Let the following denote the set of points on the $n$-dimensional hyperboloid manifold
$$ \mathbb{H}^n_K=\left\{x\in\mathbb{R}^{n+1}\,\middle|\,\langle x,x\rangle_*=-r^2=\frac{1}{K} \,\land\,x_1>0\right\} $$
where $K<0$ is the sectional curvature of the hyperboloid, and $r>0$ can be viewn as the analogon of a radius for hyperboloids. The inner product $\langle \cdot,\cdot\rangle_*$ is the Minkowski inner product and $||\cdot||_*$ the induced norm.
Now, let $\mathbf{a}\in\mathbb{R}^{n+1}$ be any point in ambient space. Give a formula for the orthogonal projection of $\mathbf{a}$ onto $\mathbb{H}^n_K$.
The analogous for a $n$ dimensional ball with radius $r$ would be: $$ proj(v)=r\cdot \frac{v}{||v||}. $$