circ.cent.bastri: Circumcenter of a basic triangle

Description

Returns the circumcenter of a basic triangle \(T_b=T((0,0),(1,0),(c_1,c_2))\) given \(c_1\), \(c_2\) where \(c_1\) is in \([0,1/2]\), \(c_2>0\) and \((1-c_1)^2+c_2^2 \le 1\).

Any given triangle can be mapped to the basic triangle by a combination of rigid body motions (i.e., translation, rotation and reflection) and scaling, preserving uniformity of the points in the original triangle. Hence basic triangle is useful for simulation studies under the uniformity hypothesis.

See (weisstein-tri-centers,ceyhan:comp-geo-2010;textualpcds) for triangle centers and (ceyhan:arc-density-PE,ceyhan:arc-density-CS,ceyhan:dom-num-NPE-Spat2011;textualpcds) for the basic triangle.

Usage

circ.cent.bastri(c1, c2)

Arguments

c1, c2

Positive real numbers representing the top vertex in 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 \((1-c_1)^2+c_2^2 \le 1\).

Value

circumcenter of the basic triangle \(T_b=T((0,0),(1,0),(c_1,c_2))\) given \(c_1\), \(c_2\) as the arguments of the function

References

Examples

Run this code

# NOT RUN {
c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);  #the vertices of the basic triangle Tb
Tb<-rbind(A,B,C)
CC<-circ.cent.bastri(c1,c2)  #the circumcenter
CC

D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2; #midpoints of the edges
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tb[,1])
Ylim<-range(Tb[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

oldpar <- par(no.readonly = TRUE)
par(pty="s")
plot(A,pch=".",asp=1,xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
points(rbind(CC))
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)

txt<-rbind(Tb,CC,D1,D2,D3)
xc<-txt[,1]+c(-.03,.04,.03,.06,.06,-.03,0)
yc<-txt[,2]+c(.02,.02,.03,-.03,.02,.04,-.03)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)
par(oldpar)

#for an obtuse triangle
c1<-.4; c2<-.3;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);  #the vertices of the basic triangle Tb
Tb<-rbind(A,B,C)
CC<-circ.cent.bastri(c1,c2)  #the circumcenter
CC

D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2; #midpoints of the edges
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tb[,1],CC[1])
Ylim<-range(Tb[,2],CC[2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

oldpar <- par(no.readonly = TRUE)
par(pty="s")
plot(A,pch=".",asp=1,xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
points(rbind(CC))
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)

txt<-rbind(Tb,CC,D1,D2,D3)
xc<-txt[,1]+c(-.03,.03,.03,.07,.07,-.05,0)
yc<-txt[,2]+c(.02,.02,.04,-.03,.03,.04,.06)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)
par(oldpar)

# }