Three geometries – Euclidean, Hyperbolic, Elliptical
There are precisely three different classes of three-dimensional constant-curvature geometry: Euclidean, hyperbolic and elliptic geometry. The three geometries are all built on the same first four axioms, but each has a unique version of the fifth axiom, also known as the parallel postulate. The 1868Essay on an Interpretation of Non-Euclidean Geometryby Eugenio Beltrami (1835 – 1900) proved the logical consistency of the two Non-Euclidean geometries, hyperbolic and elliptical.
The parallel postulate is as follows for the corresponding geometries.
Euclidean geometry:Playfair’s version: “Given a lineland a pointPnot onl, there exists a unique linemthroughPthat is parallel tol.” Euclid’s version: “Suppose that a linelmeets two other linesmandnso that the sum of the interior angles on one side oflis less than 180°. Thenmandnintersect in a point on that side ofl.” These two versions are equivalent; though Playfair’s may be easier to conceive, Euclid’s is often useful for proofs.
Hyperbolic geometry:Given an arbitrary infinite lineland any pointPnot onl, there exist two or more distinct lines which pass throughPand are parallel tol.
Elliptic geometry:Given an arbitrary infinite lineland any pointPnot onl, there does not exist a line which passes throughPand is parallel tol.