# 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.

## The Parallel Postulate[edit | edit source]

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 | edit source]

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 | edit source]

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.

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*.

4. **The Hyperboloid Model**. The hyperbolic plane is represented on one of the sheets of a 2-sheeted hyperboloid. This model is used in modern physics to represent velocity space.

### Defining *Parallel*[edit | edit source]

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 | edit source]

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 | edit source]

Elliptic geometry may be first considered as rotation geometry in 3D space: every rotation has both an axis (say specified by a unit vector *r*) and a turn, generally ranging from 0 degrees to 180 degrees. A turn in the interval (180, 360) may be interpreted as being about the opposite axis −*r* with turn taken as the complement in 360.

w:William Rowan Hamilton practiced celestial geometry as an astronomer. He invented 4D quaternion geometry which has a 3D sphere of versors that represent the sides of spherical triangles. The versor algebra has products corresponding to composed rotations. Elliptic geometry looks at this product as a spherical triangle: a side of the triangle is a versor, and quaternion multiplication relates two sides to the third as follows:

To get a versor, start with the formula of Euler

that uses an "imaginary unit" i with i^{2} = − 1. Now imagine an unit sphere of such units. Call a generic point on this sphere *r* so r^{2} = −1. For three points i, j, k at right angles to eachother, the unit vector may be written Hamilton's convention has i, j, and k anticommute, so ij = −ji, etcetera. To this space of imaginaries, Hamilton adjoined a real number axis to form a real quaternion For a given *r*, its versors lie on a circle through 1 and −1. As *r* ranges over S^{2} these circles form the three-sphere. The elliptic geometry of rotations in 3-space has this hypersphere of versors as points. To obtain the distance between versors *v* and *w*, first find the versor then use its turn for the distance.

The connection between rotations and versor operators is shown in Associative Composition Algebra/Quaternions.