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comparison of common geometries
In this entry, the most common models of the three most common twodimensional geometries (Euclidean, hyperbolic, and spherical) will be considered.
The following abbreviations will be used in this entry:

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

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

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

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

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

$S^{2}$ 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 finite 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 finite 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  $\infty$  0 
passing through a point not on the given line  
entire space has infinite area with respect  yes  yes  no 
to its own geometry  
lines have infinite length  yes  yes  no 
number of centers of a circle  1  1  2 
angle sum $\Sigma$ of triangles (in radians)  $\Sigma=\pi$  $0<\Sigma<\pi$  $\pi<\Sigma<3\pi$ 
ASA holds  yes  yes  yes 
SAS holds  yes  yes  yes 
SSS holds  yes  yes  yes 
AAS holds  yes  yes  no 
AAA holds  no  yes  yes 
Related:
EuclideanGeometry, NonEuclideanGeometry, Geometry
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