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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?
thanks in advance.
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?
thanks in advance.
