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Find the hyperbolic distances between the following pairs of points

( 4/3, 0,5/3) , (3/4, 0,5/4) ∈ H^+

where H^+ = {x,y,z ∈ R^3 | z^2-y^2-x^2=1 and z>0}

Own work:

The formula I have been given is

d(w,z) = ln({1 + |(w-z)/(1-($\bar w$z)|}/{1-|(w-z)/(1-($\bar w$z)|})

where d(w,z) is the distance between to complex numbers w and z.

Im struggling on how to use this formula when I have been given coordinates rather than two complex numbers.

Any help appreciated.

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user407151
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1 Answers1

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You may map $(x,y,z)$ in your hyperboloid representation to the complex unit disk (Poincaré disk) by $$ \left(x,y,z=\sqrt{1+x^2+y^2}\right) \mapsto \frac{ x + i y}{1+z}$$ You should then calculate (through the formula you state) the distance in the Poincaré disk. You may google on "Poincaré disk model" to get further information.

H. H. Rugh
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  • Thank you this really helped – user407151 Feb 25 '17 at 13:31
  • Given that you use the hyperboloid coordinates, but the Poincaré metric, I wouldn't know of any other way. But your two points correspond to simple nice (real-valued) points in the unit disk: $1/2$ and $1/3$ so very easy to calculate their distance. – H. H. Rugh Feb 25 '17 at 13:32
  • are the hyoerboloid coordinates the ones provided in the question? – user407151 Feb 25 '17 at 13:35
  • Yes, you may have a look at: https://en.wikipedia.org/wiki/Poincar%C3%A9_disk_model (around two thirds down the page: Relation to the hyperboloid model) – H. H. Rugh Feb 25 '17 at 13:36