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:

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$E^{2}$ for the Euclidean plane (the most common model for two-dimensional Euclidean geometry);
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$\\mathbb{H}^{2}$ for two-dimensional hyperbolic geometry;
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$B K$ for the Beltrami-Klein model of $\\mathbb{H}^{2}$ ;
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$P D$ for the Poincaré disc model of $\\mathbb{H}^{2}$ ;
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$U H P$ for the upper half plane model of $\\mathbb{H}^{2}$ ;
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$S^{2}$ for the unit sphere (the most common model for two-dimensional spherical geometry).

单词	ComparisonOfCommonGeometries
释义	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 2 Comparison of Properties of the Geometries
随便看	opposing angles in a cyclic quadrilateral are supplementary Opposite opposite OppositeGroup opposite group OppositeNumber opposite number OppositePolynomial opposite polynomial OppositeRing opposite ring OptionalProcess optional process OpusNumber opus number Oracle oracle Orbifold orbifold Orbit orbit OrbitsOfANormalSubgroupAreEqualInSizeWhenTheFullGroupActsTransitively orbits of a normal subgroup are equal in size when the full group acts transitively OrbitstabilizerTheorem orbit-stabilizer theorem

comparison of common geometries

1 Comparison of Properties of the Models

2 Comparison of Properties of the Geometries