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.
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.