Hyperbolic Distance and double Cross Ratio.

1. Jul 19, 2008

MathematicalPhysicist

The question is as follows (by the way I'm asking here, cause the calculus and beyond forum seems to be primarily concerned with Calculus,DE, LA and AA):
let f(z)=(2z+1)/(z+1) be an isometry of the hyperbolic plane H={z| Im(z)>0}.
let l be a hyperbolic line in H which is invariatn under f, calculate the hyperbolic distance:
p(z,f(z)) for some z in l.

Now I want to use here the definition of the hyperbolic distance given by the cross ratio.

So I found the fixed points of f, which are: w=(1+-(sqrt(5))/2, those points are in l (or so I think), from here we can use the cross ration definition, i.e:
p(z,f(z))=$$log(D(w_1,z,f(z),w_2)$$ where D is the double cross ration defined by:
D(z1,z2,z3,z4)=(z1-z3)(z2-z4)/((z1-z4)(z2-z3))
from here just plug and go, but is my appraoch correct?