non-Euclidean geometry

A non-Euclidean geometry is a 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. (http://planetmath.org/Ie), 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 is called hyperbolic (or Bolyai-Lobachevski). In these of , the sum of the angles of a triangle is strictly in $0$ 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 (http://planetmath.org/Circle)) boundary. This is the Beltrami-Klein model for $\mathbb{H}^{2}$ . It is relatively easy to see that, in this , 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 is called spherical (or elliptic). In these of , the sum of the angles of a triangle is strictly in $\pi$ 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 (http://planetmath.org/Defect) and area of a spherical triangle for more details.)

As an example, consider the surface of the unit sphere (http://planetmath.org/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 , 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 is semi-Euclidean geometry, in which the axiom of Archimedes fails.

Title	non-Euclidean geometry
Canonical name	NonEuclideanGeometry
Date of creation	2013-03-22 13:54:51
Last modified on	2013-03-22 13:54:51
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	22
Author	Wkbj79 (1863)
Entry type	Definition
Classification	msc 51-00
Classification	msc 51M10
Related topic	Sphere
Related topic	ComparisonOfCommonGeometries
Defines	hyperbolic geometry
Defines	Bolyai-Lobachevski geometry
Defines	elliptic geometry
Defines	spherical geometry
Defines	semi-Euclidean geometry