## Hyperbolic translation in Poincare disc model

Hyperboloid model ${\mathbb{H}^{n+1}=(x_0,x_1,\cdots,x_{n+1})}$ is hyperquadric in ${\mathbb{L}^{n+2}}$, with

$\displaystyle -x_0^2+\sum_{i=1}^{n+1} x_i^2=-1$

Half-space model ${\mathbb{R}^{n+1}_+=\{(y_1,y_2,\cdots,y_{n+1}),y_{n+1}>0\}}$. Denote ${(y_1,\cdots,y_{n},y_{n+1})=(y,y_{n+1})}$,

Poincaré ball model ${\mathbb{B}^{n+1}=(u_1,\cdots,u_{n+1})\subset\mathbb{R}^{n+1}}$ There are some transformation formula between these hyperbolic models, say

$\Psi_{12}:\mathbb{H}^{n+1}\longmapsto \mathbb{R}^{n+1}_+$

$(x_0,\cdots,x_{n+1})\longmapsto \frac{1}{x_0+x_{n+1}}\left(x_1,\cdots,x_{n},1\right)$

$\Psi_{21}:\mathbb{R}^{n+1}_+\longmapsto \mathbb{H}^{n+1}$

$(y_1,\cdots,y_{n+1})\longmapsto \left(\frac{1+|y|^2+y_{n+1}^2}{2y_{n+1}},\frac{y}{y_{n+1}},\frac{1-|y|^2-y_{n+1}^2}{2y_{n+1}}\right)$

$\Psi_{23}:\mathbb{R}^{n+1}_+\longmapsto \mathbb{B}^{n+1}$

$(y_1,\cdots,y_{n+1})\longmapsto\frac{1}{|y|^2+(y_{n+1}+1)^2}(2y,|y|^2+y_{n+1}^2-1)$

$\Psi_{32}:\mathbb{B}^{n+1}\longmapsto\mathbb{R}^{n+1}_+$
$(u_1,\cdots, u_{n+1})\longmapsto \frac{1}{|u|^2+(u_{n+1}-1)^2}\left(2u,2(1-u_{n+1})\right)-(0,\cdots,0,1)$

On ${\mathbb{R}^{n+1}_+}$, there are vertical scaling transformations which are isometries of ${\mathbb{R}^n_+}$

$\displaystyle \tau(y_1,\cdots,y_{n+1})= (ty_1,\cdots, ty_{n+1}),\quad t>0$

On ${\mathbb{B}^{n+1}}$, they are called translation in ${e_{n+1}}$ direction through the map

$\displaystyle (u,u_{n+1})\mapsto\Psi_{23}\circ\tau\circ\Psi_{32}(u,u_{n+1})$

In component,

$\displaystyle u_i\mapsto \frac{4tu_i}{(1-t)^2|u|^2+[u_{n+1}(t-1)+t+1]^2}$
$\displaystyle u_{n+1}\mapsto \frac{(t^2-1)|u|^2+4t^2u_{n+1}+(t^2-1)(u_{n+1}-1)^2}{(1-t)^2|u|^2+[u_{n+1}(t-1)+t+1]^2}$

On $\mathbb{H}^{n+1}$,

$(x_0,x,x_{n+1})\longmapsto \Psi_{21}\circ \tau\circ\Psi_{12}(x_0,x,x_{n+1})$

In component,

$\displaystyle x_0=\frac{(1+t^2)x_0^2+(1-t^2)x_{n+1}^2}{2t}$

$x_i=x_i,i=1,\cdots,n$

$\displaystyle x_{n+1}=\frac{(1-t^2)x_0^2+(1+t^2)x_{n+1}^2}{2t}$