I had thought the Non-Euclidean geometry is an old fashion. Today, however, hyperbolic geometry is an essntial part of topology and knot theory. It is applied to even cosmology. Also they say it has relations to Fermart's Last theorem. You live on the earth and you know it is a sphere.
Some scientists say, with evidence, that our universe has hyperbolic geometry.
One of the most difficulties is that we have to treat infinity. I am not a geometer but a simple hobbyist. I began writing this web-page wthout knowledge of hyperbolic geometry. I pursued whatever comes to mind on a given occasion. Say, this way or that.