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I'm trying to derive the distance between two distinct points in hyperbolic space and I'm working on the upper half plane. So, with the parametrization

$\sigma(t): x=r\cos(t), y=r\sin(t),\; \alpha\leq t\leq \beta$

the distance is just

$$\mathrm{dist}((x_{1},y_{1}),(x_{2},y_{2}))=\int_{\alpha}^{\beta}\rho(\sigma(t))|\sigma'(t)|dt=\int_{\alpha}^{\beta}\frac{dt}{\sin(t)}=\ln\left|\frac{\tan\frac{\beta}{2}}{\tan\frac{\alpha}{2}}\right|$$

We also know that from http://en.wikipedia.org/wiki/Poincar%C3%A9_half-plane_model that

$$\mathrm{dist}((x_{1},y_{1}),(x_{2},y_{2}))=\mathrm{arccosh}\left(1+\frac{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}{2y_{1}y_{2}}\right)$$

And we know from http://en.wikipedia.org/wiki/Poincar%C3%A9_metric that

$$\mathrm{dist}((z_{1},z_{2}))=2\mathrm{arctanh}\frac{|z_{1}-z_{2}|}{|z_{1}-\overline{z_{2}}|}=\log\frac{|z_{1}-\overline{z_{2}}|+|z_{1}-z_{2}|}{|z_{1}-\overline{z_{2}}|-|z_{1}-z_{2}|}$$

Also from Expression of the Hyperbolic Distance in the Upper Half Plane that the last three formulation are equal. Question is I cannot find the equal connection between my calculation and the other three Can anyone help?

Elif.

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  • $\begingroup$ I've improved your question's formatting; apologies if I changed your meaning. You can see here how I edited your question. Please see here for a guide to writing math with MathJax, and see here for a guide to formatting posts with Markdown. $\endgroup$ – Zev Chonoles Feb 18 '14 at 7:58
  • $\begingroup$ Thank you very much! I'm new at writing in Latex so your improvement will help. $\endgroup$ – user129164 Feb 18 '14 at 8:06
  • $\begingroup$ See math.stackexchange.com/questions/160338/…. $\endgroup$ – Martín-Blas Pérez Pinilla Feb 18 '14 at 9:02
  • $\begingroup$ You can already see this link in my question. But this is not the answer! $\endgroup$ – user129164 Feb 18 '14 at 9:32
  • $\begingroup$ In the first formula for distance, how are $\alpha$ and $\beta$ related to $x_1,y_1,x_2,y_2$? This would be the first thing to look in. $\endgroup$ – user127096 Feb 20 '14 at 1:40

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