Why do we need non-euclidean geometry? Don’t Euclidean shapes like sphere and hyperbola suffice?

Question

Why do they have to be geometries of their own kind? For example, instead of talking about lines in spherical geometry, can’t we just imagine circles on a literal sphere in Euclidean space?

In: 4

Answer 1

We actually don’t need it for most purposes. On Earth, it’s mostly to help map longitude and latitude. Non-euclidian geometries really come into their own when mapping out Gravity since Gravity warps space. The very idea of straight lines gets a bit wonky.