Main Question or Discussion Point
In introductions to non Euclidean geometry, examples are often given in the form of measuring angles on a 2D surface embedded in a 3D space, such as the surface of a sphere or on a saddle surface. This gave me the initial impression that 3D non Euclidean geometry would have to be embedded in 4 or more spatial dimensions. However, it then occurred to me that if I can map 3D Euclidean coordinates to 3D non Euclidean coordinates in a one to one relationship in an unambiguous manner, then I can describe any set of Euclidean 3D coordinates in terms of non Euclidean coordinates, without having to invoke any higher spatial dimensions. Is the one to one mapping relationship true and is the conclusion true?