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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: $$ proj(v)=r\cdot \frac{v}{||v||}. $$

ndrizza
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  • What do you mean by "orthogonal projection"? It's usually defined for planes, not for curved surfaces. But the obvious answer is $r,\vec a/\lVert\vec a\rVert$, at least when $\vec a\cdot\vec a<0$. – mr_e_man May 13 '19 at 04:17
  • I also thought about the rescaling. However, that won't work when the vector points into a direction which has no intersection with the hyperboloid. I don't know what would be the most sensible way to project such a vector. I just guessed that it would be the best to project it orthogonally onto the hyperboloid surface. In other words: project a point onto the closest point onto the hyperboloid in terms of the euclidean distance. However, this approach is very different from the rescaling. Maybe there's an approach which treats both cases more uniformly (e.g., flipping signs?). – ndrizza May 14 '19 at 12:09

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