NPEbasic.tri {pcds}

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 \ge 1 and vertex regions based on center M=(m_1,m_2) in Cartesian coordinates or M=(\alpha,\beta,\gamma) 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 (2005, 2010)).

Usage

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

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 \ge 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 \le 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).

Value

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

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

NPEtri, NAStri, NCStri, and IarcPEbasic.tri

Examples

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.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.2)

r<-2

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

#or try
Rv<-rel.vert.basic.tri(P1,c1,c2,M)$rv
NPEbasic.tri(P1,r,c1,c2,M,Rv)

P1<-c(1.4,1.2)
P2<-c(1.5,1.26)
NPEbasic.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