceyhan:Phd-thesis,ceyhan:comp-geo-2010,ceyhan:arc-density-CS;textualpcds).

Usage

IarcCSstd.tri(p1, p2, t, M = c(1, 1, 1), re = NULL)

Arguments

p1	A 2D point whose CS proximity region is constructed.
p2	A 2D point. The function determines whether p2 is inside the CS proximity region of p1 or not.
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.
re	The index of the edge region in T_e containing the point, either 1,2,3 or NULL (default is NULL).

Value

I(p2 is in N_{CS}(p1,t)) for points p1 and p2, that is, returns 1 if p2 is in N_{CS}(p1,t), returns 0 otherwise

Author(s)

Elvan Ceyhan

References

\insertAllCited

See Also

IarcCStri, IarcCSbasic.tri, and IarcPEstd.tri

Examples

A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)
n<-3

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

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2) or M=(A+B+C)/3

IarcCSstd.tri(Xp[1,],Xp[3,],t=2,M)
IarcCSstd.tri(c(0,1),Xp[3,],t=2,M)

#or try
Re<-rel.edge.tri(Xp[1,],Te,M) $re
IarcCSstd.tri(Xp[1,],Xp[3,],t=2,M,Re)

pcds documentation built on May 29, 2024, 9:36 a.m.