comparison of common geometries

In this entry, the most common models of the three most common two-dimensional 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:

•

$E^{2}$ for the Euclidean plane (the most common model for two-dimensional Euclidean geometry);
•

$\mathbb{H}^{2}$ for two-dimensional hyperbolic geometry;
•

$B K$ for the Beltrami-Klein model of $\mathbb{H}^{2}$ ;
•

$P D$ for the Poincaré disc model of $\mathbb{H}^{2}$ ;
•

$U H P$ 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).

1 Comparison of Properties of the Models

property	$E^{2}$	$B K$	$P D$	$U H P$	$S^{2}$
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	$\infty$	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 $\Sigma$ 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 (http://planetmath.org/AASIsNotValidInSphericalGeometry)
AAA holds	no	yes	yes

Title	comparison of common geometries
Canonical name	ComparisonOfCommonGeometries
Date of creation	2013-03-22 17:13:06
Last modified on	2013-03-22 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 51-01
Classification	msc 51-00
Related topic	EuclideanGeometry
Related topic	NonEuclideanGeometry
Related topic	Geometry