| rel.vert.basic.tri {pcds} | R Documentation |
The index of the vertex region in a standard basic triangle form that contains a given point
Description
Returns the index of the related vertex
in the standard basic triangle form
whose region contains point p.
The standard basic triangle form is T_b=T((0,0),(1,0),(c_1,c_2))
where c_1 is in [0,1/2], c_2>0 and (1-c_1)^2+c_2^2 \le 1..
Vertex regions are based on the general center M=(m_1,m_2)
in Cartesian coordinates or
M=(\alpha,\beta,\gamma)
in barycentric coordinates
in the interior of the standard basic triangle form T_b.
Vertices of the standard basic triangle form T_b are labeled
according to the row number the
vertex is recorded, i.e., as 1=(0,0), 2=(1,0),and 3=(c_1,c_2).
If the point, p, is not inside T_b,
then the function yields NA as output.
The corresponding vertex region is the polygon
with the vertex, M, and projections from M
to the edges on the lines joining vertices and M.
That is, rv=1 has vertices (0,0),D_3,M,D_2;
rv=2 has vertices (1,0),D_1,M,D_3; and
rv=3 has vertices (c_1,c_2),D_2,M,D_1
(see the illustration in the examples).
Any given triangle can be mapped to the standard basic triangle form by a combination of rigid body motions (i.e., translation, rotation and reflection) and scaling, preserving uniformity of the points in the original triangle. Hence, standard basic triangle form is useful for simulation studies under the uniformity hypothesis.
See also (Ceyhan (2005, 2010)).
Usage
rel.vert.basic.tri(p, c1, c2, M)
Arguments
p |
A 2D point for which |
c1, c2 |
Positive real numbers
which constitute the vertex of the standard basic triangle form
adjacent to the shorter edges;
|
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 basic triangle form. |
Value
A list with two elements
rv |
Index of the vertex whose region contains point, |
tri |
The vertices of the standard basic triangle form, |
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 (2012).
“An investigation of new graph invariants related to the domination number of random proximity catch digraphs.”
Methodology and Computing in Applied Probability, 14(2), 299-334.
See Also
rel.vert.basic.triCM, rel.vert.tri,
rel.vert.triCC, rel.vert.basic.triCC,
rel.vert.triCM, and rel.vert.std.triCM
Examples
c1<-.4; c2<-.6
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C);
M<-c(.6,.2)
P<-c(.4,.2)
rel.vert.basic.tri(P,c1,c2,M)
n<-20 #try also n<-40
set.seed(1)
Xp<-runif.basic.tri(n,c1,c2)$g
M<-as.numeric(runif.basic.tri(1,c1,c2)$g) #try also M<-c(.6,.2)
Rv<-vector()
for (i in 1:n)
{ Rv<-c(Rv,rel.vert.basic.tri(Xp[i,],c1,c2,M)$rv)}
Rv
Ds<-prj.cent2edges.basic.tri(c1,c2,M)
Xlim<-range(Tb[,1],Xp[,1])
Ylim<-range(Tb[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
if (dimension(M)==3) {M<-bary2cart(M,Tb)}
#need to run this when M is given in barycentric coordinates
plot(Tb,pch=".",xlab="",ylab="",axes=TRUE,
xlim=Xlim+xd*c(-.1,.1),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
points(Xp,pch=".",col=1)
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)
xc<-Tb[,1]+c(-.04,.05,.04)
yc<-Tb[,2]+c(.02,.02,.03)
txt.str<-c("rv=1","rv=2","rv=3")
text(xc,yc,txt.str)
txt<-rbind(M,Ds)
xc<-txt[,1]+c(-.02,.04,-.03,0)
yc<-txt[,2]+c(-.02,.02,.02,-.03)
txt.str<-c("M","D1","D2","D3")
text(xc,yc,txt.str)
text(Xp,labels=factor(Rv))