NPEbas.tri: The vertices of the Proportional Edge (PE) Proximity Region in a standard basic triangle

Description

Returns the vertices of the PE proximity region (which is itself a triangle) for a point in the standard basic triangle $T_{b} = T ((0, 0), (1, 0), (c_{1}, c_{2})) =$ (rv=1,rv=2,rv=3).

PE proximity region is defined with respect to the standard basic triangle $T_{b}$ with expansion parameter $r \geq 1$ and vertex regions based on center $M = (m_{1}, m_{2})$ in Cartesian coordinates or $M = (α, β, γ)$ in barycentric coordinates in the interior of the basic triangle $T_{b}$ or based on the circumcenter of $T_{b}$ ; default is $M = (1, 1, 1)$ i.e., the center of mass of $T_{b}$ .

Vertex regions are labeled as $1, 2, 3$ rowwise for the vertices of the triangle $T_{b}$ . rv is the index of the vertex region p resides, with default=NULL. If p is outside of tri, it returns NULL for the proximity region.

See also (ceyhan:Phd-thesis,ceyhan:comp-geo-2010,ceyhan:mcap2012;textualpcds).

Usage

NPEbas.tri(p, r, c1, c2, M = c(1, 1, 1), rv = NULL)

Value

Vertices of the triangular region which constitutes the PE proximity region with expansion parameter r and center M for a point p

Arguments

p: A 2D point whose PE proximity region is to be computed.
r: A positive real number which serves as the expansion parameter in PE proximity region; must be $\geq 1$ .
c1, c2: Positive real numbers representing the top vertex in standard basic triangle $T_{b} = T ((0, 0), (1, 0), (c_{1}, c_{2}))$ , $c_{1}$ must be in $[0, 1 / 2]$ , $c_{2} > 0$ and $(1 - c_{1})^{2} + c_{2}^{2} \leq 1$ .
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 basic triangle $T_{b}$ or the circumcenter of $T_{b}$ which may be entered as "CC" as well; default is $M = (1, 1, 1)$ i.e., the center of mass of $T_{b}$ .
rv: Index of the M-vertex region containing the point p, either 1,2,3 or NULL (default is NULL).

Author

Elvan Ceyhan

References

Examples

Run this code

if (FALSE) {
c1<-.4; c2<-.6
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C);

M<-as.numeric(runif.bas.tri(1,c1,c2)$g)  #try also M<-c(.6,.2)

r<-2

P1<-as.numeric(runif.bas.tri(1,c1,c2)$g)  #try also P1<-c(.4,.2)
NPEbas.tri(P1,r,c1,c2,M)

#or try
Rv<-rv.bas.tri.cent(P1,c1,c2,M)$rv
NPEbas.tri(P1,r,c1,c2,M,Rv)

P1<-c(1.4,1.2)
P2<-c(1.5,1.26)
NPEbas.tri(P1,r,c1,c2,M) #gives an error if M=c(1.3,1.3)
#since center is not the circumcenter or not in the interior of the triangle
}

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