pcds (version 0.1.8)

circumcenter.basic.tri: Circumcenter of a standard basic triangle form

Description

Returns the circumcenter of a standard basic triangle form \(T_b=T((0,0),(1,0),(c_1,c_2))\) 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 standard basic triangle form 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, standard basic triangle form 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 standard basic triangle form.

Usage

circumcenter.basic.tri(c1, c2)

Value

circumcenter of the standard basic triangle form \(T_b=T((0,0),(1,0),(c_1,c_2))\)

given \(c_1\), \(c_2\) as the arguments of the function.

Arguments

c1, c2

Positive real numbers representing the top vertex in standard basic triangle form \(T_b=T((0,0),(1,0),(c_1,c_2))\), \(c_1\) must be in \([0,1/2]\), \(c_2>0\) and \((1-c_1)^2+c_2^2 \le 1\).

Author

Elvan Ceyhan

References

See Also

circumcenter.tri

Examples

# \donttest{
c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
#the vertices of the standard basic triangle form Tb
Tb<-rbind(A,B,C)
CC<-circumcenter.basic.tri(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(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)

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

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)
# }