complex geometry – possible research with rigid movements in the plane and in space

What are the current studies and future directions of work to construct Mobius transformations with Sterrographic projections from the Riemann sphere to the complex plane, rigid motions that lead one Riemann sphere to another, and to regress that second sphere to the complex plane?

Even if not, which directions of research can be carried out in space using Mobius in-plane transformations and rigid (or isometric) motions?

Thank you for your time and suggestions.