>>13967948Non euclidean geometry is simply geometry where one (typically the fifth) or more of euclids postulates for geometry don't hold.
The fifth postulate (if the others are kept) reduces to the fact that every line has a unique parallel line passing through a given point not on the line.
If we replace this postulate with P2, that there passes more than one parallel line through the point, we obtain hyperbolic geometry, if we replace it by P3, that all lines intersect, we obtain eliptic geometry.
In 2 dimensions non euclidean geomtry can be modeled by a surface inbedded in 3D euclidean space. Here "straight lines" correspond to geodesics in the surfaces, and the wierd properties are caused by curvature.
However, when speaking of non euclidean geometry the types of surfaces we deal with will typically have constant curvature (think of a sphere) and in order to make our lives easier we project the surfaces (or parts of them) to a plane or part of a plane. This is good since we don't want to be reliant on the way the surface is embedded in space, and it also becomes easier to do calculations.
