So I'm currently studying the hyperbolic plane with curvature $\kappa$. I'm mainly working with the hyperboloid model, which is defined as
$\mathbb{H}(\kappa) = \{(x,y,z) \in \mathbb{R}^3 ~|~ z > 0\}$ with metric $ds^2 = dx^2+dy^2-dz^2$.
Now I'm also interested in some different models, mainly the Poincare disk and Poincaré half-plane models. I have found plenty of references that give their definitions and explain the relation between the models but they are only defined in the case where $\kappa = -1$.
I think that the 'generalization' of the upper half-plane model is just the upper half-plane but now equipped the metric $ds^2 = \dfrac{dx^2+dy^2}{-\kappa y^2}$. However, I'm mainly struggling with how I can define the Poincaré disk model in the general case, as well as how the stereographic projection would then be defined.