First we load the pcds package:
Recall that a Delaunay cell is an interval in 1D space, a triangle in 2D space, and a tetrahedron in 3D space. Extreme points or extrema are defined in a local or restricted sense, e.g., points closest to the center in a vertex or edge region or closest to the opposite edge in a vertex region, etc of a Delaunay cell. Such points are usually the best candidates for the (minimum) dominating sets for PCDs.
First we define the same triangle \(T\) used in the previous sections
And we generate \(n=\) 10 uniform points in it
using the function runif.tri.
Here an extrema point in each \(M\)-vertex region in a triangle \(T\), where \(M\) is a center,
is the point in the given data set, \(\mathcal{X}_n\) closest to the opposite edge.
That is, for the triangle \(T(A,B,C)\), if the \(M\)-vertex region for vertex \(A\) is \(V(A)\),
then the opposite edge is edge \(BC\), and closest point
in \(\mathcal X_n \cap V(A)\) to edge \(BC\) is found (if there are no data points in \(V(A)\),
the function returns NA for the vertex region \(V(A)\)).
These extrema are used to find the minimum dominating set and hence the domination number of the PE-PCD,
since a subset (possibly all) of this type of extrema points constitutes a minimum dominating set
(see Ceyhan and Priebe (2007), Ceyhan and Priebe (2005), and Ceyhan (2011) for more details).
With M=CC (i.e., when the center is the circumcenter),
the extrema point here is the closest \(\mathcal{X}\) point in CC-vertex
region to the opposite edge, and is found by the function cl2edgesCCvert.reg
which is an object of class Extrema
and has arguments Xp,tri where
Xp, a set of 2D points representing the set of data points andtri, a \(3 \times 2\) matrix with each row representing a vertex of the triangle.Its call (with Ext in the below script) just returns the type of the extrema.
Its summary returns the type of the extrema, the extrema points,
distances between the edges and the closest points to the edges in CC-vertex regions,
vertices of the support of the data points.
The plot function (or plot.Extrema) would return
the plot of the triangle together with the CC-vertex regions and
the generated points with extrema being marked with red crosses.
Ext<-cl2edgesCCvert.reg(Xp,Tr)
Ext
#> Call:
#> cl2edgesCCvert.reg(Xp = Xp, tri = Tr)
#>
#> Type:
#> [1] "Closest Points in CC-Vertex Regions of the Triangle with Vertices A=(1,1), B=(2,0), and C=(1.5,2) \n to the Opoosite Edges"
summary(Ext)
#> Call:
#> cl2edgesCCvert.reg(Xp = Xp, tri = Tr)
#>
#> Type of the Extrema
#> [1] "Closest Points in CC-Vertex Regions of the Triangle with Vertices A=(1,1), B=(2,0), and C=(1.5,2) \n to the Opoosite Edges"
#>
#> Extremum Points: Closest Points in CC-Vertex Regions of the Triangle to its Edges
#> (Row i corresponds to vertex region i for i=1,2,3)
#> [,1] [,2]
#> [1,] 1.723711 0.8225489
#> [2,] 1.794240 0.2158873
#> [3,] 1.380035 1.5548904
#> [1] "Vertex labels are A=1, B=2, and C=3 (correspond to row number in Extremum Points)"
#>
#> Distances between the Edges and the Closest Points to the Edges in CC-Vertex Regions
#> (i-th entry corresponds to vertex region i for i=1,2,3)
#> [1] 0.06854235 1.06105561 0.66109225
#>
#> Vertices of the Support Triangle
#> [,1] [,2]
#> A 1.0 1
#> B 2.0 0
#> C 1.5 2
plot(Ext)With M=CM (i.e., when the center is the center of mass),
the extrema point here is the closest \(\mathcal{X}\) point in CM-vertex
region to the opposite edge, and is found by the function cl2edgesCMvert.reg
which is an object of class Extrema.
Its arguments, call, summary and plot are as in cl2edgesCCvert.reg.
Ext<-cl2edgesCMvert.reg(Xp,Tr)
Ext
#> Call:
#> cl2edgesCMvert.reg(Xp = Xp, tri = Tr)
#>
#> Type:
#> [1] "Closest Points in CM-Vertex Regions of the Triangle with Vertices A=(1,1), B=(2,0), and C=(1.5,2) \n to the Opposite Edges"
summary(Ext)
#> Call:
#> cl2edgesCMvert.reg(Xp = Xp, tri = Tr)
#>
#> Type of the Extrema
#> [1] "Closest Points in CM-Vertex Regions of the Triangle with Vertices A=(1,1), B=(2,0), and C=(1.5,2) \n to the Opposite Edges"
#>
#> Extremum Points: Closest Points in CM-Vertex Regions of the Triangle to its Edges
#> (Row i corresponds to vertex region i for i=1,2,3)
#> [,1] [,2]
#> [1,] 1.316272 1.0372685
#> [2,] 1.687023 0.7682074
#> [3,] 1.482080 1.1991317
#> [1] "Vertex labels are A=1, B=2, and C=3 (correspond to row number in Extremum Points)"
#>
#> Distances between the Edges and the Closest Points to the Edges in CM-Vertex Regions
#> (i-th entry corresponds to vertex region i for i=1,2,3)
#> [1] 0.4117393 0.7181527 0.4816895
#>
#> Vertices of the Support Triangle
#> [,1] [,2]
#> A 1.0 1
#> B 2.0 0
#> C 1.5 2
plot(Ext)With a general center M in the interior of the triangle \(T\),
the extrema point here is the closest \(\mathcal{X}\) point in \(M\)-vertex
region to the opposite edge,
and is found by the function cl2edgesMvert.reg
which is an object of class Extrema and has arguments Xp,tri,M,alt where
Xp,tri are as in cl2edgesCCvert.reg.M, a 2D point in Cartesian coordinates or a 3D point in barycentric coordinates
which serves as a center in the interior of the triangle tri or the circumcenter of tri;
which may be entered as “CC” as well;alt, a logical argument for alternative method of finding the closest points to the edges, default alt=FALSE.
When alt=FALSE, the function sequentially finds the vertex region of the data point and then the minimum distance
to the opposite edge and the relevant extrema objects, and when alt=TRUE,
it first partitions the data set according which vertex regions they reside, and
then finds the minimum distance to the opposite edge and the relevant extrema on each partition.Its call, summary and plot are as in cl2edgesCCvert.reg.
We comment the plot(Ext) out below for brevity in the exposition,
as the plot is similar to the one in cl2edgesCMvert.reg above.
M<-c(1.6,1.0) #try also M<-as.numeric(runif.tri(1,Tr)$g)
Ext<-cl2edgesMvert.reg(Xp,Tr,M)
Ext
#> Call:
#> cl2edgesMvert.reg(Xp = Xp, tri = Tr, M = M)
#>
#> Type:
#> [1] "Closest Points in M-Vertex Regions of the Triangle with Vertices A=(1,1), B=(2,0), and C=(1.5,2)\n to Opposite Edges"
summary(Ext)
#> Call:
#> cl2edgesMvert.reg(Xp = Xp, tri = Tr, M = M)
#>
#> Type of the Extrema
#> [1] "Closest Points in M-Vertex Regions of the Triangle with Vertices A=(1,1), B=(2,0), and C=(1.5,2)\n to Opposite Edges"
#>
#> Extremum Points: Closest Points in M-Vertex Regions of the Triangle to its Edges
#> (Row i corresponds to vertex region i for i=1,2,3)
#> [,1] [,2]
#> [1,] 1.482080 1.1991317
#> [2,] 1.687023 0.7682074
#> [3,] 1.380035 1.5548904
#> [1] "Vertex labels are A=1, B=2, and C=3 (correspond to row number in Extremum Points)"
#>
#> Distances between the Edges and the Closest Points to the Edges in M-Vertex Regions
#> (i-th entry corresponds to vertex region i for i=1,2,3)
#> [1] 0.2116239 0.7181527 0.6610922
#>
#> Vertices of the Support Triangle
#> [,1] [,2]
#> A 1.0 1
#> B 2.0 0
#> C 1.5 2
plot(Ext)There is also the function cl2edges.vert.reg.basic.tri which takes arguments Xp,c1,c2,M
and finds the closest points in a data set Xp
in the \(M\)-vertex regions to the corresponding (opposite) edges in the standard basic triangle, \(T_b\)
where the standard basic triangle is \(T_b=T((0,0),(1,0),(c_1,c_2))\)
with \(0 < c_1 \le 1/2\), \(c_2>0\),
and \((1-c_1)^2+c_2^2 \le 1\).
Here an extrema point in each CC-vertex region
is the point in the given data set, \(\mathcal{X}_n\) closest to the circumcenter (CC).
That is, for the triangle \(T(A,B,C)\), if the \(CC\)-vertex region is \(V(A)\),
the closest point in \(\mathcal X_n \cap V(A)\) to the circumcenter is found
(if there are no data points in \(V(A)\), the function returns NA for the vertex region \(V(A)\)).
The description and summary of extrema points and their plot can be obtained using the function cl2CCvert.reg
which is an object of class Extrema and takes arguments Xp,tri,ch.all.intri where
Xp,tri are as in cl2edgesCCvert.reg andch.all.intri is a logical argument (default=FALSE) to check whether all data points are inside
the triangle tri. So, if it is TRUE,
the function checks if all data points are inside the closure of the triangle (i.e., interior and boundary
combined) else it does not.Its call, summary and plot are as in cl2edgesCCvert.reg, except in summary
distances between the circumcenter and the closest points to the circumcenter in CC-vertex regions are provided.
Ext<-cl2CCvert.reg(Xp,Tr)
Ext
#> Call:
#> cl2CCvert.reg(Xp = Xp, tri = Tr)
#>
#> Type:
#> [1] "Closest Points in CC-Vertex Regions of the Triangle with Vertices A=(1,1), B=(2,0), and C=(1.5,2) to its Circumcenter"
summary(Ext)
#> Call:
#> cl2CCvert.reg(Xp = Xp, tri = Tr)
#>
#> Type of the Extrema
#> [1] "Closest Points in CC-Vertex Regions of the Triangle with Vertices A=(1,1), B=(2,0), and C=(1.5,2) to its Circumcenter"
#>
#> Extremum Points: Closest Points in CC-Vertex Regions of the Triangle to its Circumcenter
#> (Row i corresponds to vertex i for i=1,2,3)
#> [,1] [,2]
#> [1,] 1.723711 0.8225489
#> [2,] 1.794240 0.2158873
#> [3,] 1.529720 1.5787125
#> [1] "Vertex labels are A=1, B=2, and C=3 (correspond to row number in Extremum Points)"
#>
#> Distances between the Circumcenter and the Closest Points to the Circumcenter in CC-Vertex Regions
#> (i-th entry corresponds to vertex i for i=1,2,3)
#> [1] 0.4442261 0.9143510 0.7428921
#>
#> Vertices of the Support Triangle
#> [,1] [,2]
#> A 1.0 1
#> B 2.0 0
#> C 1.5 2
plot(Ext) There is also the function cl2CCvert.reg.basic.tri which takes arguments Xp,c1,c2,ch.all.intri and
finds the closest points in a data set Xp in the CC-vertex regions to the circumcenter in the standard basic triangle, \(T_b\).
Here an extrema point in each CC-vertex region
is the point in the given data set, \(\mathcal{X}_n\) furthest to the corresponding vertex.
That is, for the triangle \(T(A,B,C)\), if the \(CC\)-vertex region is \(V(A)\),
the furthest point in \(\mathcal X_n \cap V(A)\) from the vertex \(A\) is found
(if there are no data points in \(V(A)\), the function returns NA for the vertex region \(V(A)\)).
The description and summary of extrema points and their plot can be obtained using the function fr2vertsCCvert.reg
which is an object of class Extrema.
Its arguments, call, summary and plot are as in cl2edgesCCvert.reg,
except in summary distances between the vertices and
the furthest points in the CC-vertex regions are provided.
We comment the plot(Ext) out below for brevity in the exposition,
as the plot is a special case of the one in kfr2vertsCCvert.reg with \(k=1\) below.
Ext<-fr2vertsCCvert.reg(Xp,Tr)
Ext
#> Call:
#> fr2vertsCCvert.reg(Xp = Xp, tri = Tr)
#>
#> Type:
#> [1] "Furthest Points in CC-Vertex Regions of the Triangle with Vertices A=(1,1), B=(2,0), and C=(1.5,2) from its Vertices"
summary(Ext)
#> Call:
#> fr2vertsCCvert.reg(Xp = Xp, tri = Tr)
#>
#> Type of the Extrema
#> [1] "Furthest Points in CC-Vertex Regions of the Triangle with Vertices A=(1,1), B=(2,0), and C=(1.5,2) from its Vertices"
#>
#> Extremum Points: Furthest Points in CC-Vertex Regions of the Triangle from its Vertices
#> (Row i corresponds to vertex i for i=1,2,3)
#> [,1] [,2]
#> [1,] 1.723711 0.8225489
#> [2,] 1.794240 0.2158873
#> [3,] 1.380035 1.5548904
#> [1] "Vertex labels are A=1, B=2, and C=3 (correspond to row number in Extremum Points)"
#>
#> Distances between the vertices and the furthest points in the vertex regions
#> (i-th entry corresponds to vertex i for i=1,2,3)
#> [1] 0.7451486 0.2982357 0.4609925
#>
#> Vertices of the Support Triangle
#> [,1] [,2]
#> A 1.0 1
#> B 2.0 0
#> C 1.5 2
plot(Ext) There is also the function fr2vertsCCvert.reg.basic.tri which takes the same arguments as cl2CCvert.reg.basic.tri and
returns the furthest points in a data set in the vertex regions from the vertices in the standard basic triangle, \(T_b\).
Here extrema points in each CC-vertex region
are the \(k\) most furthest points in the given data set, \(\mathcal{X}_n\) to the corresponding vertex.
That is, for the triangle \(T(A,B,C)\), if the \(CC\)-vertex region is \(V(A)\),
the \(k\) most furthest points in \(\mathcal X_n \cap V(A)\) from the vertex \(A\) are found
(if there are \(k'\) with \(k'< k\) data points in \(V(A)\),
the function returns \(k-k'\) NA’s for at the end of the extrema vector for the vertex region \(V(A)\)).
The description and summary of extrema points and their plot can be obtained using the function kfr2vertsCCvert.reg.
which is an object of class Extrema and takes arguments Xp,tri,k,ch.all.intri
where Xp,tri,ch.all.intri are as in fr2vertsCCvert.reg and k represents the number of furthest points from each vertex.
Its call, summary and plot are also as in fr2vertsCCvert.reg,
except in summary distances between the vertices and
the \(k\) most furthest points to the vertices in the CC-vertex regions are provided.
k=3
Ext<-kfr2vertsCCvert.reg(Xp,Tr,k)
Ext
#> Call:
#> kfr2vertsCCvert.reg(Xp = Xp, tri = Tr, k = k)
#>
#> Type:
#> [1] "3 Furthest Points in CC-Vertex Regions of the Triangle with Vertices A=(1,1), B=(2,0), and C=(1.5,2) from its Vertices"
summary(Ext)
#> Call:
#> kfr2vertsCCvert.reg(Xp = Xp, tri = Tr, k = k)
#>
#> Type of the Extrema
#> [1] "3 Furthest Points in CC-Vertex Regions of the Triangle with Vertices A=(1,1), B=(2,0), and C=(1.5,2) from its Vertices"
#>
#> Extremum Points: 3 Furthest Points in CC-Vertex Regions of the Triangle from its Vertices
#> (Row i corresponds to vertex i for i=1,2,3)
#> [,1] [,2]
#> 1. furthest from vertex 1 1.723711 0.8225489
#> 2. furthest from vertex 1 1.687023 0.7682074
#> 3. furthest from vertex 1 1.482080 1.1991317
#> 1. furthest from vertex 2 1.794240 0.2158873
#> 2. furthest from vertex 2 NA NA
#> 3. furthest from vertex 2 NA NA
#> 1. furthest from vertex 3 1.380035 1.5548904
#> 2. furthest from vertex 3 1.529720 1.5787125
#> 3. furthest from vertex 3 1.477620 1.7224190
#> [1] "Vertex labels are A=1, B=2, and C=3 (where vertex i corresponds to row numbers 3(i-1) to 3i in Extremum Points)"
#>
#> Distances between the vertices and the 3 furthest points in the vertex regions
#> (i-th entry corresponds to vertex i for i=1,2,3)
#> [,1] [,2] [,3]
#> [1,] 0.3686516 0.7250712 0.3511223
#> [2,] 0.2982357 NA NA
#> [3,] 0.4609925 0.4223345 0.2784818
#>
#> Vertices of the Support Triangle
#> [,1] [,2]
#> A 1.0 1
#> B 2.0 0
#> C 1.5 2
plot(Ext)There is also the function kfr2vertsCCvert.reg.basic.tri which takes arguments Xp,c1,c2,k,ch.all.intri and
finds the \(k\) most furthest points in a data set Xp in the vertex regions from
the corresponding vertices in the standard basic triangle, \(T_b\).
Here extrema points are the points in the given data set, \(\mathcal{X}_n\) closest to the edges of the the standard equilateral triangle \(T_e=T(A,B,C)\) with vertices \(A=(0,0)\), \(B=(1,0)\), and \(C=(1/2,\sqrt{3}/2)\). That is, for the triangle \(T_e\) the closest points in \(\mathcal X_n \cap T_e\) to the edges of \(T_e\) are found.
First we generate \(n=20\) uniform points in the standard equilateral triangle \(T_e\).
The description and summary of extrema points and their plot can be obtained using the function cl2edges.std.tri
which is an object of class Extrema with arguments Xp,ch.all.intri which are as in cl2CCvert.reg.
Its call, summary and plot are as in cl2edgesCCvert.reg,
except in summary distances between the edges and the closest points to the edges in the standard equilateral triangle.
We used the option asp=1 in plotting so that \(T_e\) is actually depicted as an equilateral triangle.
Ext<-cl2edges.std.tri(Xp)
Ext
#> Call:
#> cl2edges.std.tri(Xp = Xp)
#>
#> Type:
#> [1] "Closest Points in the Standard Equilateral Triangle Te=T(A,B,C) with Vertices A=(0,0), B=(1,0), and C=(1/2,sqrt(3)/2) to its Edges"
summary(Ext)
#> Call:
#> cl2edges.std.tri(Xp = Xp)
#>
#> Type of the Extrema
#> [1] "Closest Points in the Standard Equilateral Triangle Te=T(A,B,C) with Vertices A=(0,0), B=(1,0), and C=(1/2,sqrt(3)/2) to its Edges"
#>
#> Extremum Points: Closest Points in the Standard Equilateral Triangle to its Edges
#> (Row i corresponds to edge i for i=1,2,3)
#> [,1] [,2]
#> [1,] 0.4763512 0.7726664
#> [2,] 0.4763512 0.7726664
#> [3,] 0.7111212 0.1053883
#> [1] "Edge labels are AB=3, BC=1, and AC=2 (correspond to row number in Extremum Points)"
#>
#> Distances between Edges and the Closest Points in the Standard Equilateral Triangle
#> (Row i corresponds to edge i for i=1,2,3)
#> [1] 0.06715991 0.02619907 0.10538830
#>
#> Vertices of the Support Standard Equilateral Triangle
#> [,1] [,2]
#> A 0.0 0.0000000
#> B 1.0 0.0000000
#> C 0.5 0.8660254
plot(Ext,asp=1)Here an extrema point in each CM-edge region,
is the point in the given data set, \(\mathcal{X}_n\) furthest to the corresponding edge in the standard equilateral triangle.
That is, for the triangle \(T_e\), if the \(CM\)-edge region is \(E(AB)\),
the furthest point in \(\mathcal X_n \cap E(AB)\) from the edge \(AB\) is found
(if there are no data points in \(E(AB)\),
the function returns NA for the edge region \(E(AB)\)).
The description and summary of extrema points and their plot can be obtained using the function fr2edgesCMedge.reg.std.tri
which is an object of class Extrema which takes the same arguments as cl2edges.std.tri.
Its call, summary and plot are as in cl2edgesCCvert.reg,
except in summary distances between the edges and the furthest points in the CM-edge regions to the edges are provided.
We used the option asp=1 in plotting so that \(T_e\) is actually depicted as an equilateral triangle.
Ext<-fr2edgesCMedge.reg.std.tri(Xp)
Ext
#> Call:
#> fr2edgesCMedge.reg.std.tri(Xp = Xp)
#>
#> Type:
#> [1] "Furthest Points in the CM-Edge Regions of the Standard Equilateral Triangle T=(A,B,C) with A=(0,0), B=(1,0), and C=(1/2,sqrt(3)/2) from its Edges"
summary(Ext)
#> Call:
#> fr2edgesCMedge.reg.std.tri(Xp = Xp)
#>
#> Type of the Extrema
#> [1] "Furthest Points in the CM-Edge Regions of the Standard Equilateral Triangle T=(A,B,C) with A=(0,0), B=(1,0), and C=(1/2,sqrt(3)/2) from its Edges"
#>
#> Extremum Points: Furthest Points in the CM-Edge Regions of the Standard Equilateral Triangle from its Edges
#> (Row i corresponds to edge i for i=1,2,3)
#> [,1] [,2]
#> [1,] 0.6508705 0.2234491
#> [2,] 0.4590657 0.2878622
#> [3,] 0.7111212 0.1053883
#> [1] "Edge Labels are AB=3, BC=1, and AC=2 (correspond to row number in Extremum Points)"
#>
#> Distances between the edges and the furthest points in the edge regions
#> (i-th entry corresponds to edge i for i=1,2,3)
#> [1] 0.1906305 0.2536315 0.1053883
#>
#> Vertices of the Support Standard Equilateral Triangle
#> [,1] [,2]
#> A 0.0 0.0000000
#> B 1.0 0.0000000
#> C 0.5 0.8660254
plot(Ext,asp=1)We use the same interval \((a,b)=(0,10)\) as in the previous sections.
And we generate \(n_x=\) 10 uniform points in it by using the basic R function runif.
Here an extrema point in each \(M_c\)-vertex region, where \(M_c=a+c(b-a)\) for the interval \((a,b)\),
is the point in the given data set, \(\mathcal{X}_n\) closest to center \(M_c\).
For the interval \((a,b)\), if the \(M_c\)-vertex region is \(V(a)\),
the closest point in \(\mathcal X_n \cap V(a)\) to the center \(M_c\) is found.
That is, for the interval \((a,b)\),
the closest data point in \((M_c,b)\) to \(M_c\) and
closest data point in \((a,Mc)\) to \(M_c\) are found.
If there are no data points in \(V(A)\), the function returns NA for the vertex region \(V(a)\).
These extrema are used in finding the minimum dominating set
and hence the domination number of the PE-PCDs in the 1D setting,
since a subset (possibly all) of these extrema constitutes a minimum dominating set
(see Priebe, DeVinney, and Marchette (2001), Ceyhan (2020), and Ceyhan (2008) for more details).
The description and summary of extrema points and their plot can be obtained using the function cl2Mc.int
which is an object of class Extrema and takes arguments Xp,int,c where
Xp, a set or vector of 1D points from which closest points to \(M_c\) are found
in the interval int.int, a vector of two real numbers representing an interval.c, a positive real number in \((0,1)\) parameterizing the center inside int\(=(a,b)\).
For the interval, int\(=(a,b)\), the parameterized center is \(M_c=a+c(b-a)\).
.
Its call, summary and plot are as in cl2edgesCCvert.reg,
except in summary distances between the center \(M_c\) and
the closest points to \(M_c\) in \(M_c\)-vertex regions are provided.Ext<-cl2Mc.int(Xp,int,c)
Ext
#> Call:
#> cl2Mc.int(Xp = Xp, int = int, c = c)
#>
#> Type:
#> [1] "Closest Points in Mc-Vertex Regions of the Interval (a,b) = (0,10) to its Center Mc = 4"
summary(Ext)
#> Call:
#> cl2Mc.int(Xp = Xp, int = int, c = c)
#>
#> Type of the Extrema
#> [1] "Closest Points in Mc-Vertex Regions of the Interval (a,b) = (0,10) to its Center Mc = 4"
#>
#> Extremum Points: Closest Points in Mc-Vertex Regions of the Interval to its Center
#> (i-th entry corresponds to vertex i for i=1,2)
#> [1] 2.454885 4.100841
#> [1] "Vertex Labels are a=1 and b=2 for the interval (a,b)"
#>
#> Distances between the Center Mc and the Closest Points to Mc in Mc-Vertex Regions
#> (i-th entry corresponds to vertex i for i=1,2)
#> [1] 1.5451149 0.1008408
#>
#> Vertices of the Support Interval
#> a b
#> 0 10
plot(Ext)We illustrate the extrema for a tetrahedron which is obtained by slightly jittering the vertices of
the standard regular tetrahedron (for better visualization) and
generate \(n=20\) uniform \(\mathcal{X}\) points in it using the function runif.tetra.
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
set.seed(1)
tetra<-rbind(A,B,C,D)+matrix(runif(12,-.25,.25),ncol=3)
n<-10 #try also n<-20
Cent<-"CC" #try also "CM"
n<-10 #try also n<-20
Xp<-runif.tetra(n,tetra)$g #try also Xp<-cbind(runif(n),runif(n),runif(n))Here an extrema point in each \(M\)-vertex region, where \(M\) is a center of the tetrahedron,
is the point in the given data set, \(\mathcal{X}_n\) closest to the opposite face.
That is, for the tetrahedron \(T(A,B,C,D)\), if the \(M\)-vertex region is \(V(A)\),
the opposite face is the triangular face \(T(B,C,D)\), and closest point
in \(\mathcal X_n \cap V(A)\) to face \(T(B,C,D)\) is found
(if there are no data points in \(V(A)\), the function returns NA for the vertex region \(V(A)\)).
With M=CC (i.e., when the center is the circumcenter), the extrema point here is the closest \(\mathcal{X}\) point in CC-vertex
region to the opposite face, and is found by the function cl2faces.vert.reg.tetra
which is an object of class Extrema and takes arguments Xp,th,M where
M1, the center to be used in the construction of the vertex regions in the tetrahedron,th. Currently it only takes“CC”for circumcenter and“CM”for center of mass; default=“CM”`.This function is the 3D version of the function cl2edgesCCvert.reg
Its call, summary and plot are as in cl2edgesCCvert.reg,
except in summary distances between the faces and
the closest points to the faces in CC-vertex regions are provided.
Ext<-cl2faces.vert.reg.tetra(Xp,tetra,Cent)
Ext
#> Call:
#> cl2faces.vert.reg.tetra(Xp = Xp, th = tetra, M = Cent)
#>
#> Type:
#> [1] "Closest Points in CC-Vertex Regions of the Tetrahedron with Vertices A=(-0.12,-0.15,0.06), B=(0.94,0.2,-0.22), C=(0.54,1.09,-0.15), and D=(0.7,0.37,0.65) to the Opposite Faces"
summary(Ext)
#> Call:
#> cl2faces.vert.reg.tetra(Xp = Xp, th = tetra, M = Cent)
#>
#> Type of the Extrema
#> [1] "Closest Points in CC-Vertex Regions of the Tetrahedron with Vertices A=(-0.12,-0.15,0.06), B=(0.94,0.2,-0.22), C=(0.54,1.09,-0.15), and D=(0.7,0.37,0.65) to the Opposite Faces"
#>
#> Extremum Points: Closest Points in CC-Vertex Regions of the Tetrahedron to its Faces
#> (Row i corresponds to face i for i=1,2,3,4)
#> [,1] [,2] [,3]
#> [1,] 0.1073281 0.0109421 0.19871179
#> [2,] 0.6535570 0.2922984 0.15795015
#> [3,] 0.5199352 0.6610763 0.08954581
#> [4,] 0.5127296 0.5449680 0.24057920
#> [1] "Vertex labels are A=1, B=2, C=3, and D=4 (correspond to row number in Extremum Points)"
#>
#> Distances between Faces and the Closest Points to the Faces in CC-Vertex Regions
#> (i-th entry corresponds to vertex region i for i=1,2,3,4)
#> [1] 0.7554212 0.2773495 0.4803672 0.3509001
#>
#> Vertices of the Support Tetrahedron
#> [,1] [,2] [,3]
#> A -0.1172457 -0.1491590 0.06455702
#> B 0.9360619 0.1991948 -0.21910686
#> C 0.5364267 1.0883630 -0.14701271
#> D 0.7041039 0.3690740 0.65477496
plot(Ext)References