I'm trying to derive the distance between two distinct points in hyperbolic space and I'm working on the Poincare 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 https://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 https://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.