comparison of common geometries

In this entry, the most common models of the three most common two-dimensional geometriesMathworldPlanetmathPlanetmath (Euclidean (, hyperbolic (, and spherical ( will be considered.

The following abbreviations will be used in this entry:

1 Comparison of Properties of the Models

property E2 BK PD UHP S2
model has area when no yes yes no yes
considered as a subset of a
Euclidean space
lines in model look like lines line segmentsMathworldPlanetmath 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

2 Comparison of Properties of the Geometries

property E2 2 S2
two distinct points determine a unique line yes yes no
(yes if points are not antipodal)
parallel linesMathworldPlanetmath exist yes yes no
number of lines parallelMathworldPlanetmath to a given line and 1 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 ( of a circle 1 1 2
angle sum Σ of triangles (in radians) Σ=π 0<Σ<π π<Σ<3π
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
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