| IedgeCSstd.tri {pcds.ugraph} | R Documentation |
The indicator for the presence of an edge from a point to another for the underlying or reflexivity graphs of Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case
Description
Returns I(p1p2 is an edge
in the underlying or reflexivity graph of CS-PCDs )
for points p1 and p2 in the standard equilateral triangle.
More specifically, when the argument ugraph="underlying", it returns
the edge indicator for points p1 and p2
in the standard equilateral triangle,
for the CS-PCD underlying graph,
that is, returns 1 if p2 is
in N_{CS}(p1,t) or p1 is in N_{CS}(p2,t),
returns 0 otherwise.
On the other hand,
when ugraph="reflexivity", it returns
the edge indicator for points p1 and p2
in the standard equilateral triangle,
for the CS-PCD reflexivity graph,
that is, returns 1 if p2 is
in N_{CS}(p1,t) and p1 is in N_{CS}(p2,t),
returns 0 otherwise.
In both cases 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 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.
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)).
Usage
IedgeCSstd.tri(
p1,
p2,
t,
M = c(1, 1, 1),
ugraph = c("underlying", "reflexivity")
)
Arguments
p1 |
A 2D point whose CS proximity region is constructed. |
p2 |
A 2D point. The function determines
whether there is an edge from |
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 |
ugraph |
The type of the graph based on CS-PCDs,
|
Value
Returns 1 if there is an edge between points p1 and p2
in the underlying or reflexivity graph of CS-PCDs
in the standard equilateral triangle, and 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 (2016).
“Edge Density of New Graph Types Based on a Random Digraph Family.”
Statistical Methodology, 33, 31-54.
See Also
IedgeCSbasic.tri, IedgeCStri,
and IarcCSstd.tri
Examples
#\donttest{
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<-pcds::runif.std.tri(n)$gen.points
M<-as.numeric(pcds::runif.std.tri(1)$g)
IedgeCSstd.tri(Xp[1,],Xp[3,],t=1.5,M)
IedgeCSstd.tri(Xp[1,],Xp[3,],t=1.5,M,ugraph="reflexivity")
P1<-c(.4,.2)
P2<-c(.5,.26)
t<-2
IedgeCSstd.tri(P1,P2,t,M)
IedgeCSstd.tri(P1,P2,t,M,ugraph = "reflexivity")
#}