# Geometry/Hyperbolic and Elliptic Geometry

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 elliptic.

## Contents

## The Parallel Postulate[edit]

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*.

## Hyperbolic Geometry[edit]

Hyperbolic geometry is also known as saddle geometry or Lobachevskian geometry. It differs in many ways to Euclidean geometry, often leading to quite counter-intuitive results. Some of these remarkable consequences of this geometry's unique fifth postulate include:

1. The sum of the three interior angles in a triangle is strictly less than 180°. Moreover, the angle sums of two distinct triangles are not necessarily the same.

2. Two triangles with the same interior angles have the same area.

### Models of Hyperbolic Space[edit]

The following are four of the most common models used to describe hyperbolic space.

1. **The Poincaré Disc Model**. Also known as the conformal disc model. In it, the hyperbolic plane is represented by the interior of a circle, and lines are represented by arcs of circles that are orthogonal to the boundary circle and by diameters of the boundary circle. Preserves hyperbolic angles.

2. **The Klein Model**. Also known as the Beltrami-Klein model or projective disc model. In it, the hyperbolic plane is represented by the interior of a circle, and lines are represented by chords of the circle. This model gives a misleading visual representation of the magnitude of angles.

3. **The Poincaré Half-Plane Model**. The hyperbolic plane is represented by one-half of the Euclidean plane, as defined by a given Euclidean line *l*, where *l* is not considered part of the hyperbolic space. Lines are represented by half-circles orthogonal to *l* or rays perpendicular to *l*. Preserves hyperbolic angles.

4. **The Lorentz Model**. Spheres in Lorentzian four-space. The hyperbolic plane is represented by a two-dimensional hyperboloid of revolution embedded in three-dimensional Minkowski space.

### Defining *Parallel*[edit]

Based on this geometry's definition of the fifth axiom, what does *parallel* mean? The following definitions are made for this geometry. If a line *l* and a line *m* do not intersect in the hyperbolic plane, but intersect at the plane's boundary of infinity, then *l* and *m* are said to be **parallel**. If a line *p* and a line *q* neither intersect in the hyperbolic plane nor at the boundary at infinity, then *p* and *q* are said to be **ultraparallel**.

### The Ultraparallel Theorem[edit]

For any two lines *m* and *n* in the hyperbolic plane such that *m* and *n* are ultraparallel, there exists a unique line *l* that is perpendicular to both *m* and *n*.

## Elliptic Geometry[edit]

Elliptic geometry differs in many ways to Euclidean geometry, often leading to quite counter-intuitive results. For example, directly from this geometry's fifth axiom we have that there exist no parallel lines. Some of the other remarkable consequences of the parallel postulate include: The sum of the three interior angles in a triangle is strictly greater than 180°.

### Models of Elliptic Space[edit]

Spherical geometry gives us perhaps the simplest model of elliptic geometry. Points are represented by points on the sphere. Lines are represented by circles through the points.