        \Pi_{\mathbb{D}^{d, k} \rightarrow \mathbb{H}^{d d, 1}}\left(x_{1}, \ldots, x_{d}\right)=\frac{\sqrt{k} \left(1+|| \mathbf{x}||_{2}^{2}, 2 x_{1}, \ldots, 2 x_{d}\right)}{1-\|\mathbf{x}\|_{2}^{2}}
