# The distance between two distinct points in the upper half plane

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.

• 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. – Zev Chonoles Feb 18 '14 at 7:58
• Thank you very much! I'm new at writing in Latex so your improvement will help. – user129164 Feb 18 '14 at 8:06
• – Martín-Blas Pérez Pinilla Feb 18 '14 at 9:02
• You can already see this link in my question. But this is not the answer! – user129164 Feb 18 '14 at 9:32
• 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. – user127096 Feb 20 '14 at 1:40