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

In pcds: Proximity Catch Digraphs and Their Applications

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 (\insertCiteweisstein-tri-centers,ceyhan:comp-geo-2010;textualpcds) for triangle centers and (\insertCiteceyhan: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)

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.

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.

Author(s)

Elvan Ceyhan

References

\insertAllCited

See Also

circumcenter.tri

Examples

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)

pcds documentation built on May 29, 2024, 9:36 a.m.