Euclidian geometry is used for things on a globe.

non-euclidian spaces are those that are not spherical. Such as a flat earth.

This is incorrect. Euclidean geometry deals with planar geometry such as that which cartesian coordinates are used to describe. I mean, here's a quote from Wikipedia:

More generally, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n.

Spherical surfaces are even used as kind of the classical example of non-Euclidean geometry. For example, you can form a triangle along great circles on the surface of a sphere and have all three angles be right angles (90-90-90); something not possible in Euclidean/planer geometry. See the linked text.