An object of class "NumEdges".
Returns the number of edges of
the underlying or reflexivity graph of
Arc Slice Proximity Catch Digraph (AS-PCD)
and various other quantities and vectors such as
the vector of number of vertices (i.e., number of data points)
in the Delaunay triangles,
number of data points in the convex hull of Yp points,
indices of the Delaunay triangles for the data points, etc.
AS proximity regions are defined with respect to the
Delaunay triangles based on Yp points
and vertex regions in each triangle are based on the center M="CC"
for circumcenter of each Delaunay triangle
or \(M=(\alpha,\beta,\gamma)\) in barycentric coordinates in the
interior of each Delaunay triangle;
default is M="CC", i.e., circumcenter of each triangle.
Each Delaunay triangle is first converted to
a (nonscaled) basic triangle so that M will be the same
type of center for each Delaunay triangle
(this conversion is not necessary when M is \(CM\)).
Convex hull of Yp is partitioned
by the Delaunay triangles based on Yp points
(i.e., multiple triangles are the set of these Delaunay triangles
whose union constitutes the
convex hull of Yp points).
For the number of edges,
loops are not allowed so edges are only possible
for points inside the convex hull of Yp points.
See (ceyhan:Phd-thesis,ceyhan:stamet2016;textualpcds.ugraph) for more on AS-PCDs. Also, see (okabe:2000,ceyhan:comp-geo-2010,sinclair:2016;textualpcds.ugraph) for more on Delaunay triangulation and the corresponding algorithm.
num.edgesAS(Xp, Yp, M = "CC", ugraph = c("underlying", "reflexivity"))A list with the elements
A short description of the output: number of edges and related quantities for the induced subgraphs of the underlying or reflexivity graphs (of AS-PCD) in the Delaunay triangles
Type of the graph as "Underlying" or "Reflexivity" for the AS-PCD
Total number of edges in all triangles, i.e., the number of edges for the entire underlying or reflexivity graphs of the AS-PCD
Number of Xp points
in the convex hull of Yp points
The vector of number of Xp points
in the Delaunay triangles based on Yp points
The vector of the areas of
Delaunay triangles based on Yp points
The vector of the number of edges
of the components of the AS-PCD in the
Delaunay triangles based on Yp points
A matrix of indices of vertices of
the Delaunay triangles based on Yp points,
each column corresponds to the vector of
indices of the vertices of one triangle.
A vector of indices of vertices of
the Delaunay triangles in which data points reside,
i.e., column number of del.tri.ind for each Xp point.
Points on which the tessellation of the study region is performed,
here, tessellation is the Delaunay triangulation based on Yp points.
Vertices of the underlying or reflexivity graph, Xp.
A set of 2D points which constitute the vertices of the underlying or reflexivity graph of the AS-PCD.
A set of 2D points which constitute the vertices of the Delaunay triangles.
The center of the triangle.
"CC" stands for circumcenter of each Delaunay triangle
or 3D point in barycentric
coordinates which serves as a center
in the interior of each Delaunay triangle;
default is M="CC", i.e., the circumcenter of each triangle.
The type of the graph based on AS-PCDs,
"underlying" is for the underlying graph, and "reflexivity" is for
the reflexivity graph (default is "underlying").
Elvan Ceyhan
num.edgesAStri, num.edgesPE,
num.edgesCS, and num.arcsAS
#\donttest{
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-5;
set.seed(1)
Xp<-cbind(runif(nx),runif(nx))
Yp<-cbind(runif(ny,0,.25),
runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
pcds::plotDelaunay.tri(Xp,Yp,xlab="",ylab="")
M<-c(1,1,1)
Nedges = num.edgesAS(Xp,Yp,M)
Nedges
summary(Nedges)
plot(Nedges)
#}
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