A non-Euclidean geometry is a geometry in which at least one of the axioms from Euclidean geometry fails. Within this entry, only geometries that are considered to be two-dimensional will be considered.

The most common non-Euclidean geometries are those in which the parallel postulate fails; i.e., there is not a unique line that does not intersect a given line through a point not on the given line. Note that this is equivalent to saying that the sum of the angles of a triangle is not equal to $\pi$ radians.

If there is more than one such parallel line, the geometry is called hyperbolic (or Bolyai-Lobachevski). In these types of geometries, the sum of the angles of a triangle is strictly in between $0$ and $\pi$ radians. (This sum is not constant as in Euclidean geometry; it depends on the area of the triangle. See the entry regarding defect for more details.)

As an example, consider the disc $\{(x,y) \in \mathbb{R}^2 : x^2+y^2<1 \}$ in which a point is similar to the Euclidean point and a line is defined to be a chord (excluding its endpoints) of the (circular) boundary. This is the Beltrami-Klein model for $\mathbb{H}^2$ . It is relatively easy to see that, in this geometry, given a line and a point not on the line, there are infinitely many lines passing through the point that are parallel to the given line.

If there is no parallel line, the geometry is called spherical (or elliptic). In these types of geometries, the sum of the angles of a triangle is strictly in between $\pi$ and $3\pi$ radians. (This sum is not constant as in Euclidean geometry; it depends on the area of the triangle. See the entries regarding defect and area of a spherical triangle for more details.)

As an example, consider the surface of the unit sphere $\{(x,y,z) \in \mathbb{R}^3 : x^2+y^2+z^2=1 \}$ in which a point is similar to the Euclidean point and a line is defined to be a great circle. (Note that, when a sphere serves as a model of spherical geometry, its radius is typically assumed to be 1.) It is relatively easy to see that, in this geometry, given a line and a point not on the line, it is impossible to find a line passing through the point that does not intersect the given line.

Note also that, in spherical geometry, two distinct points do not necessarily determine a unique line; however, two distinct points that are not antipodal always determine a unique line.

One final example of a non-Euclidean geometry is semi-Euclidean geometry, in which the axiom of Archimedes fails.