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?
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> Don’t Euclidean shapes like sphere and hyperbola suffice?
Many non-Euclidean geometries can be embedded in higher-dimensional Euclidean spaces, but sometimes you need to add a lot more dimensions, and often there are aspects of the geometry that are more convenient to describe using the lower-dimensional non-Euclidean description. And nowadays there are lots of complicated geometries people study that simply cannot be described using a higher-dimensional Euclidean space.
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