[olddustytrail]

It's not too complicated. Get a round ball of some sort. When you draw on the surface, that's a "non-Euclidean space".

Take a straight line down from the "north pole" of your ball to its equator. Draw another straight line around a quarter of the equator. Draw a third line back to the pole. You've just drawn a triangle with 3 straight lines and the angles add to 270 degrees.

A non straight line is just not the shortest distance between two points on that surface.

[posterboy]

Calling that a "straight" line with a straight face is so Euclidean. Get over it.

Shortest distance between two points is what it is.