The following are statements that I can't easily confirm, so first things first, tagging Devac. School me good, in any way, if I'm not 100% correct. Either of you.

Basically whenever you are projecting something that is 3D onto a flat surface or vice versa you are using non-Euclidean geometry.

I don't think this is technically true(?). This is the case if you were to attempt to translate spherical or cylindrical surfaces into 2D projections at the scale of taking a pocketknife to the surface of a globe or donut or drumstick and attempting to flatten it, but there are relatively neat mathematical ways of doing such a thing, handled by the differential path length treatment.

Non-Euclidean geometry arises when path length and topology (space-time, in this instance) are a function of something, e.g. position, and thus the transformation tensors vary from point to point. Frame dragging, gravitational space-time dilation, and the event horizon are all related concepts.

Edit: I think I'm actually incorrect, and that spherical coordinates do seem to violate "Euclideanism", which seems like a dumb formulation anyway. Leaving my original statements, because we can easily reformulate the violations of Euclideanism, like a triangle on a spherical surface with three 90 degree corners, into orthogonal coordinates by imposing a set of conditions, and then we're "Euclidean" once more. This is kinda a waste of time, debating the origins of geometry. Saying that, but acknowledging that I'm spoiled to have been exposed to these ideas casually in my education.