NAStri {pcds}

The vertices of the Arc Slice (AS) Proximity Region in a general triangle

Description

Returns the end points of the line segments and arc-slices that constitute the boundary of AS proximity region for a point in the triangle tri=T(A,B,C)=(rv=1,rv=2,rv=3).

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 circumcenter of tri; default is M="CC", i.e., circumcenter of tri. rv is the index of the vertex region p1 resides, with default=NULL.

If p is outside of tri, it returns NULL for the proximity region. dec is the number of decimals (default is 4) to round the barycentric coordinates when checking the points fall on the boundary of the triangle tri or not (so as not to miss the intersection points due to precision in the decimals).

See also (Ceyhan (2005, 2010)).

Usage

NAStri(p, tri, M = "CC", rv = NULL, dec = 4)

Arguments

p

A 2D point whose AS proximity region is to be computed.

tri

Three 2D points, stacked row-wise, each row representing a vertex of the triangle.

M

The center of the triangle. "CC" stands for circumcenter of the triangle tri or 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="CC" i.e., the circumcenter of tri.

rv

Index of the M-vertex region containing the point p, either 1,2,3 or NULL (default is NULL).

dec

a positive integer the number of decimals (default is 4) to round the barycentric coordinates when checking whether the end points fall on the boundary of the triangle tri or not.

Value

A list with the elements

L, R

End points of the line segments on the boundary of the AS proximity region. Each row in L and R constitute a pair of points that determine a line segment on the boundary.

arc.slices

The end points of the arc-slices on the circular parts of the AS proximity region. Here points in rows 1 and 2 constitute the end points of the first arc-slice, points on rows 3 and 4 constitute the end points for the next arc-slice and so on.

Angles

The angles (in radians) between the vectors joining arc slice end points to the point p with the horizontal line crossing the 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

NASbasic.tri, NPEtri, NCStri and IarcAStri

Examples

A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

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

P1<-as.numeric(runif.tri(1,Tr)$g)  #try also P1<-c(1.3,1.2)
NAStri(P1,Tr,M)

#or try
Rv<-rel.vert.triCC(P1,Tr)$rv
NAStri(P1,Tr,M,Rv)

NAStri(c(3,5),Tr,M)

P2<-c(1.5,1.4)
NAStri(P2,Tr,M)

P3<-c(1.5,.4)
NAStri(P3,Tr,M)

if (dimension(M)==3) {M<-bary2cart(M,Tr)}
#need to run this when M is given in barycentric coordinates

CC<-circumcenter.tri(Tr)  #the circumcenter

if (isTRUE(all.equal(M,CC)) || identical(M,"CC"))
{cent<-CC
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)
cent.name<-"CC"
rv<-rel.vert.triCC(P1,Tr)$rv
} else
{cent<-M
cent.name<-"M"
Ds<-prj.cent2edges(Tr,M)
rv<-rel.vert.tri(P1,Tr,M)$rv
}
RV<-Tr[rv,]
rad<-Dist(P1,RV)

Int.Pts<-NAStri(P1,Tr,M)

#plot of the NAS region
Xlim<-range(Tr[,1],P1[1]+rad,P1[1]-rad)
Ylim<-range(Tr[,2],P1[2]+rad,P1[2]-rad)
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",asp=1,xlab="",ylab="",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
#asp=1 must be the case to have the arc properly placed in the figure
polygon(Tr)
points(rbind(Tr,P1,rbind(Int.Pts$L,Int.Pts$R)))
L<-rbind(cent,cent,cent); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
interp::circles(P1[1],P1[2],rad,lty=2)
L<-Int.Pts$L; R<-Int.Pts$R
segments(L[,1], L[,2], R[,1], R[,2], lty=1,col=2)
Arcs<-Int.Pts$a;
if (!is.null(Arcs))
{
  K<-nrow(Arcs)/2
  for (i in 1:K)
  {A1<-Int.Pts$arc[2*i-1,]; A2<-Int.Pts$arc[2*i,];
  angles<-angle.str2end(A1,P1,A2)$c

  test.ang1<-angles[1]+(.01)*(angles[2]-angles[1])
  test.Pnt<-P1+rad*c(cos(test.ang1),sin(test.ang1))
  if (!in.triangle(test.Pnt,Tr,boundary = TRUE)$i) {angles<-c(min(angles),max(angles)-2*pi)}
  plotrix::draw.arc(P1[1],P1[2],rad,angle1=angles[1],angle2=angles[2],col=2)
  }
}

#proximity region with the triangle (i.e., for labeling the vertices of the NAS)
IP.txt<-intpts<-c()
if (!is.null(Int.Pts$a))
{
 intpts<-unique(round(Int.Pts$a,7))
  #this part is for labeling the intersection points of the spherical
  for (i in 1:(length(intpts)/2))
    IP.txt<-c(IP.txt,paste("I",i+1, sep = ""))
}
txt<-rbind(Tr,P1,cent,intpts)
txt.str<-c("A","B","C","P1",cent.name,IP.txt)
text(txt+cbind(rep(xd*.02,nrow(txt)),rep(-xd*.03,nrow(txt))),txt.str)

P1<-c(.3,.2)
NAStri(P1,Tr,M)