An object of class "UndPCDs".
Returns edges of the underlying or reflexivity graph of AS-PCD
as left and right end points
and related parameters and the quantities of these graphs.
The vertices of these graphs are the data points in Xp
in the multiple triangle case.
AS proximity regions are defined
with respect to the Delaunay triangles based on
Yp points, i.e.,
AS proximity regions are defined only for Xp points
inside the convex hull of Yp points.
That is, edges may exist for points only
inside the convex hull of Yp points.
It also provides various descriptions
and quantities about the edges of the AS-PCD
such as number of edges, edge density, etc.
Vertex regions 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.
M must be entered in barycentric coordinates
unless it is the circumcenter.
The different consideration of circumcenter vs
any other interior center of the triangle is because
the projections from circumcenter are orthogonal to the edges,
while projections of M on the edges are on the extensions
of the lines connecting M and the vertices.
Each Delaunay triangle is first converted to
an (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 the AS-PCDs. Also, see (okabe:2000,ceyhan:comp-geo-2010,sinclair:2016;textualpcds.ugraph) for more on Delaunay triangulation and the corresponding algorithm.
edgesAS(Xp, Yp, M = "CC", ugraph = c("underlying", "reflexivity"))A list with the elements
A description of the underlying or reflexivity graph of the digraph
Parameters of the underlying or reflexivity graph of the digraph,
here, it is only the center M used to
construct the vertex regions.
Tessellation points, i.e., points on which the tessellation
of the study region is performed,
here, tessellation is the Delaunay triangulation based on Yp points.
Name of the tessellation points tess.points
Vertices of the digraph, Xp points
Name of the data set which constitute the vertices of the digraph
Left and right end points of the edges of
the underlying or reflexivity graph of AS-PCD for 2D data set Xp
as vertices of the underlying or reflexivity graph of the digraph
Text for "main" title
in the plot of the underlying or reflexivity graph of the digraph
Various quantities for the underlying or reflexivity graph of the digraph: number of vertices, number of partition points, number of intervals, number of edges, and edge density.
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" represents the 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.
M must be entered in barycentric coordinates
unless it is the circumcenter.
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
edgesAStri, edgesPE,
edgesCS, and arcsAS
#\donttest{
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-5;
set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
M<-c(1,1,1)
Edges<-edgesAS(Xp,Yp,M)
Edges
summary(Edges)
plot(Edges)
#}
Run the code above in your browser using DataLab