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

Parallelism in the three geometries

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

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

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

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

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

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 $\exp(i\theta )=\cos \theta +i\sin \theta$

that uses an "imaginary unit" i with i² = − 1. Now imagine an unit sphere $S^{2}\subset \Re ^{3}$ of such units. Call a generic point on this sphere r so r² = −1. For three points i, j, k at right angles to eachother, the unit vector may be written $r=xi+yj+zk.$ 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 $q=w+xi+yj+zk.$ For a given r, its versors lie on a circle through 1 and −1. As r ranges over S² 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 $v^{-1}w,$ then use its turn for the distance.

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