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comparison of common geometries
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This entry is best viewed in page images mode, but should look somewhat decent in html mode.
In this entry, the most common models of the three most common two-dimensional geometries (Euclidean, hyperbolic, and spherical) will be considered.
The following abbreviations will be used in this entry:
| property |
$E^2$ |
$BK$ |
$PD$ |
$UHP$ |
$S^2$ |
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| model has finite area when |
no |
yes |
yes |
no |
yes |
| considered as a subset of a |
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| Euclidean space |
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| lines in model look like |
lines |
line segments |
some line segments, |
some vertical rays, |
circles |
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some arcs of circles |
some semicircles |
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| lines have finite length when |
no |
yes |
yes |
yes for semicircles, |
yes |
| considered as a subset of a |
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no for vertical rays |
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| Euclidean space |
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| angles are preserved in |
yes |
no |
yes |
yes |
yes |
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| property |
$E^2$ |
$\mathbb{H}^2$ |
$S^2$ |
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| two distinct points determine a unique line |
yes |
yes |
no |
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(yes if points are not antipodal) |
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| parallel lines exist |
yes |
yes |
no |
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| number of lines parallel to a given line and |
1 |
$\infty$ |
0 |
| passing through a point not on the given line |
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| entire space has infinite area with respect |
yes |
yes |
no |
| to its own geometry |
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| lines have infinite length |
yes |
yes |
no |
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| number of centers of a circle |
1 |
1 |
2 |
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| angle sum $\Sigma$ of triangles (in radians) |
$\Sigma=\pi$ |
$0<\Sigma<\pi$ |
$\pi<\Sigma<3\pi$ |
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| ASA holds |
yes |
yes |
yes |
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| SAS holds |
yes |
yes |
yes |
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| SSS holds |
yes |
yes |
yes |
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| AAS holds |
yes |
yes |
no |
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| AAA holds |
no |
yes |
yes |
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"comparison of common geometries" is owned by Wkbj79.
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(view preamble | get metadata)
Cross-references: AAA, AAS, SSS, SAS, ASA, radians, triangles, angle sum, infinite, passing through, parallel, number, parallel lines, antipodal, points, angles, length, arcs, circles, rays, line segments, lines, Euclidean space, subset, area, property, spherical geometry, unit sphere, upper half plane model, Poincaré disc model, Beltrami-Klein model, hyperbolic geometry, Euclidean geometry, Euclidean plane, geometries
This is version 14 of comparison of common geometries, born on 2007-06-06, modified 2007-06-24.
Object id is 9543, canonical name is ComparisonOfCommonGeometries.
Accessed 1062 times total.
Classification:
| AMS MSC: | 51-00 (Geometry :: General reference works ) | | | 51-01 (Geometry :: Instructional exposition ) | | | 51M05 (Geometry :: Real and complex geometry :: Euclidean geometries and generalizations) | | | 51M10 (Geometry :: Real and complex geometry :: Hyperbolic and elliptic geometries and generalizations) |
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Pending Errata and Addenda
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