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;

• $BK$ 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).

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 $\infty$ 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