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comparison of common geometries

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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:

$E^2$ for the Euclidean plane (the most common model for two-dimensional Euclidean geometry);
$\mathbb{H}^2$ for two-dimensional hyperbolic geometry;
$BK$ for the Beltrami-Klein 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 two-dimensional spherical geometry).

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

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

comparison of common geometries is owned by Warren Buck.

Warren Buck. "comparison of common geometries" (version 14). PlanetMath.org. Freely available at http://planetmath.org/ComparisonOfCommonGeometries.html.

AMS MSC:	51-00 (Geometry :: General reference works (handbooks, dictionaries, bibliographies, etc.))
	51-01 (Geometry :: Instructional exposition (textbooks, tutorial papers, etc.))
	51M05 (Geometry :: Real and complex geometry :: Euclidean geometries (general) and generalizations)
	51M10 (Geometry :: Real and complex geometry :: Hyperbolic and elliptic geometries (general) and generalizations)

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