cl2edgesMvert.reg: The closest points among a data set in the vertex regions to the respective edges in a triangle

Description

An object of class "Extrema". Returns the closest data points among the data set, Xp, to edge \(i\) in M-vertex region \(i\) for \(i=1,2,3\) in the triangle tri\(=T(A,B,C)\). Vertex labels are \(A=1\), \(B=2\), and \(C=3\), and corresponding edge labels are \(BC=1\), \(AC=2\), and \(AB=3\).

Vertex regions are based on center \(M=(m_1,m_2)\) in Cartesian coordinates or \(M=(\alpha,\beta,\gamma)\) in barycentric coordinates in the interior of the triangle tri or based on the circumcenter of tri.

Two methods of finding these extrema are provided in the function, which can be chosen in the logical argument alt, whose default is alt=FALSE. When alt=FALSE, the function sequentially finds the vertex region of the data point and then updates 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. Both options yield equivalent results for the extrema points and indices, with the default being slightly ~ 20

See also (ceyhan:Phd-thesis,ceyhan:comp-geo-2010,ceyhan:dom-num-NPE-Spat2011;textualpcds).

Usage

cl2edgesMvert.reg(Xp, tri, M, alt = FALSE)

Value

A list with the elements

txt1: Vertex labels are \(A=1\), \(B=2\), and \(C=3\) (correspond to row number in Extremum Points).
txt2: A short description of the distances as "Distances to Edges in the Respective \eqn{M}-Vertex Regions".
type: Type of the extrema points
desc: A short description of the extrema points
mtitle: The "main" title for the plot of the extrema
ext: The extrema points, here, closest points to edges in the respective vertex region.
ind.ext: The data indices of extrema points, ext.
X: The input data, Xp, can be a matrix or data frame
num.points: The number of data points, i.e., size of Xp
supp: Support of the data points, here, it is tri
cent: The center point used for construction of vertex regions
ncent: Name of the center, cent, it is "M" or "CC" for this function
regions: Vertex regions inside the triangle, tri, provided as a list
region.names: Names of the vertex regions as "vr=1", "vr=2", and "vr=3"
region.centers: Centers of mass of the vertex regions inside tri
dist2ref: Distances of closest points in the M-vertex regions to corresponding edges.

Arguments

Xp: A set of 2D points representing the set of data points.
tri: A \(3 \times 2\) matrix with each row representing a vertex of the triangle.
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.

Author

Elvan Ceyhan

References

Examples

Run this code

# \donttest{
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);

Tr<-rbind(A,B,C);
n<-20  #try also n<-100

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

Ext<-cl2edgesMvert.reg(Xp,Tr,M)
Ext
summary(Ext)
plot(Ext)

cl2e<-Ext

Ds<-prj.cent2edges(Tr,M)

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

if (dimension(M)==3) {M<-bary2cart(M,Tr)}
#need to run this when M is given in barycentric coordinates

plot(Tr,pch=".",xlab="",ylab="",
main="Closest Points in M-Vertex Regions \n to the Opposite Edges",
axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp,pch=1,col=1)
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(cl2e$ext,pch=3,col=2)

xc<-Tr[,1]+c(-.02,.03,.02)
yc<-Tr[,2]+c(.02,.02,.04)
txt.str<-c("A","B","C")
text(xc,yc,txt.str)

txt<-rbind(M,Ds)
xc<-txt[,1]+c(-.02,.05,-.02,-.01)
yc<-txt[,2]+c(-.03,.02,.08,-.07)
txt.str<-c("M","D1","D2","D3")
text(xc,yc,txt.str)
# }

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