Distance matrix hyperbolic space

Question

I would like to do what I usually do in an euclidean space: create a distance matrix. This time I would like to have my multidimensional data in a hyperbolic space and then to create a distance matrix. Could you please please help me to understand how to it? My input data is a matrix with multidimensional vector. Thank you.

Answer 1

This may be helpful to you, as it gives translation matrices in hyperbolic space along "principal axes":

A dis­cus­sion of matri­ces for hyper­bolic geom­e­try

You might want to think a bit about which model of hyperbolic geometry you'd like to work with. The above discussion gives matrices in Minkowski space, which is nice because the matrices look similar to those for Euclidean transformations.

The paper Non-euclidean virtual reality I: explorations of H^3 gives matrices that work in the hyperboloid model in Minkowski space for arbitrary directions (but in small steps).

But depending on your goals, another model of hyperbolic space like the Poincaré ball model might be preferable.