comparison of common geometries
In this entry, the most common models of the three most common twodimensional geometries^{} (Euclidean (http://planetmath.org/EuclideanGeometry), hyperbolic (http://planetmath.org/HyperbolicGeometry), and spherical (http://planetmath.org/SphericalGeometry)) will be considered.
The following abbreviations will be used in this entry:

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${E}^{2}$ for the Euclidean plane^{} (the most common model for twodimensional Euclidean geometry);

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${\mathbb{H}}^{2}$ for twodimensional hyperbolic geometry;

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$BK$ for the BeltramiKlein model of ${\mathbb{H}}^{2}$;

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$PD$ for the Poincaré disc model of ${\mathbb{H}}^{2}$;

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$UHP$ for the upper half plane model of ${\mathbb{H}}^{2}$;

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${S}^{2}$ for the unit sphere^{} (the most common model for twodimensional spherical geometry).
1 Comparison of Properties of the Models
property  ${E}^{2}$  $BK$  $PD$  $UHP$  ${S}^{2}$ 

model has area when  no  yes  yes  no  yes 
considered as a subset of a  
Euclidean space  
lines in model look like  lines  line segments^{}  some line segments,  some vertical rays,  circles 
some arcs of circles  some semicircles  
lines have length when  no  yes  yes  yes for semicircles,  yes 
considered as a subset of a  no for vertical rays  
Euclidean space  
angles are preserved in  yes  no  yes  yes  yes 
model 
2 Comparison of Properties of the Geometries
property  ${E}^{2}$  ${\mathbb{H}}^{2}$  ${S}^{2}$ 

two distinct points determine a unique line  yes  yes  no 
(yes if points are not antipodal)  
parallel lines^{} exist  yes  yes  no 
number of lines parallel^{} to a given line and  1  $\mathrm{\infty}$  0 
passing through a point not on the given line  
space has infinite area with respect  yes  yes  no 
to its own geometry  
lines have infinite length  yes  yes  no 
number of centers (http://planetmath.org/Center8) of a circle  1  1  2 
angle sum $\mathrm{\Sigma}$ of triangles (in radians)  $\mathrm{\Sigma}=\pi $  $$  $$ 
ASA holds  yes  yes  yes 
SAS holds  yes  yes  yes 
SSS holds  yes  yes  yes 
AAS holds  yes  yes  no (http://planetmath.org/AASIsNotValidInSphericalGeometry) 
AAA holds  no  yes  yes 
Title  comparison of common geometries 
Canonical name  ComparisonOfCommonGeometries 
Date of creation  20130322 17:13:06 
Last modified on  20130322 17:13:06 
Owner  Wkbj79 (1863) 
Last modified by  Wkbj79 (1863) 
Numerical id  17 
Author  Wkbj79 (1863) 
Entry type  Topic 
Classification  msc 51M10 
Classification  msc 51M05 
Classification  msc 5101 
Classification  msc 5100 
Related topic  EuclideanGeometry 
Related topic  NonEuclideanGeometry 
Related topic  Geometry 