# 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 $$ 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 |