Covert the distance matrix into Hyperbolic coordinate?

Question

I know that we can use some method to covert the distance matrix into coordinate in Euclidean space, but can we do the same thing in hyperbolic space?

If we know the distance of several points $x_1,\cdots,x_n$, can we get the position of them in the Hyperbolic space and represent them in polar coordinates? The distance in hyperbolic space is:

$$d(x_i,x_j) = d((r_i, \phi_i),(r_j, \phi_j)) =\text{arccosh}(\cosh r_i\cosh r_j -\sinh r_i \sinh r_j \cos \Delta \phi_{ij})$$

Is there any reference? Thanks?

Answer 1

Promoting my comment to an answer.

Usual approach would be

1. Pick first point arbitrarily.
2. Pick second point right distance away from first, i.e. on circle around first.
3. Pick third point in one of the two possible positions where two circles intersect.
4. All other points are fixed as the only point of intersection of 3 circles defined by distances to first 3 points. These must not be collinear.

All of this would work just as well in hyperbolic geometry, although getting circle equations there takes a bit more work, and it's also harder to check for consistency up front.