In the hyperbolic plane, we have this following result :
On the geodesic boundary $\left(v_{1}, v_{2}\right)$, for example, we have, for $v \in\left(v_{1}, v_{2}\right)$, $$ v=\frac{\sinh \left(c-d_{\mathbb{H}^{2}}\left(v_{2}, v\right)\right)}{\sinh (c)} v_{1}+\frac{\sinh \left(d_{\mathbb{H}^{2}}\left(v_{1}, v\right)\right)}{\sinh (c)} v_{2}, $$ where $c$ is the geodesic edge lengths between $v_{1}$ and $v_{2}$ .
I am trying to find a similar one, in the Poincaré disk, so let $v_1,v_2 \in \mathbb D$ and $v\in (v_1,v_2)$. Now, the Möbius transformation : $$w=R(z) = \frac{z-v_1}{1-\bar{v_1}z} \Leftrightarrow z=R^{-1}(w) = \frac{w+v_1}{1+\bar{v_1}w}$$ is an isometry that maps $v_1$ to $0$, $R(v_2)$ to $r_2=\frac{v_2-v_1}{1-\bar{v_1}v_2}$, and $R(v)=r_1=\frac{v-v_1}{1-\bar{v_1}v}$. Since $r_1=\alpha r_2$, where $\alpha=d_{D}(0,r_1)/ d_D(0,r_2)$, we have : \begin{align*} v&=R^{-1}(r_1)=\frac{\alpha r_2+v_1}{1+\alpha r_1\bar v_1} \\ &=\frac{1-\bar v_1 v_2-\alpha}{1-\bar v_1 v_2+\alpha \bar v_1(v_2-v_1)}v_1+\frac{\alpha}{1-\bar v_1v_2+ \alpha(\bar v_1 v_2- |v_1|^2)} v_2 \\&=\lambda v_1+\mu v_2 \end{align*} The problem is to write $\lambda$ and $\mu$ as a hyperbolic function ($\tanh, \cosh$, or $\sinh$) of something. I think I should use this following theorem to rearrange the formulas, but I still cannot.
Theorem: Let $z, w \in \mathbb{D}$, and $d_{\mathbb{D}}$ the Poincaré distance, so we have :
- $\tanh \frac{1}{2} d_{\mathrm{D}}(z, w)=\left|\frac{z-w}{1-\bar{w} z}\right|$;
- $\cosh ^{2} \frac{1}{2} d_{\mathbb{D}}(z, w)=\frac{|1-z \bar{w}|^{2}}{\left(1-|z|^{2}\right)\left(1-|w|^{2}\right)}$;
- $\sinh ^{2} \frac{1}{2} d_{\mathbb{D}}(z, w)=\frac{|z-w|^{2}}{\left(1-|z|^{2}\right)\left(1-|w|^{2}\right)}$.
Any help is really appreciated !
