| IarcPEstd.tri {pcds} | R Documentation |
The indicator for the presence of an arc from a point to another for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard equilateral triangle case
Description
Returns I(p2 is in N_{PE}(p1,r))
for points p1 and p2
in the standard equilateral triangle,
that is, returns 1 if p2 is in N_{PE}(p1,r),
and returns 0 otherwise,
where N_{PE}(x,r) is the PE proximity region
for point x with expansion parameter r \ge 1.
PE 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 (Ceyhan (2005, 2010); Ceyhan et al. (2007)).
Usage
IarcPEstd.tri(p1, p2, r, M = c(1, 1, 1), rv = NULL)
Arguments
p1 |
A 2D point whose PE proximity region is constructed. |
p2 |
A 2D point. The function determines
whether |
r |
A positive real number
which serves as the expansion parameter in PE proximity region;
must be |
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 |
rv |
The index of the vertex region in |
Value
I(p2 is in N_{PE}(p1,r))
for points p1 and p2
in the standard equilateral triangle,
that is, returns 1 if p2 is in N_{PE}(p1,r),
and returns 0 otherwise.
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, Priebe CE, Marchette DJ (2007).
“A new family of random graphs for testing spatial segregation.”
Canadian Journal of Statistics, 35(1), 27-50.
See Also
IarcPEtri, IarcPEbasic.tri,
and IarcCSstd.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)
IarcPEstd.tri(Xp[1,],Xp[3,],r=1.5,M)
IarcPEstd.tri(Xp[1,],Xp[3,],r=2,M)
#or try
Rv<-rel.vert.std.triCM(Xp[1,])$rv
IarcPEstd.tri(Xp[1,],Xp[3,],r=2,rv=Rv)
P1<-c(.4,.2)
P2<-c(.5,.26)
r<-2
IarcPEstd.tri(P1,P2,r,M)