IarcPEset2pnt.tri: The indicator for the presence of an arc from a point in set `S` to the point `p` for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one triangle case

Description

Returns \(I(\)p in \(N_{PE}(x,r)\) for some \(x\) in S\()\), that is, returns 1 if p is in \(\cup_{x in S}N_{PE}(x,r)\), and returns 0 otherwise.

PE proximity region is constructed with respect to the triangle tri with the expansion parameter \(r \ge 1\) and vertex regions are based on the 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; default is \(M=(1,1,1)\), i.e., the center of mass of tri. Vertices of tri are also labeled as 1, 2, and 3, respectively.

If p is not in S and either p or all points in S are outside tri, it returns 0, but if p is in S, then it always returns 1 regardless of its location (i.e., loops are allowed).

Usage

IarcPEset2pnt.tri(S, p, tri, r, M = c(1, 1, 1))

Value

\(I(\)p is in \(U_{x in S} N_{PE}(x,r))\), that is, returns 1 if p is in S

or inside \(N_{PE}(x,r)\) for at least one \(x\) in S, and returns 0 otherwise, where PE proximity region is constructed with respect to the triangle tri

Arguments

S: A set of 2D points. Presence of an arc from a point in S to point p is checked by the function.
p: A 2D point. Presence of an arc from a point in S to point p is checked by the function.
tri: A \(3 \times 2\) matrix with each row representing a vertex of the triangle.
r: A positive real number which serves as the expansion parameter in PE proximity region constructed in the triangle tri; must be \(\ge 1\).
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; default is \(M=(1,1,1)\), i.e., the center of mass of tri.

Author

Elvan Ceyhan

Examples

Run this code

# \donttest{
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

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

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

r<-1.5

S<-rbind(Xp[1,],Xp[2,])  #try also S<-c(1.5,1)

IarcPEset2pnt.tri(S,Xp[3,],Tr,r,M)
IarcPEset2pnt.tri(S,Xp[3,],r=1,Tr,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,])
IarcPEset2pnt.tri(S,Xp[3,],Tr,r,M)

S<-rbind(c(.1,.1),c(.3,.4),c(.5,.3))
IarcPEset2pnt.tri(S,Xp[3,],Tr,r,M)

P<-c(.4,.2)
S<-Xp[c(1,3,4),]
IarcPEset2pnt.tri(Xp,P,Tr,r,M)
# }