Non-Euclidean geometry

Remarks

In the hyperbolic case, all axioms of absolute geometry hold, and only the parallel axiom is replaced. In the elliptic case, lines are closed curves, so the axioms of order must also be changed. Both geometries have Riemannian models of constant Gaussian curvature: $-1$ for the hyperbolic plane and $+1$ for the elliptic plane.

Example: the Poincaré half-plane model is a hyperbolic plane. Through $(1, 1)$, both the vertical line $x = 1$ and the semicircle centred at $(2, 0)$ of radius $\sqrt 2$ miss the line $x = 0$.

Non-example: in the Euclidean plane $\R^2$, the only line through $(1, 1)$ that misses $x = 0$ is $x = 1$. So the parallel postulate holds there, and the Euclidean plane is not non-Euclidean.