IndNCStriSet: The indicator for the presence of an arc from a point in set `S` to the point `p` for Central Similarity Proximity Catch Digraphs (CS-PCDs) - one triangle case

Description

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

CS proximity region is constructed with respect to the triangle tri with the expansion parameter \(t>0\) and edge 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; default is \(M=(1,1,1)\) i.e., the center of mass of tri.

Edges of tri\(=T(A,B,C)\), \(AB\), \(BC\), \(AC\), are also labeled as edges 3, 1, and 2, 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

IndNCStriSet(S, p, tri, t, M = c(1, 1, 1))

Value

I(p is in \(\cup_{x in S} N_{CS}(x,t)\)), that is, returns 1 if p is in S or inside \(N_{CS}(x,t)\) for at least one \(x\) in S, returns 0 otherwise where CS 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.
t: A positive real number which serves as the expansion parameter in CS proximity region constructed in the triangle tri.
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; default is \(M=(1,1,1)\) i.e., the center of mass of tri.

Author

Elvan Ceyhan

Examples

Run this code

if (FALSE) {
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

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

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

tau<-.5

IndNCStriSet(S,Xp[3,],Tr,tau,M)
IndNCStriSet(S,Xp[3,],Tr,t=1,M)
IndNCStriSet(S,Xp[3,],Tr,t=1.5,M)

S<-rbind(c(.1,.1),c(.3,.4),c(.5,.3))
IndNCStriSet(S,Xp[3,],Tr,tau,M)
}