What precisely is the difference between Euclidean Geometry, and non-Euclidean Geometry?

I was wondering, what it is precisely which defines the difference between Euclidean and non-Euclidean Geometry, in a few words/equations/diagrams? Would I be correct in understanding that non-Euclidean Geometry is just a more relaxed version of Euclidean Geometry?

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In euclidean geometry the fifth axiom of Euclid holds. In the non - euclidean geometry it doesn't. It means in the euclidean geometry to a point outside of a straight line passes exactly one line parallel to the line. In non - euclidean geometry this isn't true. – Riccardo.Alestra Nov 11 '13 at 11:21
Your recent questions have been about tilings. You may want to investigate Klein's Erlangen Program, a treatise that characterizes geometries by their isometry groups. – Blue Nov 11 '13 at 12:11
Blue is right. Speaking of homogeneous spaces, it is the isometry group of the space that defines it. Euclidean space has a certain group of isometries distinct from the other spaces. There are flat homogeneous spaces that are non-Euclidean. – Andrey Sokolov Nov 13 '13 at 4:13
Non-euclidean geometry is more general.. that held euclidean geometry as a special case of perceived relaxed development for 19 centuries. – Narasimham Jul 22 '15 at 6:40

Let me make an example of something usual in our world, maybe you can find out a sort of differences better. That is the curvature. We are accustomed to think about the flat things on Earth,i.e. the curvature is zero. Making circles, triangles and many many shapes are drew on a piece of a blank paper. Indeed, the curvature of a paper when you put it on a table is $0$. And that is why we learn Ed. Geometry for students on blackboard. This is absolutely OK. But what is the real story? if you want to explain this concept as it is in the real world, you can consider an orange. A surface of an orange (its cover) inside it and outside of it can give us what is the main point about the curvature. Geometrically, when we are working with any spaces with positive curvature,i.e. outside the cover of an orange, if you draw two greater circle, they intersect themselves so, there are no lines parallel to a given line through an outside point (Rejecting the 5-th postulate of Euclidean Geometry).

On the other hand, while in a space of negative curvature, like the surface of an hyperbolic paraboloid or inside the cover an orange , we can draw many lines parallel to a given line through an outside point (Again rejecting the 5-th postulate of Euclidean Geometry). However, when sketching shapes on a blank paper, you are experiencing the zero curvature and inn this case you can easily draw just a single line parallel to a given line through an outside point.

The following figs are from Google's images:

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Nice exposition and graphics! +1 – amWhy Nov 11 '13 at 14:16
@amWhy: Thanks. Honestly, I am so worried about the writing. I hope she could read what I was trying to tell here correctly. :-) – Babak S. Nov 11 '13 at 14:19
I think so: your exposition reads nicely! – amWhy Nov 11 '13 at 14:20
Thanks for an excellent illustration of how Euclidean and non-Euclidean Geometry differs, and you are very clear! – Seraphina Nov 11 '13 at 21:50

If we consider spaces of constant curvature, than Euclidean space has curvature zero, whereas the hyperbolic space has curvature $-1$, and the sphere has curvature $+1$. See the book "Spaces of constant curvature" by Joseph Wolf. This gives perhaps a good idea what non-Euclidean geometry looks like.

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The sum of the interior angles of a (geodesic) triangle is constant, namely $\pi$.

A colleague of mine once stated: “Would we live on a planet which has a ratio diameter : mass much smaller the earth's one the sum of interior angles would be much smaller than $\pi$, thus no one would have came up with the idea we would live in an Euclidian space.

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No, there is spherical excess and pseudospherical defect compared to $\pi$ in the two types (elliptic,hyperbolic) non-euclidean geometries. – Narasimham Jul 23 '15 at 6:50

While going away from flatness essentially generalization of curved lines is about

1. Straightness

and their

1. Parallelism
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