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Let b>a>0 real numbers, and z a point in H={z in CU{infinity}| Im(z)>0}
let I be in H, a euclidean interval which connects az with bz. find the point w in I such that it divides I into two intervals with the same hyperbolic length.
Now what I did is as follows:
let f be a mobius transformation such that: f(az)=|z|(ia) and f(bz)=|z|(ib)
it keeps the length of I, which is: (b-a)|z| (a euclidean length), now because f is an isometry in H, the f(w) divides the interval on {z=Im(z)>0} to two parts.
now the hyperbolic length should be calculated by using the hyperbolic riemanian metric, after some manipulations I get that: (f(w))^2=(ab)|z|^2 f(w)=sqrt(ab)|z|.
now we know that f(w)=(a'w+b')/(c'w+d') for a'd'-b'c'=1 all in CU{inifnity}.
now from f(az)=|z|(ia) and f(bz)=|z|(ib) we have two equations with four variables: (a',b',c',d') so we can choose two of them arbitrarily and from the last equations get the other two (we can also use the fact on a'd'-b'c'=1), now if I choose: d'=1,c'=0 then i get that a'=1 and b'=|z|(ia)-az=|z|(ib)-bz
from here I get that: w=sqrt(ab)|z|-|z|(ia)+az
not sure if this is correct cause, from the next to last equation i get that: z=i|z| which means that z=Im(z).
any hints?
let I be in H, a euclidean interval which connects az with bz. find the point w in I such that it divides I into two intervals with the same hyperbolic length.
Now what I did is as follows:
let f be a mobius transformation such that: f(az)=|z|(ia) and f(bz)=|z|(ib)
it keeps the length of I, which is: (b-a)|z| (a euclidean length), now because f is an isometry in H, the f(w) divides the interval on {z=Im(z)>0} to two parts.
now the hyperbolic length should be calculated by using the hyperbolic riemanian metric, after some manipulations I get that: (f(w))^2=(ab)|z|^2 f(w)=sqrt(ab)|z|.
now we know that f(w)=(a'w+b')/(c'w+d') for a'd'-b'c'=1 all in CU{inifnity}.
now from f(az)=|z|(ia) and f(bz)=|z|(ib) we have two equations with four variables: (a',b',c',d') so we can choose two of them arbitrarily and from the last equations get the other two (we can also use the fact on a'd'-b'c'=1), now if I choose: d'=1,c'=0 then i get that a'=1 and b'=|z|(ia)-az=|z|(ib)-bz
from here I get that: w=sqrt(ab)|z|-|z|(ia)+az
not sure if this is correct cause, from the next to last equation i get that: z=i|z| which means that z=Im(z).
any hints?