non-Euclidean geometry

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

π

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

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

)

∈

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

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

π

and

3

⁢

π

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

)

∈

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

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