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 1868 *Essay on an Interpretation of Non-Euclidean Geometry* by 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 line *l* and a point *P* not on *l*, there exists a unique line *m* through *P* that is parallel to *l*.” Euclid’s version: “Suppose that a line *l* meets two other lines *m* and *n*so that the sum of the interior angles on one side of *l*is less than 180°. Then *m* and *n* intersect in a point on that side of *l*.” 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 line *l* and any point *P* not on *l*, there exist two or more distinct lines which pass through *P* and are parallel to *l*.

**Elliptic geometry:** Given an arbitrary infinite line *l*and any point *P* not on *l*, there does not exist a line which passes through *P* and is parallel to *l*.

**Source and more info: https://en.m.wikibooks.org/wiki/Geometry/Hyperbolic_and_Elliptic_Geometry**