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

Description

Returns \(I(\)p in \(N_{PE}(x,r)\) for some \(x\) in S\()\) for S, in the standard equilateral triangle, 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 standard equilateral triangle \(T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2))\) with the expansion parameter \(r \ge 1\) and 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 \(T_e\); default is \(M=(1,1,1)\), i.e., the center of mass of \(T_e\) (which is equivalent to the circumcenter for \(T_e\)).

Vertices of \(T_e\) 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 \(T_e\), 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.std.tri(S, p, r, M = c(1, 1, 1))

Value

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

for S in the standard equilateral triangle, 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. PE proximity region is constructed with respect to the standard equilateral triangle \(T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2))\)

with M-vertex regions

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.
r: A positive real number which serves as the expansion parameter in PE proximity region in the standard equilateral triangle \(T_e=T((0,0),(1,0),(1/2,\sqrt{3}/2))\); 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 standard equilateral triangle \(T_e\); default is \(M=(1,1,1)\) i.e., the center of mass of \(T_e\).

Author

Elvan Ceyhan

Examples

Run this code

# \donttest{
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-10

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

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

r<-1.5

S<-rbind(Xp[1,],Xp[2,])  #try also S<-c(.5,.5)
IarcPEset2pnt.std.tri(S,Xp[3,],r,M)
IarcPEset2pnt.std.tri(S,Xp[3,],r=1,M)

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

IarcPEset2pnt.std.tri(S,Xp[6,],r,M)
IarcPEset2pnt.std.tri(S,Xp[6,],r=1.25,M)

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