How can we do stereographic projection using de Sitter space $\Bbb S^2_1$ and the hyperbolic plane $\Bbb H^2$, in Lorentz-Minkowski space $\Bbb L^3$.
For $\Bbb S^2_1$ it is not clear what point should be used as a pole, and for $\Bbb H^2$ it seems that using $(0,0, 1)$ doesn't work (I put the minus sign last in the metric here). I don't think it is much related to the isometry between the hyperboloid model and the Klein model, either.
I looked around a bit but found nothing. It would be nice if there's a way to generalize the stereographic projection to pseudo-spheres in spaces of arbitrary index $\Bbb R^n_\nu$, but I'll be happy with the cases above. I want to use this to study Weierstrass representations of critical surfaces in $\Bbb L^3$. Thanks.