Poincaré disc/ball model Consider the central projection with center $s$ of $H^n$ onto the plane $x_{n+1}=0$. This map can be described by \begin{align*}\sigma\colon H^n &\to D=\{u\in\R^n:(u,u)<1\}\\ x&\mapsto\frac1{x_{n+1}+1}\begin{pmatrix} x_1\\\vdots\\x_n\end{pmatrix}\,. \end{align*} The inverse is given by \[\sigma^{-1}(u)=\frac1{1-(u,u)}\begin{pmatrix} 2u\\1+(u,u)\end{pmatrix}\,.\] For $s+t\left(\begin{pmatrix}u\\ 0\end{pmatrix}-s\right)=\begin{pmatrix}tu\\-1+t\end{pmatrix}$ is … Continue reading Lecture 22