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单词 NonEuclideanGeometry
释义

non-Euclidean geometry


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

The most common non-Euclidean geometries are those in which the parallel postulateMathworldPlanetmath 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 equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to saying that the sum of the angles of a triangleMathworldPlanetmath is not equal to π radians.

If there is more than one such parallel lineMathworldPlanetmath, the is called hyperbolic (or Bolyai-Lobachevski). In these of , the sum of the angles of a triangle is strictly in 0 and π 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)2:x2+y2<1} in which a point is similarMathworldPlanetmath to the EuclideanMathworldPlanetmathPlanetmath point and a line is defined to be a chord (excluding its endpointsMathworldPlanetmath) of the (circular (http://planetmath.org/Circle)) boundary. This is the Beltrami-Klein model for 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 parallelMathworldPlanetmath 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 π and 3π 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 sphereMathworldPlanetmath (http://planetmath.org/Sphere) {(x,y,z)3:x2+y2+z2=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.

Titlenon-Euclidean geometry
Canonical nameNonEuclideanGeometry
Date of creation2013-03-22 13:54:51
Last modified on2013-03-22 13:54:51
OwnerWkbj79 (1863)
Last modified byWkbj79 (1863)
Numerical id22
AuthorWkbj79 (1863)
Entry typeDefinition
Classificationmsc 51-00
Classificationmsc 51M10
Related topicSphere
Related topicComparisonOfCommonGeometries
Defineshyperbolic geometry
DefinesBolyai-Lobachevski geometry
Defineselliptic geometry
Definesspherical geometry
Definessemi-Euclidean geometry
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