pcds (version 0.1.4)

NAStri: 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="CC" for circumcenter of tri; or \(M=(m_1,m_2)\) in Cartesian coordinates or \(M=(\alpha,\beta,\gamma)\) in barycentric coordinates in the interior of the triangle tri; default is M="CC" the circumcenter of tri. rv is the index of the vertex region pt1 resides, with default=NULL.

If pt 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:Phd-thesis,ceyhan:comp-geo-2010,ceyhan:mcap2012;textualpcds).

Usage

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

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 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 row 1 and row 2 constitute the end points of one arc-slice, points on row 3 and row 4 constitute the end points for the next arc-slice and so on.

Arguments

pt

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 pt, 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.

Author

Elvan Ceyhan

References

See Also

NASbastri, NPEtri, NCStri and IndNAStri

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<-rv.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<-circ.cent.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<-rv.triCC(P1,Tr)$rv
} else
{cent<-M
cent.name<-"M"
Ds<-cp2e.tri(Tr,M)
D1<-Ds[1,]; D2<-Ds[2,]; D3<-Ds[3,]
rv<-rv.tri.cent(P1,Tr,M)$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 = T)$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)