Canonical projections for $\mathbb{H}^2_1$ and $\mathbb{S}^2_1$

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We know that there exists a canonical projection $\pi:\mathbb{S}^2\to\mathbb{R}\mathbb{P}^2$. Are there similar transformations for hyperbolic plane $\mathbb{H}^2_1$ or pseudosphere $\mathbb{S}^2_1$ in Minkowski 3-space?