# non-Euclidean geometry

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.)

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