Is there a way to normalize a vector such that it lives on a Poincare Disk?

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If I have a vector, by normalizing its length I can make sure such vectors lives on a hypersphere.

Is there a way to do this for Poincare Disk or other hyperbolic geometry?

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I guess the open disk is homeomorphic to the Euclidean plane. I found my answer on Wikipedia (https://en.wikipedia.org/wiki/Unit_disk).